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INTRODUCTION Three decades have passed since a new line of scientific research called "chaos theory" was begun. As opposed to what happens in other fields of physics, like quantum mechanics, research on the fundamental particles that comprise matter or theories concerning the origin of the Universe, one is trying to apply this "science of chaos" to many events directly linked with habitual human experience, and so explain phenomena as dissimilar as arrhythmias in the functioning of the heart, or aspects of the economy like the fluctuations of the stock market, or also the appearance of life upon the Earth, in addition to the behavior of dynamic physical systems with a high number of components which could be the atmosphere or a liquid in a state of turbulence. The physicist Joseph Ford, in an article in the book, The New Physics, proclaimed the new chaos science to be "the beginning of the third revolution in the physics of the present century," the other two previous ones being the theory of relativity and quantum theory. It still seems premature, nevertheless, the give it the designation of third revolution in physics, since as opposed to the other two cases is is not correct to speak of a "chaos theory," a theory that still does not exist. It refers to a new and very promising way of applying the known laws of physics, with the fundamental assistance of the computer, to very diverse phenomena that include, in addition to those traditional in physics, those occurring in the biological sciences and social sciences, whenever they can be confronted as if dealing with complex dynamic systems. In physics there is work with growing intensity upon the topic of disordered sets and their specific properties, of great scientific and technological interest. But one must avoid the confusions that can be generated around this topic, particularly by the expectations which the study of chaos can awaken in those who work in other fields of knowledge. Actually a problem of interpretation appears here: among those who are not familiar with the physical sciences or mathematics--and due in part to the declarations of certain scientists--a sort of mythology of Chaos or Disorder has been installed, which assigns a transcendent significance to the accidents (real or apparent) of nature, and that proclaims the definitive demise of determinism, when everything indicates that for chaotic systems determinism will continue being valid, if indeed one requires a probabilistic description of their behavior. In a similar manner to what occurred with the catastrophes theory developed by René Thom--who made a first attempt to study certain complex phenomena mathematically--there are those who hope the study of chaos will help explain the mysteries of the great social transformations or that of the relations between the neuronal nets and psychology, and it has unleashed formidable speculations about the meaning of time and of disorder in the Universe. Of course the extension of discoveries achieved in one field of knowledge to other areas is very beneficial, yet when they deal, for example, with human behavior, individual or collective, which has a complexity incomparably greater than that of physical systems, that extension should be done with much prudence, and in general can only have the character of an analogy. Thus, it may be fruitful to utilize the psychology of a familiar group applying certain guidelines that have analogies with those of dynamic physical systems, and with difficulty one might work on such a theme applying the mathematics of chaos, and seeking fractal dimensions and strange attractors. One should avoid using a language that seems to attribute magic powers to "chaos": in scientific texts, this concept has a precise meaning, which leads us to complex phenomena, particularly difficult to formulate mathematically, yet which do not in principle manifest any connection with the primordial Chaos conceived by the ancient mythologies. It is important to make clear that the fundamental laws of physics continue to rule and that the fact we utilize, as we shall see, statistical characteristics to predict behavior is not an "epistemological drama" as some have suggested. The strong emotional charge that the word, chaos has is in part the cause of the above-mentioned confusions, and have contributed to them the fact that the name of this new discipline is still not definitively established. 30 years ago, when it began to develop, one spoke of the "science of chaos," that soon came to be called "deterministic chaos," to distinguish it from the chaos produced by pure chance. Now support for the word, "complexity" tends to strengthen, which designates the study of those dynamic systems that are at some intermediate point between the order in which nothing changes, which might be that of crystalline structures, and the state of total disorder or chaos such as might be an ideal gas in thermodynamic equilibrium. The phenomena of "deterministic chaos" or of "complexity" refer to many systems that exist in nature whose behavior keeps changing with the passage of time (dynamic systems). Such phenomena appear when systems become extremely sensitive to their initial conditions of position, velocity, et cetera, such that very small alterations in their causes are capable of provoking great differences in the effects. As a consequence of this it is not possible to exactly predict how such systems will behave beyond a certain time, because they seem to follow no law, as if they were ruled by chance. But the researchers have discovered that dynamic systems in these conditions show signs of collective regularity although it may not be possible to distinguish the individual behavior of each one of their components. It has been determined that there are certain common characteristics that permit including in the study of complex processes not only physical systems and inert chemicals but also living organisms, all approached through common mathematical tools. The fundamental tool is, of course, the computer, without which it would have been impossible to develop this new focus upon dynamic systems. In a manner similar to the impulse given to science by the use of the telescope and the microscope in the 17th and 18th centuries, the use of this machine enormously eases the testing of theories through experimentation, thanks to the immense increment in calculating capacity and the possibility of making simulations of real processes and of creating models of complex systems. Without their existence concepts like that of strange attractors, fractals or algorithmic complexity could not have been developed. In fact, this entire vast field of non-linear phenomena, with high sensitivity to the initial conditions, was not unknown to the great mathematicians and physicists of the 19th century; but in that era the solution of the corresponding systems of equations would have required numerical calculations so extensive as to be impractical, so that we had to wait for the second half of this century, with the apparition of fast computers, to be able to confront it. One of the most positive results due to the emergence of this new field of research is that interdisciplinary groups have been formed--integrating, for example, biologists, physicists, mathematicians, or sociologists, economists and computer experts--to study the problems inherent in complex dynamic systems. These range from turbulent liquids to ecological systems or economic models of society. Various centers at a high level have emerged, in particular in the United States, Russia and France; thus can be mentioned the Santa Fe Institute in New Mexico, and departments dedicated to the subject at Los Alamos National Laboratory (Center for Non-Linear Studies), the Georgia Institute of Technology, the University of California at Berkeley, and the Center for Research in Saclay, France, in addition to the group at the Free University of Brussels, Belgium. A major role in the study of these phenomena was that filled by the Russian school of physics and mathematics, like Landau, Kolmogorov, Andronov, Lyapunov, and others, who developed the necessary techniques long before deterministic chaos was to become a fashionable topic. The above school continues making great contributions to this important area. In the last years the advance of this activity is displayed in the growing number of congresses regarding its diverse aspects. The possibilities of application to the most diverse sciences were placed in relief at the First Conference on Experimental Chaos, held in the United States in October of 1991, where it was shown that there are cases when one can profit from chaos instead of avoiding it, and obtain thereby systems of greater flexibility than those which for being ordered have a "good" behavior, or are predictable. This book has the intentions of describing the fundamental characteristics of complex systems and the methods which are used to study their behavior. I wish, finally, to express my gratitude to Marcos Saraceno for his valuable suggestions concerning the development of this theme and the presentation of the text. I.The Universe, does it function like a clock?COMMON TO many creation myths is the concept of an eternal battle between Order and Chaos, which suggests that those two concepts are profoundly rooted in the human mind. For primitive humanity, Nature was a Chaos, a capricious entity, subject to the whims of powerful and indecipherable gods, whose actions could not be predicted. Thus, the cosmologists of many cultures imagined an initial state of Disorder, or Chaos, from which things and beings emerged. InThe Theogony,Hesiod says: "Chaos was first, and then the Earth." Chaos is a word of Greek origin, equivalent to abyss, or also a formless entity: cosmos, for its part, designates order and, by extension, the Universe. A Chinese creation myth says that from chaos two principles emerged, Yin and Yang, which with their feminine and masculine aspects later created the entire Universe. In the Bible we read: "And the earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters." Chaos then is the primordial formless substance, from which the Creator molded the world. But human beings need to discover order in nature, seek the laws behind their complex behavior that will permit them to know the duration of the days and the nights, the phases of the Moon, the season for the harvests. The notion of order is thus introduced under the necessity of foreseeing in anticipation, of predicting, indispensable for survival. Slowly, over the length of many millennia, mankind kept discovering ordered, regular behaviors in nature, which they learned to register, predict and exploit. In Greek thought two different visions appear regarding what is important for our comprehension of the Universe, and which is essentially maintained in the West until our times: the ecstatic monism of Parmenides and the being in perpetual motion conceived by Heraclitus. Plato took from Parmenides the emphasis upon order, on the immutable aspects of reality, and from Pythagoras the study of the mathematical and geometrical laws that are the expressions of the "Forms," eternal ideas, of which mankind perceives only the shadows. Furthermore, like Heraclitus, he kept in mind the instability and the incessant flux of all that is manifested in nature. With this focus, Platonic thought seeks for something that is behind the process of change and the notion of the passage of time, to be able to make an intelligible description of reality. The Cosmos is rational, and if in the beginning there was Chaos, it was the Demiurge, the Supreme Orderer, who arranged the transformation of the Universe from the state of disorder to that of order. This divine Being is mathematical, and the order that it establishes also is. The field of knowledge where this concept is most clearly expressed is that of geometry, and Platonic thought makes a clear distinction between the Forms, which are idealizations, are perfect, and objects such as they appear in nature, where perfect circles, spheres or planes are never found. From this came the clear aversion of the Platonic thinkers to using tools for studying geometry--they only tolerated the use of the ruler and the compass--since every material object is imperfect, a pallid reflection of the Ideas. Due to the action of an "errant cause," matter resists being modeled by ideas, and so opposes being incarnated in all the pure forms of geometry. The result is then a Cosmos agitated by numerous small convulsions, which in modernity we could call whirlwind motions. For Plato, consequently, there is a hierarchy formed with three fundamental levels: the Ideas and mathematical Forms, which are the perfect model of all things; the original Chaos; and an intermediate state, which is our imperfect world, complex, the result of the work performed by the Demiurge parting the Chaos and modeling it upon the basis of the Ideas. In this hierarchical order, the supreme value is that of the Ideas and mathematical Forms, which express the divine qualities of simplicity, harmony, regularity. This philosophy was incorporated into medieval Christian thought, and the scientists as the Renaissance began have followed its fundamental aspects. Galileo, two thousand years after Plato, says that the book of nature is written in mathematical language, and in his era the bases for the scientific method are established, which consists in seeking the eternal laws that rule a natural phenomenon, those that should be formulable in mathematical language, and questioning nature to discover whether the facts corroborate or negate the proposed theory. For thinkers like Galileo, Kepler, Newton, or Einstein, everything occurs as if God (or nature) were to have chosen the order of the "pure forms," those which the more simple, the more beautiful and true are considered, and the ideal of science is to discover through reason that order and regularity which is behind the apparent disorder of nature. As Albert Einstein will express with so much clarity: We recognize at the root of all scientific work at a certain level a conviction comparable to a religious sense, that accepts a world based upon reason, an intelligible world. This conviction, tied to a profound sense of a superior reason which is revealed in the world of experience, expresses for me the idea of God. Science thus postulates that behind the complexity of the world there are mathematical laws that show an underlying harmony, in which there is no place for disorder or the unforeseeable; those dissipate on being illuminated by the light of reason. Yet there is another vision that, until recently, has been less propagated than the Platonic search for that which is invariant in nature. It is the perspective of Aristotle, who puts his emphasis on change in the observable processes in the world, instead of the invariant Forms behind them that cannot be observed. In this vision, his observations of the living world, for which he was very gifted, were very influential. For the advocates of this focus, nature is similar to an organism, complex, changing, and they accept as real and not appearances its aspects of disorder and unpredictability, trying to understand those characteristics without eliminating them, since no one would thus try to transform a living organism into a machine as predictable as a clock. We shall return to this focus further on, but now we will detail the characteristics of the scientific method such as it was applied at the outset of the work of the great thinkers such as Galileo, Descartes, Huyghens, and Leibniz. The scientific method applied by Galileo had its mathematical expression thanks to Descartes, who emphasized the necessity of analyzing, that is, dividing what is examined into its simplest components, so as later to recompose them in a synthesis that will permit understanding the phenomenon with certainty. He also created analytic geometry, basing it on his introduction of coordinates, thus identifying space as an immense grid every point of which can be assigned a numerical value, with which he succeeded in united geometry with algebra.NEWTON'S MESSAGEThree centuries ago the monumental work of Isaac Newton was published,Mathematical principles of natural philosophy,whose message has been decisive for the culture of the West. According to this: As we have seen, for scientists at the beginning of the Renaissance it reaffirms the Platonic focus, given that there is an order in the Universe behind its apparent complexity, with simple laws that contain immutable aspects, expressed in the famous magnitudes which remain invariant in physics: total energy, momentum, electrical charge. The laws discovered by Newton seem simple, and their application to the behavior of bodies allowed describing precisely the movement of the stars in the firmament, the fall of bodies, et cetera. Newton formulated his laws through mathematical equations, that relate the magnitudes which we can measure of a body, such as its position and its velocity, and the form in which these vary with time. The first law was the formulation of Galileo's discovery, who observed that one should attend not to the velocity of a body but to the change in that velocity with the passage of time, thus putting an end to the Aristotelian belief which had blocked the advance of physics for many centuries. Newton also applied a concept that has been essential to the scientific method: that of ideally isolating the dynamic system that one wishes to examine from the rest of the universe of which it forms a part. This allows its behavior to be understood, now that it is not necessary to consider all the infinite relationships with the universe, which would only be possible for an infinite being. It is enough then that they consider only those characteristics of the system that are relevant to the phenomenon which they desire to study. Thus, the first law asks what can be said of the movement of an isolated body, or that is, one to which no force is being applied. Aristotle had said that it ought to remain in repose. Newton, like Galileo before, establishes that, in this situation, the body can be in repose or can move in a straight line with uniform velocity. If a body falls towards the earth, this is because the force of gravity acts on it, and therefore its velocity cannot be uniform but must always be greater; therefore it is more dangerous to fall from a larger height than from a lesser. Is there something that remains constant in this process of falling? Yes, the acceleration, that is, the speed with which the velocity of the falling body increases. The second law formulates this concept, which is expressed mathematically asF = m × a(the forceFapplied to the body is proportional to the accelerationaof its movement, according to a constantm,the mass or quantity of material in the body). For any dynamic system in the Universe, the laws of motion can be expressed asF = m × a,no matter what is the origin of the applied force. It can deal with the force of gravity, the electrical or the magnetic, and for all the same surprisingly simple equation applies equally. The velocity with which a magnitude changes is determined by the difference between its values for two successive times, and thereby the term "differential" that appears in mathematical analysis: Newton's equations involve speeds of change and are, accordingly,differential equations.Algebraic equations are distinguished from differential ones because they do not involve rates of change. They are not always easy to solve. Yet to solve differential equations is, in general, much more difficult, and it becomes truly surprising that so many equations important for their applications to physics have a definite solution. A basic mathematical principle of differential equations is that their solutions, that is, their integration, are unambiguously determinate, and give a single result for each set of numerical data that are introduced into the equations; if, for example, one wishes to know the height reached by a projectile, by introducing into the equations the data of initial velocity, angle of the cannon, et cetera, a result is obtained that clearly defines this altitude; they are accordingly deterministic equations: there is a single effect for each cause. The importance of these differential equations lies in that they can be applied to dynamic systems, which is to say to any process which changes over time. Many physicists and mathematicians have marveled before the fact of how effective these equations seem in describing the structure of the physical world. In view of the complexity of the world that surrounds us, it is truly noteworthy that there are natural phenomena that allow description through simple physical laws. How is this possible? It was Isaac Newton who had the vision that opened the road for the natural sciences to have so much success over these past three centuries. This is due partly to them initially restricting their attention to the study of simple natural systems with only a few components.PHYSICAL LAWS AND INITIAL CONDITIONSIn accordance with the focus held by Newton on mechanics, a material system can be conceptually divided into:1)the "initial conditions," which specify its physical state at a certain initial time (these conditions can be, for example, the position and the velocity of a projectile, or of the Moon with respect to the Earth);2)the natural or "physical laws," that specify how this state changes. The initial conditions are usually very complicated, a complication that reflects the complexity of the world in which we live. The natural laws, on the other hand, can be and are instead more simple, and are expressed through differential equations. This division--laws and initial conditions--is maintained through today. In practice those equations can only be strictly solved that represent simple physical laws for systems with simple initial conditions: the shooting of a projectile, the movement of the Earth around the Sun without taking into account the influence of the other planets. That is, before the infinite complexity of nature, which causes each of its components to be linked with the rest by an immense quantity of relations, an abstraction is performed, ideally considering the system one wishes to study as if separated from the rest, and selecting those characteristics of the system which seem to be sufficiently important as against others that produce almost no effect upon the phenomenon being examined. So, to calculate the trajectory of a projectile only the attraction of the Earth's gravitation is considered as an influence, since the attraction that the Moon or the Sun exercise is so small that it need not be considered among the initial conditions. In like manner, to study the movement of the Moon around the Earth one will not take into account the attraction of the stars. One has then simple initial conditions because those variables have been selected which are those that most affect the phenomenon under study, and furthermore these obey laws expressed by equations in which small variations in the initial conditions yield solutions that differ little among themselves. In this way, the fact that the initial conditions are known in general with a certain margin of error affects relatively little the result which can be expected from these equations. For example, if a projectile is shot from a gun, the calculation of where it shall hit the target starting from a certain position of its barrel or from another slightly different will produce a proportionally small difference in the result. Another important aspect, also studied by the founders of mechanics, was that of reversibility in time of the trajectories of dynamic systems: the equations show that if the sign of the velocities of all the components of the system are inverted, replacingvby-v,the result is mathematically equivalent to changing the timetto-t,as if the system could flow "backwards" in time. This is the mathematical form of expressing that if starting at a certain instant there is a change in a dynamic system, another change, defined through the inversion of the velocities of the components, can restore the original conditions.EVERY PHYSICAL PROCESS IS DETERMINATE AND THEREFORE IT IS POSSIBLE TO PREDICT ITS BEHAVIORAbiding by these rules of the game, we arrive at the conclusion that in a system which responds to the laws of classical mechanics, and which, accordingly, is determinist, if the positions and the velocities of its components at an instant are known, one can calculate the positions and velocities at every following or previous instant. Thus if in the dynamic system formed by two billiard balls we know at a given initial moment the position and velocity of each one, we can through the differential equations of Newton predict their respective movements from when they received the initial impulse until they give up the effort. Even more, at the end of the 18th century the conviction arose that if one knew the position and velocity of each of the planets that comprise the solar system at a given instant, they could calculate their positions in the future, and also their positions in the past, through equations that unambiguously determine the trajectories. It was precisely the application of the equations to the movement of the stars, the celestial mechanics, which signified Newton's greatest triumph. The various scientific disciplines that continued developing over the centuries following Newton's theory studied other magnitudes in addition to the position and velocity of bodies. Yet the procedure, consisting of introducing values for those magnitudes for an initial time into a mathematical equation which, once solved, determines those values for any other time, it being the same whether one deals with the configuration of an atom, the movement of a comet, the temperature of a gas, or the voltage of an electrical circuit. Upon finalizing this process of scientific development it seems that the behavior of the entire Universe could come to be expressed mathematically, since it is determined by the immutable laws mentioned, which dictate the movement of every particle in an exact form and for always, the scientific task consisting in applying these laws to the particular phenomena. The Newtonian scheme thus made possible the construction of the majestic structures of classical mechanics, which gave humanity the vision of an ordered and predictable Universe.LAPLACE'S DEMONThis revolution in thought had its clearest expression with Pierre Simon de Laplace, who in the age of Napoleon wrote, in hisAnalytic theory of probabilities: We should consider the present state of the Universe as the effect of its previous state and as the cause of its future state. An Intelligence that, for an instant, were to understand all the forces with which nature is animated and the situation regarding the beings that comprise it, if it also were sufficiently profound as to submit those to [mathematical] analysis, it would embrace in the same formula the movements of the largest bodies in the Universe and of the quickest atom: nothing would be uncertain for it and the future, like the past, would be present before its eyes. The human spirit offers, in the perfection that it has known how to give to astronomy, a pallid example of this Intelligence. His discoveries in mechanics and in geometry, together with that of universal gravitation, have placed him in a condition of embracing in the same analytic expressions to past states and the future systems of the world. Thus he has a vision of the Universe as a gigantic mechanism that functions "like a clock" (not an electronic digital clock, but one of the classics, formed with moving parts, springs and gears). Such a mechanism is, over and above everything, absolutely deterministic and, accordingly, predictable: it is governed by eternal laws which cause that under identical circumstances the same things always result. And if the circumstances, instead of repeating themselves in an identical form, change slightly, the resultant will also change in a proportionately slight manner. A specialist in these laws of mechanics who knows its characteristics and its state at a given moment can, in principle, establish exactly what it will do at any moment of the past or future. If in practice this can be applied only to relatively simple systems, clocks, machines, planets, and there are many objects in the world that appear to have irremediably disordered, chaotic, unpredictable behavior, that relates only to something apparent, and to the degree that mathematical analysis perfects itself and the corresponding hidden physical laws are discovered, the day will come when that apparent chaos will disappear. From this viewpoint the future is rigidly determined since the beginning of the Universe. Time ceases having much physical significance, since it is as if that Intelligence proposed by Laplace which many call "Laplace's demon" (from the Greek "dáimon," a secondary divinity intermediate between the gods and man) were to have the entire history of the Universe recorded on a cinematographic film, which can be contemplated going forward or backwards in time. Since this time is reversible, it only marks the direction in which one observes a process that cannot be modified. II.Where chaos appears in the machineDURING the 18th and 19th centuries Newton's mechanics were applied with impressive success. The mechanistic point of view became popular and, combined with the experimental method, gave a great impetus to physics, chemistry and biology. Also it fundamentally transformed the new political, economic and social theories. The ancient conception of chaos as patriarch of nature, where things follow by chance, by caprice, without any relation between cause and effect, gave way to the vision of order in the world as deterministic as a fine Swiss watch. But it would come to seem that there are cyclical processes in history, or something similar to a circular spiral where a cycle does not repeat exactly, but passes to a new level. Something like this occurs with the theme of order and chaos in our vision of the Universe. The primitive chaos was replaced by the Newtonian order. However, to the extent that knowledge of nature strengthened, difficulties appeared for the mechanistic model. In the second half of the 19th century it became clear what the limits of classical mechanics were: its validity did not extend to extremely large velocities or for the extremely minute world. As a product of this crisis there emerged, in the second half of the 20th century, two new branches of physics which studied, respectively, the theory of relativity and quantum mechanics. Today the scientists of numerous disciplines begin to persuade themselves there is a third limit to the possibility of knowledge of nature, and which is further valid for the world of our everyday experience: in many circumstances not only can the behavior not be predicted of the individual components in complex dynamic systems, which involve the interactions of a large number of components, but also the same can even occur in the case of simple systems, formed by a few components that are subjected to the action of two or more forces. We all know that the world we live in is complex, and no one is surprised at the low success for predictions about the economy of a country, or the meteorological outlook, or the behavior of any human being or of living organisms in general. This has always been taken as an indication that if there are laws for this very complicated world, they would have to be complicated, and not like those that rule in dynamic systems which physics has studied, which exhibit characteristics of order and predictability. But now it turns out that even simple physical systems, subject to simple laws, can have unpredictable, chaotic behavior. Thereby we are presented with a dilemma: our complex world is governed by simple laws that we shall progressively be discovering through the methods developed by science, in accord with a vision that we have called Platonic? Or do we adopt a vision that resembles the Aristotelian, inasmuch as it puts the emphasis on the processes of change with the passage of time, accepting that in many cases the behavior of such processes cannot be predicted exactly using simple laws which govern behind the phenomena? To reply to this dilemma it is necessary to examine in what conditions one or the other focus seems valid. Many dynamic systems, whatever be their nature (physical, chemical, electromechanical, biological) are extremely sensitive to the values of their initial conditions, such as the position, the velocity, et cetera. This places a limit on the possibility of predicting the future state of the system, given that, as we have seen, such a prediction is based on the supposition that small causes produced effects also small and that, hence, a small change in the initial values of the differential equations that describe the behavior of the system will produce a proportionally small change in the solution of these equations that allow us to know the future state. We must distinguish here between linear and non-linear differential equations. The solution of a differential equation is called integration; the most important class among integrable equations is that of linear equations. The most simple among them depends upon one single variable, and its solution will be graphically represented by a straight line; thus the name of linear equation. The family of linear equations have the characteristic that the solutions obtained by solving them for different numerical values of the variables can be summed among themselves, also giving a solution as the result. A simple example is that of the linear equation for waves, that describes the movement upon a liquid surface of waves of small amplitude. The equation has many different solutions, each one with different wave amplitudes and lengths, and these can be added thereby yielding a new solution of the equation. This expresses mathematically the physical fact that, as we see in the water of a lake, various different waves can be superimposed, and these superimpositions also correspond to a solution which is the sum of the solutions of the linear wave equations. In general, linear equations are much easier to solve than non-linear, and therefore they have been studied more; also, when a physical phenomenon might require being expressed through a difficult to solve non-linear equation, the usual procedure is to linearize it, eliminating those terms that influence it least, that is, a linear approximation is made. Yet in nature, the majority of phenomena are expressible through non-linear equations. The well-behaved equations of classical mechanics, such as those that determine the movement of the Moon, and which permit predicting the future of the system with exactitude, are the exceptions, not the rule. Recently it became possible to find the solution of any non-linear equation with the advent of the computer, since these permit numerical analysis of all types of equation--linear or non-linear--however complicated they may be. Today then, one can, thanks to computers, approach the study of dynamic systems whose behavior responds to non-linear equations, and which are precisely those that display sensitivity to the initial conditions. As an example of this let us consider a non-linear, unstable, system of behavior: a metal cone standing on its vertex, similar to that in figure II.1. However much we make its axis vertical, it will end by falling, and the side upon which it falls will depend on minuscule differences that break the equilibrium: a light breath, a small piece of dust. To predict on what side the cone will fall would require precise knowledge of all the forces to which it is subjected at the initial moment of equilibrium, which is equivalent to impossibility of introducing the totality of an immense quantity of parameters as initial conditions into the equations of movement. Another example of unstable non-linear systems of behavior is that of a snow-covered slope of a mountain, which can be in a state where the energy produced by emitting a shout may provoke as an effect an avalanche of many tons of snow, evidently disproportionate to its cause. If one deals, alternatively, with dynamic systems that periodically repeat a behavior, in those cases where a small change in the initial conditions is repeated or multiplied in each of the following periods, such that there is a situation describing positive augmentation, and which is described through non-linear equations, one can arrive at situations so distinct that there is no possibility of predicting them. Let us consider a dynamic system, formed by a sphere subjected to a force that causes it to revolve along a circular path that has a distance of 10 meters, as in figure II.2. To locate the position on the sphere at any moment we shall measure the length of the track from a fixed reference point which we have marked upon it. We shall suppose that the initial position on the sphere is of 1 meter with respect to that reference, and that we measure that position with a tape measure that give us an error of +/- 1 millimeter, that is, that the true position can be from 0.999 m, up to 1.001 m or any value included between these entries. We further suppose that each time the sphere passes by the reference point it receives an impulse, and that this provokes a displacement of the sphere around its orbit that increments its position 10 percent, such that if it was between 0.999 and 1.001 m at the initial moment, it is multiplied by a factor of 1.1 and this position comes now to be between 1.099 and 1.101 m at the end of the first period, and in turn repeats the multiplication by 1.1 for every one of the successive periods for which the position will be: between 1.209 and 1.211 m in the second period, between 1.329 and 1.332 m in the third, et cetera. At the end of 25 turns, the position will be practically the same as at the start: between 10.825 and 10.845 m, which since the total length of the orbit is 10 meters, locates itself on the sphere between 0.825 and 0.845 m with respect to the reference. Yet note well that if indeed the sphere has returned in period 25 to be practically in the initial position, the indefiniteness in its position has grown: it is now a zone of +/- 10 mm of length upon the track. The initial imprecision of +/- 1 mm in its location has been multiplied by 10. In figure II.2 the situation at 25 rotations ["vueltas"] and after 60 rotations is illustrated, where the sphere can be in any position between 3.81 and 4.19 meters, and at 70 rotations where it will be between 8.21 and 9.79 meters. After 97 rotations, this indefiniteness in the position will have been amplified 10 thousand times and, accordingly, will be 10 meters. Such that the sphere can be at any point on the track of 10 meters length, and we cannot know in advance what is that point through calculation. Of course, this seems easily remediable: since a tape measure has been used to measure the initial position with an error of 1 mm, it can be exchanged for a much more precise instrument, which yields an error 10 thousand times less, of only 1 tenth of a micron. Now then, with such a precise initial datum the sphere can be located through a calculation to predict its position after 97 rotations with an error no greater than 1 mm. But, what if one intends to continue predictions through calculation for a greater number of rotations? It results that even with this much more precise initial measurement, after 193 rotations it again is in the initial position: an indefiniteness of 10 meters, which does not allow predicting where the sphere is. As is obvious one cannot go on increasing the precision of measurement in an indefinite manner, for there rapidly appears the need to measure magnitudes with an error less than the size of an atom, and even if a similar marvel might be realized, a few hundred rotations more will come to the same frustrating situation. We are here between the sword and the wall: either we use an approximate measurement, which is not sufficiently exact for making predictions about any future state, or we try to achieve such an extremely precise measurement that it is impractical. A dynamic system that has this behavior is not rare in nature. It refers to systems that periodically repeat a certain state, and where this undergoes a perturbation which is multiplied every period (it is a case of resonance), being sufficient for after a certain time producing effects of a disproportionately large magnitude in comparison with the initial perturbation. In these systems, governed by strictly deterministic laws, under certain conditions calculation of their behavior can become increasingly imprecise, causing every intent of predicting the long-term future to be defeated.PHYSICAL PROCESSES ARE DETERMINATE BUT, CAN THEY ALWAYS BE PREDICTED?It now becomes necessary to distinguish between determinism and predictability, two words that since the age of Laplace were considered synonymous. We see that, under certain conditions, a dynamic system can be deterministic and, nevertheless, contain unpredictable behavior. Of course the physicists knew that the Universe demonstrates an immense complexity, but they supposed that, in general, that entity could be analyzed by decomposing it into its simple components, and that of these one could select a few variables as initial conditions to introduce into the equations, given that the immense majority of the other conditions are of such small magnitude, that not to consider them does not affect the result. But now we learn that those cases are something special, and that much more common is the situation in which even when one deals with systems for that where determinism rules, there is a limitation on the possibility of predicting future behavior which can come to be practically equivalent to a situation of chaos. This does not mean, however, that one can say nothing about such systems, since for certain initial conditions, the behavior is indeed ordered and hence predictable in the long run. Furthermore, even in the chaotic behavior state, these systems display many properties that can be understood with the help of the theory of probabilities, this mix of determinism and probability being a very fruitful form of attacking the characteristic problems of complex systems. One deals then with a new physics of non-linear phenomena whose object is the study of processes such as, for example, physico-chemical and biological turbulences and oscillations. These phenomena have an apparently random aspect, yet with unsuspected similarities in their behavior, which leads to encompassing their study in mathematical methods, where common notions result like bifurcation, strange attractors and Lyapunov's exponents. This new focus on complex phenomena implies a change of paradigm, now that that posited by Laplace with his demon capable of predicting any event has been put in a fair context: it does not try to absolutely predict all the phenomena that appear in nature using calculations based on deterministic laws; neither Laplace nor any physicist have dreamed that that would be possible. But the demon places in relief that this impossibility is due to our imperfection, as the limited beings that we are. Only an absolute Being could know the future for any term, when it always knows all the initial conditions with infinite precision and, in its calculations, also would manipulate numbers with an infinite number of digits, such as the irrational numbers. Neither human being nor computers can do this, given that they necessarily utilize a finite quantity of initial conditions and a limited number of digits in their calculations. This is the Platonic focus, which postulates that a mathematical reality exists that indeed is perfect, an ideal that sets the goal for where our efforts should be directed. Coherent action with this focus consists in concentrating effort on finding the solution of the equations that describe the phenomenon, so as to introduce in them empirical data obtained every time with greater precision, in the confidence that with the passage of time one that would thus be more and more near to predicting a growing repertoire of behaviors. In this vision of continual progress, it is a real question regarding study of complex systems, whether that sensitivity to the initial conditions may be such that no observation, no matter how precise it may be, will allow us to determine those conditions with sufficient exactness. If indeed in a deterministic system the initial state determines the final state, in these non-linear systems an approximate knowledge of the initial state does not permit deducing approximate knowledge of the final state, and it becomes meaningless to place priority upon solving the equations and introducing constantly more exact data into them. This difficulty with predictions in deterministic complex systems was already known in the previous century: the physicist James C. Maxwell, who played a fundamental role in the theory of electromagnetism, said at one of his conferences in 1873 that although the same causes may produce the same effects, when there is sensitivity to the initial conditions, similar causes may not have similar effects. He gives among other examples the explosion of cotton powder and the effect of small human actions that can unleash great social changes. The great mathematician Henri Poincaré wrote in 1908, in hisScience and methodthat "a very small cause, which eludes us, determines a considerable effect that we cannot avoid seeing so then say that that effect is due to chance." Later he extends this concept: Why do meteorologists find it so difficult to predict the weather with any certainty? Why do the rains, the storms themselves seem brought by chance, such that many people naturally believe in prayer so that rain will fall or good weather come, when in reality they would encounter ridicule with a petition for an eclipse? We see that big perturbations are in general produced in the regions where the atmosphere is in unstable equilibrium. The meteorologists see clearly that that equilibrium is unstable, that some cyclone will occur somewhere but, where? They are not in a position of saying; a tenth of a degree more or less at any point and the cyclone erupts here and not there and extends its havoc to regions that otherwise would not have been devastated. If one could have known about that tenth of a degree they might have foreseen it in advance, but their observations were neither sufficiently rigorous nor precise enough and therefore everything seems due to the intervention of chance. For Poincaré that appearance of chance in nature has two principal roots: 1) the already mentioned sensitivity to the initial conditions, which causes that even if the system's laws are known, small initial errors have an enormous influence over the final state, and 2) the complexity of the causes, for when the process of analysis has been ideally isolated, only a part of the innumerable influences to which it is subject are taken into account. This procedure is effective only for integrable systems which, in general, are linear systems. However, for the non-linear ones, this separation for analysis is performed by paying the price of diminishment of the reach of the prediction. Maxwell as well as Poincaré gave as an example of dynamic systems sensitive to initial conditions, the case of a gas formed of many molecules that fly at great speed in all directions and have many collisions among themselves. If one desires to determine the evolution of a system with a quantity of components as immense as those that form a gas, they must keep in mind that it is impossible to calculate the movement of each one of the particles with Newton's differential equations. Consider that a cubic centimeter of air contains 27 trillion (millions and millions of millions) of atoms, and the entire surface of the Earth would not suffice to write nothing more than the corresponding differential equations, without mentioning the calculation to determine with them the movement of each atom in that cubic centimeter of gas. Let us suppose that we could observe a very tiny portion of that cubic centimeter of air, a portion sufficiently limited so as to be able to register the movement of the few molecules which might be present there. We can imagine that by thus ideally isolating our zone of observation, we might be able to follow the trajectories in individual straight lines, the collisions between molecules and how they rebound, and we shall know that they follow the laws of mechanics are perfectly predictable, now that we can calculate their individual movements... Useless effort, since it will only be valid for the briefest time! It happens that more molecules will quickly arrive from outside our zone of observation, which did not figure in our initial data, and the molecules which were under observation will disappear from the zone, and the result now will be a different movement, with new unforeseen collisions, hence impossible to calculate, as unpredictable as if one dealt with a process ruled by chance. Thus we comprehend that if one observes only an infinitesimal fraction of a very complex process, formed of innumerable components, with an enormous quantity of distinct variable that act upon the system, it can appear as fortuitous, disordered. This is valid not only for a set of molecules in a gas: there is a certain parallelism with the phenomena that the social sciences study, since if one wants to examine a nation's economy or its political evolution by studying a small part isolated from the set, they move towards failure, for that part will be continually subject to the uncontrollable and unforeseen influences from the rest of the system. How then to proceed? It would seem that with detailed knowledge blocked of each individual behavior, one would have to resign themself to accepting that social, biological, economic, and physical systems with a large number of components cannot be confronted with the scientific method, given that they have behavior which apparently depends on pure chance, although one might think that it is governed by determinist laws. The answer to this question emerged in the field of the social sciences in the 19th century, when they wanted to study the health and economic characteristics of the nations. They applied to this the practical branch of probability, which is statistics. Its mathematical development was initiated by A. Quetelet in 1820, with his bookSocial mechanics,inspired by the model ofCelestial mechanicsby Laplace, and which had as its goal the development of the "moral and political sciences." The idea appears there of the average man, with a length of life, economic income, et cetera which are calculated as the average of the whole population. The experience with the study of gases teaches that, despite that on a microscopic scale the number of components is so large it is impossible to know their individual behavior, together on a macroscopic scale they display global properties such as the temperature, pressure, density, volume, that can be measured, and that relate to each other with well-defined laws. This is analogous to what happens with a human population with a great number of individuals, for whom parameters can be established like the rate of births, deaths, product per capita, et cetera, without the need of pursuing each individual history. The statistical method gave such good results for the social sciences that the physicists later applied it, thus creating statistical mechanics, a branch of physics developed around 1900 by the Austrian Ludwig Boltzmann and the North American J. Willard Gibbs. It is thus that James Clerk Maxwell proposed in 1872 the introduction in physics of the calculus of probabilities for systems with a large number of components, and wrote: "we have found a new type of regularity, the regularity of averages."GAMES OF CHANCE AND PREDICTIONSThe theory of probabilities began in games of chance. Laplace establishes in his book that: Thus for example, a six-faced die gives us a probability of 1/6 that it will fall with the 2 facing upwards, and the same probability applies to the other five faces. If a die is rolled a great number of times and statistics are kept on how many times each one of its six numbers appear, it will be proved that the higher the number of rolls the nearer the number will be to 1/6. Yet it is unknown before rolling the die which face will be up this time and this is, of course, the reason for the existence of games of chance. In theory, if one knew the forces applied upon throwing the die, the frictions, air resistance, et cetera, they would be able to predict how it would fall, but it is impossible to obtain all the necessary data and, accordingly, players can continue to enjoy the emotions of betting without knowing what chance will deal them. Thus when we speak of dynamic systems in which Newton's equations cannot be applied to determine the behavior of each individual case, the nearest to a prediction is to establish the probability of a certain event occurring. We should respond now to a new type of question. Not now, what is this particle's velocity, but instead, what is my probability of finding particles whose velocity is between such and such a value? The method was initially applied in physics in the study of gases, so that the analogy to which we have referred between the molecules of a gas and the individuals of a society is logical. This concept was introduced in thermodynamics, which emerged as a scientific development from the study of heat. At the end of the 19th century there existed then two scientific foci for the mathematical formulation of natural phenomena: analysis through differential equations for systems with few components, and statistical analysis for those systems with an elevated number of components. Yet in the scientific community the hierarchical distinction between each focus was quite clear: the highest level corresponded to the first of these visions, that which expressed the focus we have called Platonic, wherein nature is ruled by eternal laws written in mathematical language, whereas the second owes its existence to the ignorance that derives from our human limitations, which can be partially compensated through the use of probability, but which inevitably will see its field of application shrink to the extent that knowledge advances, thanks to that continual and inevitable progress which was a basic belief in the West beginning with the Enlightenment. As we saw, such a vision changed radically during the passage of the 20th century, Henri Poincaré being the first who perceived the essential characteristics of our focus. Just like Maxwell, he warned that simple systems, of few components and that, therefore, are in the first of the categories which we have described, can nevertheless behave chaotically and then require statistical methods for their description. Poincaré made this warning on examining the case of a very simple dynamic system, formed of only three bodies that are attracted by action of the force of gravity, when one of them is very small in relation to the other two. He tried to determine how it would move and to graphically represent its evolution as a trajectory in a mathematical space call "space of the phases" (to this space we shall return below). The structure of this trajectory, called a homoclinic, turned out to be so extraordinarily complicated are far from intuition that Poincaré stopped trying to draw it and wrote in hisNew methods of celestial mechanics: We attempt to imagine an idea of the figure formed from these two curves and their infinite intersections, each one of which corresponds to a doubly asymptotic solution. Such intersections form a sort of framework, a weaving, a net of infinitely fine mesh. None of these curves can cross themselves; further, they have to fold into themselves in a very complex manner to cut through the mesh of the net infinite times. The complexity of this figure that he did not even want to draw is surprising. Nothing is more relevant to give us an idea of the complexity of problems with three bodies and, in general, all the problems within Dynamics where there is not a uniform integral. This unawaited complexity, that so surprised the genial mathematician, is what is today studied under dynamic systems with structures like the homoclinics, the heteroclinics and also the strange attractors, fundamental representations for the examination of deterministic chaos. Poincaré's has been basic for the study of dynamic systems and is consecrated as the foundation of the means that today permit approaching the mathematics of deterministic chaos. However, there was a period of more than sixty years when his contributions on non-linear functions were practically ignored, and it is recently starting with the work of A. Kolmogorov and V. Arnold in Russia, S. Smale in the United States, D. Ruelle and R. Thom in France, that serious work in this field resumes. Despite Poincaré being a visionary, he did not advance in the study of those systems, which may be attributed to two principal reasons. The first is that in his era numerical calculations were done manually, which makes impossible the numerical treatment of the non-linear systems of equations, which must await the appearance of computers. The other reason is his philosophical attitude, shared with the majority of the mathematicians and physicists of his era, and which is clearly reflected in a writing by Poincaré where he notes the existence of certain mathematical functions that do not meet the classical requirements of being continuous and derivable: "Logic sometimes engenders monsters. For half a century we have seen a multitude of rare functions emerge that seem to strain to resemble as little as possible the honest functions which serve for anything. No more continuity, no more derivatives, et cetera." What would have been Poincaré's astonishment before the attractor that Lorenz discovered to model meteorological phenomena! Not for the mathematical aspect, since as we saw, he understood perfectly the impossibility of making long-term predictions, but because this strange attractor, which permits the graphing of essential characteristics of the dynamic system of the atmosphere, is derived using the "rare" mathematical monsters that he so little trusted. He had arrived at discerning the fascinating universe of the mathematics of the unpredictable, but he retreated before that lack of regularity, of continuity, which caused him so much displeasure. A consequence of that vision was that the scientists imbued with it have tended to ignore certain fields of research, like those that shelter the "monsters" which Poincaré mentioned (for example, curves without tangents and all the algorithms that today we would call "chaotic"). To consider deterministic chaos with real scientific value, one must renounce the belief in a hierarchy where the top is occupied by the perfect forms (circle, sphere, et cetera). But, why accept that a circle is superior to a complex figure, such as a fractal or any element of nature? As the mathematician B. Mandelbrot would say, "the mountains are not cones, nor are the clouds spheres, nor does the direction of a river go in a straight line; without doubt, the forms that surround us throughout our life have always been very far from the simplicity of Euclidism." This change in valuation has a cultural and philosophical origin, and if this is overlooked, one risks reducing the modern study of chaos to a mere set of new techniques. An important aspect of this new focus is the indispensable use of the computer, which permits making the calculations and additionally translating into images the behavior of dynamic systems. Also in this regard there is a clear distinction between the attitudes of the mathematicians who adhere to one or the other focus. Just as the use of tools that were not the ruler and the compass was rejected by the Greek geometricians, today the resistance of many "pure" mathematicians continues to the use of computers for mathematical demonstrations, based on the true fact that the number of decimals in any numerical result that a computer can handle is limited. Another new aspect is the deliberate introduction of randomness in many calculations, for example, through series of numbers generated as random, with the goal of modeling natural phenomena that have unpredictable components. The use of probabilities had been begun in the past, as we saw, by Quetelet in the social sciences and by Maxwell in the kinetic theory of gases, but it was considered as an index of ignorance, which indicated the limits of the zones containing the topics that true science had managed to clarify; and in those zones randomness, probabilities could only be accommodated when true science could not be applied. However in the first quarter of the century, quantum mechanics showed that the concept of probability and statistical methods are essential for the formulation of the laws of physics, at least at the atomic scale, from where appreciation began of their conceptual significance. The mathematicians and physicists were aware that there is a certain amount of disorder in nature, yet they did not believe its study constituted a true science. Now, on the other hand, chaos is fully recognized and it is accepted that even in the phenomena that exhibit order and regularity, there is a vast universe of disordered, irregular phenomena, which are not reducible to pure forms, and which also can appear unexpectedly in very simple systems, like that comprised of three bodies. Chaotic phenomena can be studied scientifically, for although there is no possibility whatsoever of foreseeing the detailed behavior of its individual components, yet a qualitative prediction can be made of the evolution of the system as a whole, and also the conditions can be sought by which a dynamic system that is in a state of order passes into chaotic behavior and vice versa. As can be seen, some of the most interesting ideas of existing science are related to those topics, which has obliged us to determine the true reach of determinism, of our capacity to predict the future and of the true significance of natural laws. Only time will give us an exact idea of the true impact of those changes upon the method and the philosophy of science. Now let us turn to examining the characteristics of dynamic systems which can exhibit chaotic behavior.IN NATURE THERE ARE MANY CYCLICAL PROCESSESWith regard to the dynamic processes that are observed in nature, when one wants to study them they should underline the universality of what are periods or oscillations, ubiquitous processes in physics, astronomy and biology, whether in the structure of the clouds in the atmosphere, the eddies in a river torrent, the melodious sound of a violin, or the flux of electric energy in an electronic circuit, and go from the know movement of pendulums and of the orbits of the planets to complex biological rhythms: fluctuations in the reproduction cycles of an animal population, respiration, cardiac rhythm, alternation of waking and sleep, neurophysiological processes, et cetera. In all of those are found this characteristic of periodicity of behavior. Since the dispositions and phenomena presented by this oscillatory behavior are legion, it is natural that the oscillator would inspire a growing scientific interest. But now we go far beyond that, for the conclusion which is extracted from the previous vision is that the properties of oscillators can be applied to the set of dynamic processes. In effect, one can describe mathematically any process that fluctuates over time like the algebraic sum of a set of periodic oscillations, by a method called Fourier analysis. As can be seen in figure II.3, if we graph the variations over time in a process O that we wish to analyze, we obtain an irregular curve, which does not repeat over time. But Fourier demonstrated that one can consider it as the resultant of summing various regular periodic curves, such as those of A, B and C, and hence any theory of dynamic processes can require application of the concept of oscillators, which in combination account for any variation over time. It is for this reason that, as an initiation into the study of dynamic systems we shall examine in detail the behavior of an elementary oscillator, such as a simple pendulum, and later we shall see what happens when two or more of these elementary oscillators are combined to form a dynamic system. III.Simple pendulums can be very complexWE SHALL ANALYZE here under what conditions the movement will be chaotic in a system as little complicated as one can be, that formed of a pair of pendulums. The pendulum is a dynamic system that repeats its behavior at regular intervals which we call periods. They have been studied at least since the age of Galileo, such that one might suppose it would have no occasion to surprise us. Furthermore, for centuries the pendulum has constituted the paradigm of predictability and regularity. Before the utilization of quartz oscillators, clocks were regulated by pendulums.CONSERVATIVE OR DISSIPATIVE DYNAMIC SYSTEMSOne should before everything distinguish betweenconservative dynamicsystems, which are those where the energy of the system remains constant because there is an absence of rubbing or internal friction, and those in which due to friction that is a continual diminution of energy, calleddissipative systems.Regarding the former they should also be called "Hamiltonian systems," since it becomes very fruitful to describe their behavior with a mathematical function developed by W. Hamilton that uses the positions as variables and, in place of velocities, momentums (the product of velocity and mass). Examples of Hamiltonian systems are the solar system and the plasma in a particle accelerator, and of the dissipative the terrestrial atmosphere, the oceans, living organisms, and all machines. We shall examine the characteristics of Hamiltonian systems, beginning with the most simple case, or that is of the ideal pendulum, which no one will find in any laboratory, yet which permits establishing the essential ideas for studying oscillators. This ideal pendulum moves in a two-dimensional space, or that is, upon a vertical plane, the thread from which it hangs being rigid and weightless, without friction in the pivot and being in a vacuum, such that there is no air resistance to its movement. Thus we deal with a simple Hamiltonian system, and it is possible to mathematically describe its movement through Newton's equation: a differential equation that links the second derivative of the angle the thread forms with the vertical, with respect to time (or that is, variation over time in the speed with which the angle changes), with other parameters like the length of the thread and the acceleration of gravity. If one wants to represent its movement graphically, we shall need to calculate for every moment its angle with respect to the vertical and its velocity, which we shall call θ (theta) andvrespectively. We wish to know how each changes to the degree that time elapses and for us it will be of supreme utility to use the space of its phases.THE SPACE OF THE PHASES AND COUPLED PENDULUMSThis is an abstract mathematical space, that should not be confused with that in which the components of the system really move, but which contain in their geometrical forms some concrete information: the variable which describe the movement of the dynamic system. We shall use here a method which is the exact opposite from that used by René Descartes when he conceived the use of coordinates. Descartes discovered how to transform the geometry into numbers by imagining space as an immense grid, the system of Cartesian coordinates, so that the position of any point in space is defined through numbers that measure its distance to these reference coordinates. Accordingly, any geometric form can be expressed in numbers through a mathematical equation which refers to the Cartesian coordinates. Now to introduce the space of the phases we do the reverse: we transform numbers into geometric forms pretending that these numbers are coordinates in such an imaginary space. The advantage of proceeding in this way is that it is enough to observe that geometrical form to know how the behavior of the system over the passage of time will vary. For that one uses as axes of coordinates the system's dynamic variables, or that is those magnitudes that are changing over time, as might the velocity of the pendulum and its angle with respect to the vertical. At an initial moment the pendulum, then, will be represented by a point in the graphic of the space of the phases which indicates to us its velocity and angle. An instant later it will have another position and velocity, and to this a different point corresponds, and thus we can create the history of the pendulum through successive points which trace a trajectory. In this manner, dynamics switches from studying long lists of numbers to visualizing how a geometric figure evolves over time. If the result after a time is a single fixed point, the system is static, does not evolve. If there is a curve and it is closed, this indicates that the system periodically repeats its behavior; if it is an open curve, its characteristics must be examined to see whether or not there are underlying regularities. It was the great mathematician Henri Poincaré who had the brilliant idea of proposing this method, by which dynamics can be made visible. Researchers try to discover the general characteristics of a system, which are better appreciated if one observes the forms that appear in the space of the phases. In general, this way of representing a dynamic system has among its advantages that the coordinates of the space of the phases can represent any characteristic of the dynamic system which varies with time, such as, for example, the electrical signals of the heart, the population of bees in a hive, the value of the dollar, et cetera. A general principle of trajectories in the space of the phases is that none can touch the other as a consequence of the determinist character of this description: if two trajectories intersected at a point, there would be two different curves starting together, which would correspond to two distinct solutions of the differential equations of the system, or that is two different behaviors at the same time. Thus, if we were representing an automobile as a dynamic system, if two trajectories were to touch at a point it would indicate that we have a single vehicle moving simultaneously at, for example, 30 and 90 km. per hour, which obviously is opposed to the unambiguous, deterministic character of these phenomena. To represent the behavior of the ideal pendulum in the space of the phases, we begin by drawing two reference axes, or coordinates, one horizontalθfor the angles, and another verticalvfor the velocities. Now we mark the path or trajectory of the pendulum onto this map as in the figure III.1babove. Let us suppose that starting at an initial instant when we provide an impetus, every tenth of a second we measure the angle and velocity, and mark the corresponding point in figure III.1b.As we can see in figure III.1a,at the initial timeT= 0, the pendulum forms an angle-θwith respect to the vertical, a situation that is represented by the pointT= 0 in figure III.1b.When you let it go it will move to the right with growing velocity, which will be at a maximum when it passes the vertical at the instantT= 1, a moment at whichθ= 0, and this is represented in figure III.1bwith the pointT= 1. Upon attaining the maximum displacement to the right, so thatθ= +θ,its velocity will have been reduced to zero, and this corresponds to the pointT= 2 in figure III.1b.Continuing this procedure one will obtain a set of points that trace the dynamic trajectory of the oscillator, a trajectory that represents the complete movement of the pendulum throughout a cycle. Given that the path is repeated cycle after cycle, the map for this simple pendulum is a single closed trajectory, also called an orbit, by analogy with the movement of the planets. If we give the pendulum a greater initial push, the maximum angle will be greater. Thus, in a single graphic we can represent the movement of the same pendulum with different initial velocities, and for each one of them there is a distinct orbit. This is one of the characteristics of conservative or Hamiltonian systems. A family of curves is obtained thereby, which can constitute the planeθ,v(figure III.2). For an ideal pendulum, and for small angles of distance from the vertical, these curves are concentric circles, which correspond to the simplest solution of the differential equation, called in this case an "equation for a simple harmonic oscillator," and which is a linear equation. As you can see, there is a central pointA,at the cross of the axes, that is, for zero values ofθandv.This represents the pendulum when it has a zero velocity and rests over the vertical, or that is when it is at rest. If now we separate the pendulum much from vertical, the relation between that force that moves it and the angle is described through a much more complicated differential equation, which is not linear. Its solution is graphed in figure III.3. The family of possible trajectories now resembles an eye. The central pointAcontinues to represent the immobile pendulum (zero velocity and angle). The concentric ellipsesBcorrespond to cycles of the pendulum more distant from vertical each time, like a swing that is given a greater push every time, until it lies 90 degrees from the vertical, and then begins to rise above the pivot. What happens when it receives a send-off such that it surpasses 180 degrees? As we know, it will not oscillate, but will rotate in a circle in one or the other dimension, like a helix. This is represented in the space of the phases as the families of curves indicated asCandD.As can be seen, those inChave a positive velocity, that is, represent complete rotations in one direction, for example in the same way as the hands of a clock, and those inDrepresent the contrary movement, that which can be indicated as a velocity with a negative sign. This method of representing the behavior of the dynamic system in the space of the phases allows us to appreciate the essential characteristics at a glance: if the graph is a point, the system is quiet; if there are concentric circles, it is oscillating to a low extent; if there are ellipses, the oscillations are broader; if they are in the zonesCorD,it is rotating instead of oscillating. The two curvesSthat form the border between theBoscillation regions andC, Dof rotation are called "separatrixes" and correspond to the pendulum positions when it is suspended exactly on the vertical at its highest position (figure III.3) or that is which has the maximum instability. If it is in that position it will fall rotating in one direction or the opposite, and will acquire an increasingly great velocity, until passing by the lowest point on the vertical and then ascending with decreasing velocity until returning to the highest point on the vertical. On the separatrix curve, the total energy available to the pendulum is exactly equal to that that it needs to leave that highest point upon the vertical and return reaching it. What happens in a more complicated Hamiltonian system, formed from two coupled ideal pendulums (AandB), that is, which mutually influence each other in their movement? Each one of the pendulums has its own oscillation period, but now this is affected by that of the other. If we ignore pendulumA,then the movement of pendulumBwill trace a closed curve in the bi-dimensional space of the phases. If we ignore pendulumB,the movement ofAwill trace another closed curve in a different bi-dimensional space. But if the two pendulums interact, now they are not independent and to represent the trajectories the two spatial planes must combine into one whose dimensions will accordingly increase from two to three. Let us suppose that periodAis, for example, nine times longer than that ofB.IfAwere independent ofB,it could be represented in figure III.4 as a closed curve upon a horizontal plane, andBas on a plane perpendicular to the horizontal. But when their movements are coupled, this is represented in the space of the phases combining both closed curves: to the extent thatAis being displaced horizontally, curveBdeviates from the horizontal plane, in a movement comparable to that of rolling a rope around an inner tube. The result of one circle rolling itself around another is the creation of a figure in the form of a solenoid ring, upon the surface of what the mathematicians call a torus (figure III.4). Here we can consider that theAcycle is the axis of the torus, and henceBis a cycle perpendicular to such an axis. Now we can see the three-dimensional torus in more detail. If the periods or frequencies of two coupled pendulums are in a simple relation, for example if one has a period nine times greater than the other, the relation is 9/1, and the line making circles around the surface of the torus is tracing a solenoid which always passes over the same points on the torus no matter how many revolutions those combined pendulums realize, which demonstrates that the dynamic system is exactly periodic. Thus, in figure III.4, the initial point whereT= 0 and the final one on completing a period,T=1, coincide. But, what happens when the periods of the coupled oscillators are incommensurable, or that is when the quotient of those periods is an irrational number? An irrational number cannot be written as a ratio and its decimal expression contains an infinite number of terms without a repetitive pattern. If the coupled system has an irrational relationship between periods, the curve in the space of the phases will proceed by wrapping around the torus passing on every turn over different points, as is seen in figure III.5, where the initial pointT= 0 and that which corresponds to a period,T= 1, do not coincide, such that the curve with the passage of time will cross every point of the surface until totally covering the torus, and never will repeat itself. A system with these characteristic is called almost periodic. Mathematicians are capable of working with toruses of any number of dimensions. Which is equivalent to it being perfectly possible to combine more than two oscillators and represent their combined movement on the surface of a multidimensional torus.POINCARÉ SECTIONS AND MULTIDIMENSIONAL SPACESIf indeed there is no restriction in principle to the number of dimensions to the space of the phases, evidently is is much easier to visualize the forms with only three or two dimensions (volumes or surfaces). There is a method, conceived by H. Poincaré, for visualizing the essential properties of complicated trajectories in spaces of three or more dimensions, that consists of lowering to one that number of dimensions. We consider the example of the trajectories that surround the torus in a bi-periodic system and which forms the surface of a tri-dimensional solenoid. We now slice the torus with a transverse plane (figure III.6) and mark the points where the trajectory intersects the plane. Since the trajectory is periodic, one point will be marked each turn, such that after a sufficiently long time one will have a true map upon the plane, which in practice is very advantageous for the simplification involved in reducing to one the number of dimensions, and also because it has passed from continuous description over the time of the trajectory in the space of the phases, to taking only the data of movement each time the trajectory crosses that section, and this implies that it is necessary to manipulate a much smaller amount of data. What can the Poincaré section tell us for this bi-periodic system? If the relation of frequencies is a rational number, the curve is fixed upon the torus, every rotation superimposing on the previous one, such that in the section there appear only the corresponding isolated points whose number and position depends on the relationship of the periods. If, however, the system is almost periodic, with a relation of frequencies that is an irrational number, the curve will pass through a different point on every turn, in time covering the whole surface of the torus, such that in the Poincaré section one has a closed curve. The analysis of more complicated oscillator systems requires the introduction of a space of the phases with more dimensions, since as we have seen, the number of dimensions depends upon the quantity of independent variables in the system, which might be the velocity or the thrust, the position, or some other dynamic characteristics that define the behavior of each of the components. To be able to confront these concepts of multidimensional space, crucial for the study of complex systems, one must make a generalization from the geometry of coordinates. Let us say that we live in a tri-dimensional space, given that our movement in space has three degrees of freedom: we can make three types of movement which have perpendicular directions between themselves (left-right, front-back, up-down) and any point in space can be reached by combining those three possible types of movement, such that its position can be indicated through three numbers that we shall call thex, y, zcoordinates of the point, and which give the distances from the point to a reference in those three perpendicular directions. We can also refer to an abstract space of four dimensions, with four coordinatesw, x, y, z,to one with five dimensions, with coordinatesv, w, x, y, z,and so on successively, always by keeping in mind that now we do not refer to the physical tri-dimensional space in which we live and move, but instead to a mathematical space. This becomes of enormous utility when one tries to comprehend the mathematics of many variables. In any problem, be it in physics, biology, economy, any significant magnitude can be considered, and visualized, as a dimension of the problem. An economist can work in a multidimensional "space" with index variables for cost of life, cost for shelter, value of the dollar, price of oil, trimesters of the past decade, et cetera. A physicist can study a dynamic system formed of three bodies, for example, where to each of them there corresponds the three positional coordinatesx, y, zplus the three coordinates of velocity or of momentum, or that is a total of 18 coordinates, such that we are dealing with a space of 18 dimensions. Nothing prevents us from continuing to augment the number of components of the system, so that we can say that, in general, a dynamic system withnindependent variables--ndegrees of freedom--can be represented in a space ofndimensions. In general, this is valid for ordered and stable systems that, though they may be formed from a great number of components, and which accordingly should be represented in a space of the phases with a great number of dimensions, in practice move in a very small sub-space of this vast multidimensional space that represents a physical state in all its slightest details, and whose volume we shall callH.Thus a solid body, for example, a rock, although it is composed of an enormous quantity of molecules, since those are rigidly linked among themselves if they move in unison, and it is enough to represent the movement with a single point: the rock's center of gravity. The evolution of the system over time is represented by the trajectory ofHin the space of the phases (figure III.7) where the points are marked which correspond to the states for the timesT= 1,T= 2,T= 3, et cetera. A dynamic system of ordered behavior can, if its characteristics are altered, come to have chaotic behavior, and the opposite process can also occur. Is it possible to represent those changes in the space of the phases? The answer is affirmative: as we shall see below, the study of the transition of an ordered dynamic system to chaos is, in a certain sense, the analysis of how a movement that can be very simple, limited and repetitive, breaks at a certain critical point, developing a new behavior which corresponds to a displacement of the system's trajectory to much vaster zones of the space of the phases. Now we shall be able to visualize the difference between predictable processes and chaotic processes. We shall represent (figure III.8) the evolution of a dynamic system of predictable behavior as a trajectory indicated by 1 in the space of the phases. If we quickly vary the initial conditions, we have trajectory 2, which remains close to that of 1, indicating that the behavior is practically the same (figure III.8a). On the other hand, in a chaotic process, the trajectories 1 and 2 that initially were nearby separate more over time, signaling the growing divergence of their behaviors (figure III.8b).CONSERVATION OF VOLUMES IN THE SPACE OF THE PHASESAs we have seen, the total energy of a Hamiltonian system is invariable. How is this expressed in the space of the phases? Let us consider representation of the evolution of a dynamic system in the space of the phases. We saw that there is a region of volumeHto represent that system, and that it displaces into the space of the phases, so tracing a trajectory which expresses its evolution over time. In the 19th century, the mathematician J. Liouville demonstrated that for every Hamiltonian system the volume of thisHregion remains constant over time, as if treating of a drop of a liquid that it is not possible to compress. This property of Hamiltonian systems is much more restrictive than simple conservation of energy or reversibility in time of the movement equations. The conservation of the volume of the region in the space of the phases can take place in two different ways:1)the region under consideration shifts around the trajectory, rotating and deforming in periodic fashion, and then the neighboring trajectories "coil up" without separating much from each other;2)the volumeHstretches over time in one direction, contracting in the perpendicular direction. While in the first case two initially close trajectories remain nearby, in the second they tend to separate. From a dynamic viewpoint, the difference is considerable. In effect, the trajectories are stable in the first instance, unstable in the second, for here a weak initial separation can be amplified with the passage of time.REVERSIBILITY IN TIMEAnother important property of the equations that describe the behavior of a Hamiltonian system is that changing the sign for time, that is, replacing +twith -thas no effect whatsoever; the equations are identical. Or say, that if the movement of an ideal pendulum is filmed, you cannot discern in which direction the film is run. One says, in general, that the conservative systems have reversible mechanics. As we shall see below, irreversiblity in time is characteristic of those systems in which energy, instead of being conserved, dissipates.WHERE IN PENDULUMS DOES CHAOS APPEARThe study of deterministic chaos began in the Sixties with the pioneering work of E. Lorenz, D. Ruelle and F. Takens, who using computers, which in those days began to show their fantastic potential, demonstrated that even simple dynamic systems, formed with only a few oscillators, can behave in an unpredictable, chaotic manner. It is easy and amusing to prove this with a system of two coupled pendulums - simple to build and even obtainable as a toy. In figure III.9, the system consists of a light pendulum formed from two small spheres united by an axis around which they can rotate. This axis hangs from a heavier pendulum, which is what imposes the basic oscillation. Both pendulums have permanent magnets attached, so that their movements are coupled. At the base of the system there is a small electromagnet fed by an electric oscillator circuit which maintains the oscillations of the main pendulum so that it does not dampen (entrained pendulum). Once an initial push is given, the heavier pendulum oscillates with a clock's regularity, while every time one of the other spheres swings near the large pendulum it receives an impulse due to the attraction between the respective magnets. Soon a surprising spectacle will occur: the light pendulum performs a strange erratic dance, oscillating at times in a rhythmic fashion, to then jump unpredictably to movements that are chaotic. How will this behavior be represented in the space of the phases? Let us take the case of the dissipative dynamic systems, which abound in this world, and whose energy diminishes through friction and other effects. We know that if we give an impetus to an actual pendulum, it will oscillate or rotate but soon, as opposed to the ideal pendulum we have studied previously, its movement will continually fade, until finally it will hang still, unless it receives new energy. This behavior can be represented as a spiral trajectory in the space of the phases, which results in pointAas the final resting position (figure III.10). No matter what initial impulse the pendulum receives, in every case the loss of energy finally immobilizes it, that is, the trajectory inevitably leads to the point of reposeA,as if this attracted the curves in the space of the phases. Hence the name of "attractor" that the fixed pointAreceives. Of course a pendulum clock would be of very little utility if this trajectory from the first impulse until rest were to take only a few minutes, and therefore diverse mechanisms have been invented that replace the energy which is lost in functioning: weights that stretch elastic springs, electromagnets that cyclically change their polarity. The result is that the pendulum moves with a regular rhythm despite friction and air resistance. Hence, in a pendulum that receives energy, the curves in the space of the phases corresponds to figure III.11. In fact, if the pendulum is given an additional push, or if it is momentarily slowed, it will eventually return to its original rhythm, which corresponds to the trajectoryC.This curve evidently constitutes a new type of attractor, since instead of the system being attracted to a fixed point, it is brought into a trajectory that forms a closed curve. Notice that there is an important difference with an ideal pendulum that moves without friction nor loss of energy: in this, the smallest perturbation caused by adding a greater impulse or slowing it causes the pendulum's orbit to change by contracting or expanding a little, or say it jumps from one concentric curve to another larger or smaller one. By contrast, the trajectory of a mechanically assisted pendulum has stability, resists small perturbations (figure III.11) and thus when it is given a greater impulse it gradually dissipates it, or if it is slowed it received energy from its source, so that in both cases it finally returns to that single closed curve, which accordingly is also an attractor, that we shall call the "attractor limit cycle." There are two basic classes of limit cycles: that which we just finished presenting for the pendulum, which is stable; the points on nearby trajectories move towards it. There also are dynamic systems with unstable limit cycles, where the point of nearby trajectories move away from this curve, which acts as a repeller. The importance of what we have analyzed for the pendulum is that it permits understanding the essential behavior characteristics of systems that act cyclically and which are so frequent in nature: an electric oscillator, the tides, the vibrations of the air in an organ pipe, the electrical impulses that cause the heart to beat, the number of individuals in an animal population... We recall the two types of attractors described up to now:1) the attractor point,that corresponds to a stationary system state, nothing happening over time;2)thecycle limitattractor, which indicates periodic behavior, which further implies that, if indeed the system is dissipative and hence is losing its energy, that is being replaced by the delivery of energy from some external source.CONTRACTION OF VOLUMES IN THE SPACE OF THE PHASESHow is the fact that we are dealing here with dissipative systems represented in the space of the phases? As we saw, the cycle limit attractor indicates that it is displaying a stable dissipative system, which replaces the energy that it loses. If one has, for example, an entrained pendulum that they perturb temporarily halting it, when the angle and the velocity diminish we obtain trajectoryB(figure III.12) but the system receives energy from its source to compensate for this diminution and the trajectory ultimately merges with cycle limitA.The same occurs is the pendulum is given an additional push that removes it from its orbit; with time, the dissipation of this excess energy will cause its trajectory to converge with cycle limitA.Accordingly, the cycle limit attractor is within a zoneCin the space of the phases such that any trajectory that is initiated from any point whatsoever inside that region will end by being inexorably drawn by the attractor. This zone is called a "basin of attraction" (figure III.12). Let us suppose that in this basinCthere is a regionRthat represents a set of initial values for positions and impulses in the system. To the extent that the trajectories are attracted by the limit cycle, they approach each other and culminate by converging upon the latter, ultimately comprising a single trajectory: there is a contraction of the regionR,which diminishes until disappearing into the attractor curve. Since a curve is a line of a single dimension, it cannot pass through all the points that comprise a volume; there will always be an infinity of points not covered by the curve and it is evident, therefore, that in a space of the phases of three dimensions an attractor should have fewer dimensions than 3, and this can be generalized: This is the principal characteristic that distinguishes dissipative systems from conservative ones. In a dissipative system all the initial conditions converge towards regions in the space of the phases which have fewer dimensions than that of the original space.DISSIPATIVE SYSTEMS FORMED FROM TWO COUPLED PENDULUMSWe turn now to study the case of a dissipative system also formed by two coupled pendulums. Here, in a similar manner to that seen in the study of conservative systems formed by combining two ideal pendulums, it will require a space of the phases of three dimensions, in which the trajectories unfold upon the surface of a torus, so that for the case of a dissipative system we have a more complicated attractor than a fixed point or a limit cycle: we deal with a curve in the shape of a solenoid that passes through all the points on the surface of the torus, thereby representing the combined behavior of two coupled pendulums with periods which are incommensurable, a that is whose quotient is an irrational number. Hence, we call it an "almost periodic attractor." The behavior of such a system is predictable, that is, that knowing the velocities and positions of its components at a given moment, one can determine them for any other instant. And even if such knowledge has a certain margin of error, or uncertainty, this remains of the same order of magnitude for all future determinations. This property translates into the space of the phases through the fact that two adjacent trajectories continue being adjacent even as time passes.STRANGE ATTRACTORS AND FRACTAL DIMENSIONSWe now return to a system that displays chaotic behavior, like that of the two pendulums coupled with a magnet which we had described. Its representation in the space of the phases will require this having three dimensions, in which the trajectory will be a continuous curve, similar to the case of the almost periodic attractor. Yet what distinguishes it from this is that if one examines two neighboring trajectories in the space of the phases, they see them diverge rapidly, continually separating more. We deal here with a "strange attractor," whose essential characteristic is the amplification of the separations, small as they may be, between trajectories in the space of the phases. This characteristic is calledsensitivity to the initial conditions.We have here the key to understanding why the determinism that governs a dynamic system does not necessarily imply predictability. If this sensitivity appears, the system is unpredictable after a certain time, no matter what the other characteristics of the space of the phases, or its number of dimensions, may be. We would only be able to predict its evolution with exactitude if we were to know with absolutely infinite precision all the factors that act upon the system, and we know that that is impossible, given that we necessarily know the initial conditions only in approximate form. However, the situations corresponding to the other attractors relate to stable and predictable states, because they do not have such a sensitivity, but indeed the contrary: they is an insensitivity to the initial conditions. This is evident because, in accordance with the above figure, there is a basin of attraction in the space of the phases within which a trajectory that starts from any initial condition inexorably ends by merging into the attractor; no matter how the process is begun, the system after a time concludes by following the stable and foreseeable behavior described by said attractor. The surprise that is felt before the appearance of chaos in a system as simple as that comprised of two coupled oscillators is natural, however no one found it novel or strange that systems as complicated as the terrestrial atmosphere or a turbulent stream have behaviors difficult to prognosticate. Nevertheless, the study of these simple systems has shed light on those constituted from a great number of components. The attractor that characterizes chaotic behavior if transformed in a spectacular and apparently counter-intuitive manner, since it should reflect that situation in its geometry. Its structure should exhibit two opposite tendencies: in honor of its name, the adjacent trajectories should converge toward the attractor and conversely, to reflect a state of sensitivity to the initial conditions, the trajectories should diverge continually separating more. The speed with which these successive trajectories diverge is measured with acoefficientcalled a"Lyapunov exponent."Furthermore, the other essential condition is that, as we have seen above, the curves formed by the trajectories cannot cross meeting at a point (a condition of determinism). An attractor then becomes very difficult to visualize, whose form cannot exist on a surface, no matter whether it is a plane or a curve like the solenoid which surrounds the torus in the simple case we examine. The reason for this impossibility is that the divergence between adjacent trajectories required by the chaotic state cannot be expressed upon the surface of the torus, because after a certain number of turns these, not being parallel, should cross to accommodate all the trajectories. The existing space on the surface is then insufficient and only one alternative solution exists, which is what appears in the attractor: the trajectories detach from the surface jumping outwards to occupy the surrounding region. Accordingly the dimensions of this attractor should be more than the two that correspond to a surface, or that is: 2 <dBut, and here the difficulty emerges, we are dealing with a simple system, in a space of three dimensions, and we have seen that, since the conditions of the dynamic system (velocities, positions, et cetera) are necessarily limited to a certain range of values, the attractor should occupy only a restricted zone of the space of the phases, in place of filling it completely, and therefore its dimensiondshould be less than 3: 2 <d< 3 Or let us say that we should imagine a geometric figure of greater than 2 and less than 3 dimensions, which is to say that it is in a situation intermediate between a surface and a volume! Obviously, such a situation is not normal in Euclid's geometry, and this may have been one of the reasons why the possibility of chaos in simple system was so recently discovered. Yet recently mathematics have been developed in which there exist irregular or fragmentary shapes that can be characterized by dimension which, as opposed to the Euclidean are not whole numbers, and which have been calledfractalsby the mathematician Benoit Mandelbrot, who has been one of the main forces for the study of these strange geometries. This attractor then is a fractal, which is located on a region that includes the surface of the torus but, because it is a curve, does not occupy all the points in the volume of that region, there always remaining an infinite number of points through which it does not pass. Other dynamic systems, such as those formed by electrical oscillators, turbulent fluids, chemical reactions, display attractors with those characteristics, which have been baptized "strange attractors" by Ruelle in 1971. A strange attractor and its Poincaré section can have a general aspect like that of figure III.13, which we might compare to a ball of thread after a puppy plays with it for several hours: if indeed it continues to be a single thread (hence, with a determinate trajectory) it will be impossible in following the turns to predict whether in one centimeter it is going to fold back, go toward the center of the ball, or towards the outside, et cetera. Since it has sensitivity to the initial conditions, the slightest alteration of these will be represented by another complex tangle whose turnings have nothing to do with the first, although the volume it occupies be practically the same. To obtain the shape of a strange attractor we shall use the following procedure: we consider the flux in three dimensions of the trajectories in the space of the phases (figure III.14). This flux is subject to two opposite influences, since it should contract because there is an attractor, and it should expand from the sensitivity to the initial conditions. To better analyze the process, we shall separate the two effects, such that the contraction from the condition of being an attractor is exercised in the vertical direction, while that of expansion is exercised in the horizontal direction. Successive transverse Poincaré sections thereby exhibit a rectangle being deformed, contracting vertically and dilating horizontally. This process of contraction and expansion should continue to the extent that the flux of trajectories traverse the attractor. If one measures the expansion on every successive turn and compares it with the previous, when it grows exponentially with a positive factor (Lyapunov exponent) there is really expansion of the trajectory. But we have seen that the volume which the flux occupies in the space of the phases should remain constant for Hamiltonian systems, because the energy is conserved, or should diminish for dissipative systems, which lose energy through friction. Accordingly, the area occupied by the successive deformations of that rectangle cannot increase, can only maintain itself practically constant if there is little dissipation of energy or diminish, as the case may be. Furthermore, here there appears an additional limitation, which is that responsible for the appearance of the fractal form: since the variables which describe the system (impulses, positions, et cetera) cannot have any value imaginable, but those being necessarily restricted, the flux should confine itself to a region of the space of the phases and, thus, there is a limit to the expansion of the rectangle. In consequence, the only solution that permits the geometry to fulfill all these conditions ensuring at the same time that the flux remains in a restricted zone, is for the rectangle to fold into itself. By maintaining in a simultaneous manner the three operations, contraction, stretching and doubling back, the rectangle is progressively transformed into a horseshoe that, in turn, will flatten, expand, fold, giving birth to a structure of a double bobby pin, and so on successively. The area of the resultant figure at each deformation keeps diminishing, or can maintain itself practically constant in certain cases, yet it cannot grow. In each expansion the distance increases inxbetween points that previously were contiguous, which then separate exponentially, as corresponds to the condition of sensitivity to the initial conditions measured by Lyapunov's coefficient. The strange attractor was thus fabricated in a similar fashion to that which the baker uses for the bread dough, and it is not surprising that its structure would therefore resemble that of a puff pastry. If one examines the Poincaré section of the attractor in the previous figure, she sees a band structure that repeats endlessly (figure III.15). Every band is comprised of sub-bands and these, in turn, by others of similar structure; without it mattering what scale we use to examine the microstructure, its aspect will be similar. The property is called "self-similarity" and is one of the characteristics of fractal forms, those which as we have seen, have dimensions that are not whole numbers. In summary, for the behavior of a dynamic system to become chaotic it is enough that it have a minimum of three degrees of freedom and, hence, a space of the phases with a minimum of three dimensions. If in the corresponding attractor we find that the Poincaré section displays structures that are self-similar, we shall know that the attractor is strange, which implies sensitivity to the initial conditions and, accordingly, impossibility of predicting the behavior of the system beyond a certain time. These strange attractors, curves with no end and located in the space of the phases are, in general, geometric figures of rare beauty. It is not possible to calculate them exactly on the basis of mathematical equations, since none can be described in a precise manner, and the only route towards constructing and visualizing their aspect in the space of the phases is through the computer, which is how they were discovered. It is important to note that the strange attractors are figures which occupy only one zone in the space of the phases, and this permits diagnosing at a glance whether a system that appears as chaotic has underlying regularities or whether its behavior is purely random. If one can construct an attractor based on data, this tells us there is some non-linear mechanism operating upon the system and that, therefore, we deal with deterministic chaos. If, however, the represented behavior shows a dispersion of points throughout the entire space of the phases, this indicates phenomena which occur by chance, those that technically are called white noise.THE FRACTAL DIMENSIONSWhat exactly is a fractal, and how is one made? This form can be very irregular, or very interrupted or fractured, thus originating its name, which B. Mandelbrot derived from the Latin "fractus" (interrupted or irregular). Its basic characteristic is the concept of self- similarity, making the geometry of fractals an indispensable tool for the study of all those phenomena that exhibit the same structure no matter the magnification at which they are examined. The property appears with surprising frequency in nature: we think of a rushing torrent with eddies, which if we examine in detail contain smaller eddies, or in many plants, ferns, trees, that branch through successive processes before reaching the leaves, or in the system of blood circulation with its ever smaller ramifications until arriving at the capillaries. Fractals are, at the same time, very complex and particularly simple. They are complex in virtue of their infinite detail and their unique mathematical properties (there are no two identical fractals), nevertheless, are simple because they can be generated by the successive application of a simple iteration, and the introduction of random elements. Mathematicians have conceived a great variety of fractals, but the oldest is probably that of the mathematician G. Cantor, in 1883. He wished to surprise his colleagues with two apparently contradictory characteristics for a set of numbers falling between 0 and 1:a)that the set would have a zero size, or that is if it were represented by points along a line, on any portion of it the points which did not belong to the set would greatly exceed those that are part of it, and simultaneously,b)that the number of members of the set would be as innumerable as the set of all the real numbers also included between 0 and 1. Many mathematicians, including Cantor himself at first, did not think that such a monster could exist, yet finally he discovered it. His construction seems surprisingly simple. We begin with a segment of straight line and mark the extremities with 0 and 1. We erase the middle third, while maintaining their extremities 1/3 and 2/3. There remain two segments with a total of four extreme points, and for each one of these we erase anew the interior third, so that we are left with four segments with two extreme points each, or that is points which correspond to the 8 numbers: 0, 1/9, 2/9, 3/9, 6/9, 7/9, 8/9, and 1. If we continue with the same proceduread infinitum,we eventually arrive at each segment being formed by a single point. In figure III.16 the first five erasing operations are shown, but there is no way to draw the final result. Cantor's set has openings at whatever scale it is examined, and is composed solely of an infinity of isolated points, none of which is adjacent to another point of the set. If one wants to measure the length that remains, they will verify it is zero by being comprised of a sum of holes, now that the segments which were being removed sum to a total length of 1 after the infinite erasure operations. The mathematicians say then that Cantor's set has a measurement of dimension of zero. Yet actually there is a new mathematical concept that has more meaning for measuring the dimensions of fractals, and in accord with this it is demonstrated that the dimension of Cantor's set is the so-called "fractal dimensionD" log 2D= -------- = 0.6309 log 3 which turns out not to be a whole number but a fraction instead, and which suggests a geometric form intermediate between points (Euclidian dimension 0) and a curve (Euclidian dimension 1). It should be clarified that not all fractals have aDwhich is a fraction; this can be a whole number, yet its form is always interrupted or irregular. Just as occurs with the dimensions of the space of the phases, it should be borne in mind that a fractal dimension does not have the same meaning as that of the dimensions of our Euclidian space, but instead is the numerical expression that permits us to measure the degree of irregularity or of fragmentation in one of these figures. Thus, aDdimension between 1 and 2 means we are dealing with certain very irregular curved planes that almost become a plane, and the surfaces which resemble puff pastries, with numerous folds which fill part of a volume have aDbetween 2 and 3. There are fractals withD= 1 and withD= 2, which nowhere appear like a line or a plane, but always are irregular or interrupted. We now return to examining the strange attractor corresponding to the transformation of the horseshoe. This transformation was developed by the mathematician S. Smale, for application to the study of coupled electronic oscillators that were used in radar installations. Upon creating a sections-s*(see figure III.17) the set of Cantor is observed. The essential properties of this attractor appear in a great number of Poincaré sections for chaotic systems. Cantor's set is an example of an interrupted fractal. A fractal, a classic example of an irregular curve, is Koch's curve, proposed by the Swiss mathematician H. von Koch in 1904. To construct it, we start with a straight segment (figure III.18) of length 1 and in its middle third an equilateral triangle is constructed. The length of this line is now 4/3. If the operation is repeated we obtain the figure with a length (4/3)^{2}or 16/9, and infinite iterations arrives at a fractal form of infinite length and whose extremes are, nevertheless, separated by the same distance as the initial generating segment of length 1. Its fractal dimension isD= 1.26... Another variant is Koch's snowflake which is constructed by the same method beginning with the equilateral triangle (figure III.19) in which, if each side of the triangle measures 1/3, afterniterations it has a perimeter of total length (1/3)^{n}which becomes infinite whennreaches infinity, such that if one wanted to virtually trace all the turns in the curve with a pencil, she would never reach the end point, despite its enclosing an hexagonal figure of a perfectly limited area. These strange figures seem to be the products of the imagination of some mathematicians disconnected from everyday life. However, as has been seen so often in the history of knowledge, creative thought, which does not seek an immediate practical utility, turns out to be very fruitful. Thus, this new geometry conceived at the beginning of the 20th century today becomes indispensable for the dynamic that we are studying, and which has so many applications. Now we can understand why fractals and strange attractors are found so intimately connected. As we have seen, a strange attractor is traversed by the point representing the dynamic system, which advances along a curve of infinite complexity, that extends and at the same time folds and re-foldsad infinitum,and whose Poincaré section is formed by groups of points characterized by self-similarity. A strange attractor is, accordingly, a fractal curve. IV.Deterministic chaos in the heavensONE COULD IMAGINE Newton's amazement were he to read the article in the magazineScienceof July 3rd of 1992, where it is confirmed that the behavior of the solar system as a whole exhibits signs of chaos. Thanks to modern computers they have been able to confront the old problem of how stable that system is, already set forth during the era of the great mathematician and physicist Henri Poincaré. One characteristic of the planetary system, like the other systems which celestial mechanics studies, is that for the times under consideration, which may be up to billions of years, there is practically no dissipation of energy through friction, given that the celestial bodies move in an almost perfect vacuum, and the losses of energy through other effects, such as radiation, are not important either, such that one has here bodies that form conservative or Hamiltonian dynamic systems. For the solar system, the results are surprising: G. Sussman and J. Wisdom in the article mentioned describe the calculation through numerical integration of the evolution of the overall system for the next hundred million years, which required a month of specialized computer time for each run of the program. It turned out that the behavior of the nine planets beginning the next four million years reveals that the planetary system is in a chaotic state. For our own tranquility, this does not mean the chaos in the solar system is of such characteristics as to be annihilated within a short while, with planets colliding among themselves, or fleeing toward other galaxies, but instead that the orbits are unpredictable when calculated for times in the order of a hundred million years, and hence one can only anticipate that they will move in space within determinate zones. These do not overlap, so this does not presage collisions among planets, at least so far as calculated, which corresponds to the next hundred million revolutions of our planet around the Sun. They also discovered that the sub-system formed by Jupiter and its satellites is chaotic, and the same for the orbit of the planet Pluto. The researchers estimated the error with which they know the initial position of a planet in its orbit as only 0.00000001%. One might then expect that if two possible orbits are calculated that differ initially in their position by that percentage, and orbits stay on course while the mentioned time elapses, it would give a distance between them along the order of 10 m to 1 km. But they discovered that the distance between those two alternative orbits multiplied in the calculation by 3 every 4 million years and, therefore, became 9 times greater after 8 million years, 27 at 12 million years, and so on successively; at 100 million years, then, the position of the planet could differ by 100% yielding the chaotic region, also called "of resonance." In particular, for the Earth, a measurement error for the initial location in its orbit of only 15 meters makes it impossible to predict in what position in its orbit it will be after a hundred million years. In fact the calculus shows that these chaotic regions, called of resonance, are restricted to a portion of space, such that in the case of the solar system, there is no evidence for future collisions among planets. Yet the importance of this discovery is something more than an astronomical curiosity of interest only to specialists. Among the work's consequences, one is eminently practical: across the paths of the sky, there not only circulate "heavy vehicles" like the planets and their satellites, but also it is crossed by myriad fragments such as asteroids and comets. Through observation with telescopes and calculation based upon the equations of celestial mechanics, it is possible to foresee where and when the paths of those objects will cross. Thus it is known that in July of 1994 a fragment of considerable size will collide with Jupiter, falling onto the face that at that moment will not be visible from Earth. Also it is known that the asteroid 1989 AC, which apparently is in resonance with Jupiter, will cross the Earth's orbit in the year 2004, passing at a distance from our planet of 0.011 astronomical units, or that is 1,650,000 km. Fortunately the equations allow defining with precision the location of the planet for within a few revolutions around the Sun, so that humanity awaits this event with total tranquility. Yet as we have seen, the capacity to predict whether or not there will be collisions becomes lost in a type of fog to the degree that we advance into the future. How far away is the concept of the laws of celestial mechanics that permit predicting forever the movement of the heavenly bodies! But, does this not contradict the fact that we have known the ordered movement of the planets for millennia, and that the Babylonians could exactly predict an eclipse with years of anticipation? Did the Newtonian revolution perhaps not really consist of the discovery of the immutable laws that govern all the dynamic systems which appear in nature? Now we understand that that order is such for an observational time of a few dozen thousand turns of Earth around the Sun, but that it vanishes for time scales thousands of times greater.THE DISCOVERIES OF POINCARÉThe existence of this problem was also suspected by the scientists of the past century, but it was Henri Poincaré (figure IV.1) who approached it in its true magnitude. In 1887, king Óscar II of Sweden instituted a prize of 2,500 kronor for whoever might produce an answer to a fundamental question for astronomy: is the solar system stable, defining stability as the situation in which small changes in the planetary movements only yield small alterations in the system's behavior. In that age one could entertain, for example, the suspicion that the Earth would end by falling into the Sun. In trying to resolve this case, Poincaré opened a trail for treating problems of stability in complex dynamic systems, and even if he could not resolve the problem for the ten bodies that form the solar system, he received the prize anyway for his important contributions, among them the creation of topology. The astronomers can only know in an approximate manner the initial conditions of velocities and positions of celestial bodies, but this precision limited to a certain number of decimal places has not been an obstacle for many calculations, since they normally work with equations where small variations in the initial conditions yield proportionately small effects, yet, what happens when a system simulates situations of high sensitivity to the initial conditions?THREE OR MORE BODIES CAN GENERATE CHAOSFor a Hamiltonian system of only two bodies, like the Earth and the Moon or the Earth and the Sun, Newton's equations can be solved exactly, this problem is called integrable, and its solution corresponds to an elliptical orbit. But if a third body is added (for example, upon introducing the effect that Jupiter produces, with its mass which is a thousandth part that of the Sun, into the Earth-Sun system) one must use an approximating method, called perturbations. In this method, the perturbation on the order of thousandths produced by Jupiter's gravitational attraction upon the Earth-Sun system, adds up to the solution for the case of the two bodies, Earth and Sun, thus achieving a better approximation of reality. To this result they return to add in the effect of Jupiter's perturbation, but elevated to the square, or that is a factor which is a millionth, and thus successively, in a series of approximations where each one should be of smaller magnitude than the previous. It is hoped that this series formed by the sum of terms with decreasing values will converge, that is, that if for a sum of, for example, 1,000 terms one has a certain number as a result, and by adding the 1,001st term the sum grows by 1%, the sum the 1,002nd term produces is a new increase in the sum less than that 1%, for example 0.9%, and that each time another term is added, the new sum increases in decreasing fashion, tending towards a figure practically constant for a sufficiently large number of terms. This will permit claiming that the problem of three bodies is resolved: the location of each of them in its orbit will be given by those numbers calculated with the perturbations method. Yet upon applying it to the Sun-Earth-Moon system, Poincaré discovered, to his surprise, that some orbits behaved chaotically, which is to say that their position was impossible to predict through that calculation, which implies that here there is not linear behavior of the equations. The sum of approximations diverges--its result is an increasingly large number--with the effect that these infinitesimal perturbations became amplified and in certain limit situations could come to completely remove a planet from its orbit. But Poincaré could advance no further and come to resolve the actual case of the complete solar system, due to the difficulties involved in trying to calculate using ten bodies. As we have seen, this problem had to wait a century, to be confronted with the modern tools of the computer and the calculus of numerical integration. The great mathematician demonstrated, nevertheless, the possibility that a totally deterministic dynamic system, like that of the planets in orbit, might arrive at a state of chaos where its future behavior cannot be predicted. It is enough for that that a non-linear situation occurs, in which a tiny fluctuation is amplified on being reiterated a great number of times. Furthermore, Poincaré proved that chaos can even appear in relatively simple systems, as might be one formed by only three bodies, so that the structures described in the space of the phases form the complicated geometry to which we referred in chapter 2.PHENOMENA OF RESONANCEAn important contribution of Poincaré was to demonstrate that in this system the instabilities are due to phenomena of resonance. Resonance appears when there is a simple numerical relation between rotation periods of two bodies in the solar system. For example, Pluto has a rotation period around the Sun of 248.54 years and Neptune, of 164.79 years; the relationship between periods then is 3 to 2, which is indicated as orbit- orbit resonance 3:2. Yet there can also be resonances between the orbital period of an object and that of its own rotation around its axis (spin) as, for example, the Moon, with a 1:1 spin-orbit resonance, which is why it always displays the same face to the Earth. An effect that can result from resonance is indefiniteness in a planet's position in its orbit. Such is the case with Pluto, with its 3:2 orbit-orbit resonance with Neptune. The calculation has been made via a computer of its position, running the program twice, each with a very slightly different initial position. The two orbits so obtained locate Pluto on opposite sides with respect to the Sun after four hundred million years. Another possible effect of resonance is to produce a sudden increase in the eccentricity of the orbit of an asteroid that circles a planet. It has been shown that such an effect is responsible for the existence of empty zones or "Kirkwood gaps" in the belt of asteroids which exists between Mars and Jupiter. J. Wisdom, using a method of numerical integration for this chaotic dynamic, demonstrated that the effect of resonance is to produce violent changes in the eccentricity of the orbits, those that culminated by fragmenting the asteroids and launching them against Mars, and also onto the Earth: a dramatic example of the ubiquity of chaos in the solar system. Resonances are important in Hamiltonian systems and frequently suggest chaotic situations. To understand this, we can visualize the movement of the system in the space of the phases (see figure IV.2). Here as with the ideal pendulum, we can consider small energy values which, in a similar fashion to low-amplitude oscillations of the pendulum, correspond to movements called "libration," which are small oscillations around the resting equilibrium position; and thus like for higher values of energy, the pendulum performs rotation movements, and for a Keplerian system (a unique body in orbit around the Sun) the movement is called "circulation." Between both movements, of libration and of circulation, lies the separatrix, which for the case of a perturbation (resonance) can give rise to a chaotic zone. We analyze the corresponding Poincaré section, where a point is equivalent to a periodic orbit. If two pendulums are combined, one has a two- body dynamic system. Every concentric circle that surrounds the central point in the Poincaré section corresponds to an almost periodic movement, that is a combination of the two circular movements, each with its own period. In the space of the phases we have a torus, upon whose surface the almost periodic trajectory unrolls. For there to be an almost periodic movement it is indispensable that the relation between the rotation periods be an irrational number. In this case, when the curve remains on the torus, its intersection with the Poincaré section is contained in a closed curve (an invariant curve). If the smaller period increases, we have concentric curves of growing radius. But in those zones where the relationship of periods is a rational number, like, for instance, 3:2, 5:3, et cetera (resonance conditions) chaos can appear. This is evidenced in the Poincaré section because regions of instability appear, where the successive points which mark the chaotic trajectory on crossing that section, fill in the entire unstable region in a random manner. Within this chaotic zone of resonance islands of stability appear, in each one of which there is a structure analogous to that found in the center of the Poincaré section. Around each island a chaotic zone exists in which the movement is unstable: a trajectory can rotate in the curves inside of the islands (libration) and later in the curves of the central zone (circulation). The chaotic zone corresponds to the separatrixes in the case of the pendulum, and in this zone the movement is extremely unstable, and very sensitive to the initial conditions. The origin of the chaotic movement of Mercury, Venus, Earth, and Mars is the existence of resonances between the periods of precession (or that is the rotation in space of the planetary orbit) of these planets. Calculations show that our planet is far from the central stable zone of almost periodic movements, and is more in the chaotic zone near to a chaos island (previous figure), the reason that one cannot predict its position in a hundred million years.CHAOTIC BEHAVIOR IN THE GALAXIES?Up to here the method to study the evolution of a Hamiltonian system formed by three or more bodies has been described but, what would happen if it were to be applied to a system of an immense number of components such as a galaxy, which also is a Hamiltonian system, given that for the scales of times involved, of hundreds of millions of years, the loss of energy through radiation or friction is negligible. In 1962, the astronomer M. Hénon approached this problem of trying to calculate the movement of individual stars around the center of a galaxy. One of the differences from the solar system is that, here, the center of attraction is concentrated in the Sun, a sphere around which the planets rotate in flat orbits, while for a galaxy it can be modeled as a zone of attraction in the shape of a disk around which the stars rotate in orbits on three dimensions. Hénon computed with the method of Poincaré the intersection of the successive trajectories of a star with a plane, determining what changes were produced for different system energies. As would be expected for a Hamiltonian system, the Poincaré section of the tri-dimensional torus generated by the trajectories showed concentric curves, enclosed areas proportional to the energy (see figure IV.3). But a moment arrived, for the highest energies, in which the curves would break, disappearing to be replaced by points distributed apparently randomly in zones within which other curves appeared like islands in a blustery sea. In the corresponding states of the system it then becomes impossible to precisely establish the stellar orbits. In summary: thus we see how even celestial mechanics, that is considered the best example of the predictive capacity of the physical sciences, demonstrates the limits imposed by the existence of non-linear phenomena, such as appear as much in the solar system as in galaxies. These discoveries are not only important for celestial mechanics. There are other dynamic systems of great interest, like plasmas formed from ionized gases, or that is so hot that many of its atoms lose their electrons. The principal interest in these systems is due to investigations in how to use them to make nuclear fusion reactors that provides safe and cheap energy. Studying them as Hamiltonian systems with an immense number of components, now subject not to gravitational attraction but instead to electrical and magnetic fields, Poincaré sections have been discovered with islands of regularity, similar to Hénon's figures, which confirms they can pass into a state of chaos for certain critical values. V.Chaos, entropy and the demise of the universeAS A consequence of the Industrial Revolution the necessity emerged of understanding the phenomenon of the generation and taking advantage of heat in steam engines. This involved studying systems of gases and of liquids where only certain global properties can be measured, like temperature, pressure, volume, viscosity, which do not require knowledge of the positions and velocities of each one of their atoms.LAWS FOR HEATThe first great achievement of thermodynamics was thelaw of conservation of energy,which establishes that heat is one more of the forms in which energy presents itself, and that the total energy involved in a process can change in characteristics, passing for example from caloric to kinetic, electrical or chemical, yet is never lost. In a steam engine the caloric energy is transformed into kinetic energy (of movement) of a piston, and if two bars are rubbed, one obtains heat through friction. The concepts of heat and temperature were made more precise with the development of the kinetic theory of gases: the caloric energy of a body is the sum of its movement, or kinetic, energy of all the molecules that comprise it, while the temperature is proportional to the kinetic energy of the average molecule. Our image of a gas today (figure V.1) is one of an extremely elevated number of molecules that behave like diminutive, elastic billiard balls, which move in a straight line at great speeds with no preference for any particular direction in space, until colliding with others and rebounding vigorously. The molecules that form the air in normal conditions of pressure and temperature, for example, move at an average 1,800 kilometers per hour and effect five billion collisions every second. This agitated movement never ceases and increases upon raising the temperature. We should highlight here the notable fact that the word gas was invented by the Dutch doctor Van Helmont in the 17th century. Until then one spoke of "vapors" and "spirits," but he, with rare vision, considered that those were formed of invisible particles dispersed in every direction, this being the image of chaos, the word from which the name "gas" was derived. If the gas is contained in a receptacle, its molecules will make impact in their movement with its walls, globally exercising a pressure that will be greater the more kinetic energy its molecules have, or that is the hotter the gas may be. Thermodynamics is giving us, then, two notices: one "good" and another "bad." The "good" one is that energy is inexhaustible, that there will always be energy in the Universe; the "bad" is that energy appears in two varieties, of which only one is useful to us, and that furthermore this useful energy is diminishing and some day will disappear. What example do we have of useful energy capable of performing work in an isolated system? A simple case is that of a closed container divided into two compartments through a movable partition operated by a piston (see figure V.2). Suppose that we initially fill one compartment with a hot gas, and the other with the same gas at a lesser temperature. The energy of the gas is then usable, for the hot gas contains molecules with greater energy of movement than the cold gas and, hence, exerts greater pressure against the piston, pushing it. This organized state, where there is an appreciable difference between two regions of the system, we consider to be ordered, as distinguished from the case where the receptacle is uniformly full of gas at the same temperature, which, accordingly, pushes the piston both ways and cannot move it.THE ARROW OF TIMEIf indeed molecules of a gas have individually reversible movements--a cinematographic film of its trajectories projected from beginning to end or from end to beginning will be equally valid, for Newton's laws will be the same--together the gas undergoes an irreversible process: if initially there was a difference in temperature between both compartments, this will end by disappearing, because the molecules of the hotter gas will lose part of their energy by pushing the piston and, therefore, it temperature will lower. At the same time, the molecules of the cold gas collide with the piston that advances towards them, from which they receive more kinetic energy, so that their temperature increases. After a time, both compartments will contain a warm gas, which exerts equal pressure from both sides of the piston and that, in consequence, cannot yield work. Our experience tells us that a spontaneous appearance of a temperature difference has never been seen, when the gas heats back up in one compartment and cools down in the other. It is not that there is a prohibition that emerges from the laws of physics, because for that to occur it would suffice for the majority of the molecules, which individually are moving in every direction possible without preferring any, to move in a spontaneous manner by simple coincidence all in the same direction, those of one compartment towards the piston and those of the other moving away from the piston. The probability of something like this happening has been calculated and the result is that it would require waiting the entire age of the Universe and much more, in order to produce such a phenomenon. Accordingly, this return to the initial state equivalent to a reversibility in time, in practice is so improbable that we can consider it never to occur. In this way the notion of non-reversible processes in time was introduced into physics, an asymmetry that has been called the "arrow of time," for thermodynamic processes for isolated systems only occur in one direction: that in which over time non-utilizable energy grows. Even further, if we consider the whole Universe as an isolated thermodynamic system, wherein by definition it has no exterior from where matter or energy could arrive, one comes to the conclusion that it will end with "heat-death," when all the energy it contains degrades to non-utilizable caloric energy, until attaining a final state of equilibrium. This state to which it inexorably will arrive is identified with disorder or chaos, because it is visualized as the end of the Universe as we know it, of the organization that sustains all the ordered activity we perceive, and which, from the galaxies and solar systems to the beings that inhabit them, will disappear to become a homogeneous chaos of atoms moving blindly for the rest of eternity. Thus, the most general formulation of the second law of thermodynamics states that every isolated physical system, including the Universe, spontaneously increases its disorder. In 1865, the physicist Clausius introduced the concept of entropy to express in a precise mathematical function this tendency of evolution of thermodynamic systems. The entropy function increases in an isolated system in the same manner as disorder, and is considered as a measure of that disorder. Clausius reformulated, furthermore, the two laws of thermodynamics in the following manner: These concepts were analyzed through statistical mechanics by the physicist Ludwig Boltzmann, who demonstrated that the final state of an isolated system, when there is no change over time in the macroscopic properties, such as density, pressure, temperature, et cetera, is that of thermodynamic equilibrium, finding itself, in that case, the set of its molecules in a state of maximum entropy. In 1875 he proposed his definition of entropy as proportional to the number of possible distinct configurations of the system that are compatible with the macroscopic properties. Thus, for a gas isolated in a receptacle and in equilibrium, there are a certain number of configurations, understanding by configuration each one of the possible ways that the molecules can be distributed in the receptacle, and which produce identical macroscopic properties of pressure, temperature, density, et cetera as a result. All these configurations have the same probability of presenting themselves, so that it suffices to calculate their total number to obtain the system's entropy. The greater that number is, the greater the entropy which can be given, then, as a numerical quantity for a system in equilibrium, which constitutes a great improvement compared to simply using the concepts of "order" and "disorder," which are vague. It might be thought that the disorder or chaos which reigns in a gas in equilibrium is due to the great number of molecules. The existence of deterministic chaos in systems with few degrees of freedom demonstrates that the preceding is not absolutely obvious. It is associated more with the enormous difference between the volumes occupied by ordered and by chaotic states in the space of the phases. Let us recall that that is a mathematical space, whose number of dimensions depends upon the number of independent variables or degrees of freedom to be represented for the dynamic system. If the system is comprised ofnparticles not linked to one another, there will be 3npositional coordinates and 3nforce coordinates, so that even a few cubic centimeters of any gas implies an immense number of dimensions in the space of the phases. Thus, for example, for a cubic centimeter of air, which contains some 2.7 × 10^{19}molecules, the number of dimensions in the space of the phases is around 6 × 2.7 × 10^{19}or 160,000,000,000,000,000,000. Obviously it lacks meaning to try to visualize such a "space." In any event it can be schematized as if it were tri-dimensional. What is interesting here is that in a given instant of time, this portion of air has, for example, the following observable macroscopic properties: a temperature of 20 degrees centigrade, a normal pressure of 1,013 millibars, a volume of 1 cubic centimeter; it will be represented within a certain region we shall designate asHin the space of the phases (see chapter III) and we shall not have to worry about knowing the infinite detail of all the positions and individual forces of the component molecules. Those will be shown in each of the points within the region. To each point there corresponds an instant in time with a determinate configuration, a distribution in space of the individual molecules of the gas, each with its own velocity. A distinct point symbolizes another distribution of positions and velocities, but if both points are contained in the same region the macroscopic properties are the same. The space of the phases can thus be divided into a number of regions, to each one of which there corresponds a different set of observable macroscopic characteristics. The shape of a region, which is given by the number of points that comprise it, indicates to us the quantity of different possible distributions of the molecules. And how the gas evolves over time is represented globally by a trajectory that, starting from the point which indicates the distribution of the positions and velocities of the molecules for the beginning instant, traverses the space of the phases, and emigrates from that initial region into others to the extent the macroscopic characteristics change. In figure V.3 those regions are marked by their macroscopic properties of pressurePand temperatureT.In a thermodynamic process the sizes of the different regions can be very different. For an ideal gas, which is confined in a certain volume within an isolated box and proceeds to expand until occupying the entire volume, one could estimate the relative sizes of the two corresponding regions: the initial and the final equilibrium, designated byPand_{0,}T_{0}Pin figure V.3. Upon arriving at this last region, the gas will be in thermal equilibrium, so characterized because its molecules are distributed in a uniform manner in the volume of the box and move in all directions with a range of velocities known as "Maxwell's distribution." To be able to have an idea of the major factors in play let us look at a very simplified example: we assume there are a total of six molecules inside of a box divided into two parts. There are various possibilities for their distribution in the box: two in one half and four in the other, five in one half and one in the second, et cetera. The molecules are identical, in their movement no differences exist between the two halves, and how many possible configurations there are for each distribution can be easily calculated. Twenty different combinations result that give a uniform distribution of three molecules in each half, against six where the six are in one of the two halves. But to the degree that the quantity of molecules increases, the difference increases more rapidly: for 10 molecules it is 252, against 10; for 20 molecules it is 1.3 × 10_{θ}, T_{θ}^{13}versus 20; for 100 molecules 10^{29}configurations give uniform distributions versus only 100 of those where the 100 are concentrated in one half. Keep in mind now the number of molecules there are in a liter of any gas, and you will understand that, consequently, the quantity of possible configurations in the box with a uniform distribution is immense, by comparison with those where all are found in one half. Since to each possible distribution there corresponds a point in the space of the phases, it is deduced that the region corresponding to the gas in thermal equilibrium is, far and away, the larger in the space of the phases. Suppose now that we start from the situation in which all the air is accumulated in one portion of the box. Immediately it will commence to expand continually occupying more space until filling it; after a time it will arrive at thermal equilibrium, with a uniform distribution of temperature and pressure throughout the box.ENTROPY AND THE SPACE OF THE PHASESHow shall we represent this process in the space of the phases? The point begins from a very small region - the region representing the collection of the possible initial states for when the gas is accumulated in a corner of the box. When the gas begins to expand, the trajectory of the moving point will enter into a region in the space of the phases of greater size, and later to bigger and bigger regions while the gas keeps expanding, until it finally enters into the region of the greatest volume in the space of the phases--that which corresponds to thermal equilibrium--which, because of what we have seen, occupies practically the whole space of the phases (figure V.3). Hence, the possibility that the trajectory, after having left that tiny volume of space of the phases to enter the vast domain of the region of thermal equilibrium, will return to the initial region are practically nil. It is not that there is a prohibition dictated by the laws of nature: as we have seen, all states are possible, but the probability that the trajectory enters that region anew is much less than that of finding a needle by chance in a haystack the size of the planet. It is easy to see that the entire age of the Universe would not suffice for just that configuration with the gas gathered in a portion of the box to spontaneously re-occur. What never ceases being reassuring, of which physics assures us, if indeed it is not impossible, the probability that, suddenly, all the air in the house where we are will retreat to a corner, leaving us unable to breathe, is so slight that it would not occur even once in many hundreds of billions of years, and this in practice for us is equivalent to an impossibility. From all the preceding we deduce that, once a gas has reached the state of thermal equilibrium, it does not spontaneously leave it, for even though all the other states are also possible, that one is much more probable by an immensely greater factor. Furthermore, if we consider that the entropy of the system is a measure of the volume of the corresponding region of the space of the phases, we arrive at the conclusion that if the trajectory representing the gas starts initially in a very small region of volume, or that is with minimum entropy, as time passes it will move through regions of the space of the phases with growing volumes or entropies, until arriving at the maximum of entropy in thermal equilibrium. One arrive in this way at the formulation of the second law of thermodynamics utilizing the concept of the representation of the system in the space of the phases. The entropy grows because the states outside of equilibrium are much less probable than those in equilibrium (they occupy much smaller volumes in the space of the phases). Accordingly, once the system in its long-term evolution arrives in the space of the phases at that vast region, it is very improbable, although not impossible, that it may return to that which corresponds to states out of equilibrium. VI.The behavior of systems with large numbers of componentsWE SHALL EXAMINE what occurs with systems formed from an immense quantity of component molecules such as fluids (liquids, vapors, et cetera) when they quickly pass from an ordered behavior to chaotic confusion. No one is trying here to seek some exotic phenomenon, which only can be observed through the walls of sophisticated physics laboratories; in reality, as we shall see, they are totally normal in our everyday life. Let us observe the smoke that ascends through the air from a cigarette placed in an ashtray or a just extinguished candle (figure VI.1). It ascends vertically over several centimeters, in an ordered column, almost rectilinear (laminar flow) until ever more complex whirlwinds abruptly appear in a totally disordered cloud that ends by dispersing into the air. The millions of microscopic particles of hot soot that form the smoke all move with the same velocity during the laminar flow ascent, as if one dealt with a car track with columns of autos driven by conductors respectful of the transit laws. Through the phase of eddies, they flow into the disordered cloud, and in it the ash particles are animated by movements in all directions, such that if one wished to measure the velocity of the smoke at a point in this cloud, they would verify that it varies from moment to moment in a totally random form, as unpredictable as if the data of the measurements were obtained by playing roulette.TURBULENCEThis condition is called turbulent; in physics, a flow is called turbulent if the velocity of the fluid seems to vary randomly as much in time as in space. Turbulence appears with much frequency in the fluids that we find in nature: in air currents, water streams, atmospheric processes, oceanic currents, processes in the atmosphere of planets such as Jupiter, et cetera. As we know it also constitutes one of the great problems of modern technology (in the aeronautical industry, in the subtleties of oil or water transport) and despite the efforts exerted by many scientists, they are still far from dominating its fundamental principles. Nevertheless, some of the promising avenues pass by studying what occurs in a fluid when the transition from laminar order to the chaos of turbulence is produced. Until 1970, a theory was accepted by the physicist Lev Landau, who understood that in a system formed from so many particles, when it leaves laminar flow and the first eddies begin, this is equivalent to the beginning of the oscillations in a pendulum, that is, a periodic movement appears. The initial eddies very quickly divide into smaller eddies, which implies that instabilities appear in the fluid that cause it to oscillate with another different additional period, and this in turn produces smaller eddies with new oscillations. Thus, the turbulence will initially be a superimposition of three or four periodic movements. Once it is totally developed, the number of different oscillations will be infinite, as if there were infinite mutually interacting pendulums, and this will explain the chaotic behavior of the fluid's velocity. But in 1970, the mathematicians D. Ruelle and F. Takens proposed a different interpretation from that of Landau. They agree on the first stage of the laminar flow up to the first whirlwinds. During the stage of laminar flow, all the particles move at practically the same speed in the same direction, so that representation of the process in the space of the phases is very simple: a point, which attracts the trajectories of those that deviate by small perturbations (see figure VI.2a). Here the attractor point represents the constant velocity of the fluid. Yet in reality the velocity keeps increasing, due to the rising force of the hot smoke, and when it exceeds a certain value, it generates an abrupt change: before the smallest perturbation, that could be a very light current of air, the threads and layers of smoke are deflected forming curls that turn around themselves. There here appears a rhythmic movement, with a certain period, that can be represented in the space of the phases (see figure VI.2b) in the same manner as with the case of a pendulum, with a limit cycle. This motion is relatively stable, such that it constitutes an attractor. But on continuing to move the eddy soon suffers the effect of some other air current, that simultaneously causes it to oscillate in another direction, wherein we have a case similar to that of the system formed by two pendulums of different periods of chapter III. We saw that the corresponding representation in the space of the phases is the solenoid on the surface of a tri-dimensional torus, where we find ourselves with three independent variables or degrees of freedom. What happens from then on? It is at this point that the two theories separate. For Landau, a new perturbation introduces another oscillation, which will take us to a representation in a space of the phases of four dimensions, and so on successively until reaching chaos with a space of the phases with a high number of dimensions, corresponding to the elevated number of independent variables of the turbulent fluid. For Ruelle-Takens, the chaos is produced abruptly and long before: when the eddy is deflected to simultaneously have two oscillation frequencies. The foregoing is visualized not with a new figure in a space of the phases of four dimensions, but by radically altering the shape of the attractor that was on the torus (see figure III.13 above). It is as if the solenoid that defined that trajectory had exploded, producing a figure so extravagant that Ruelle named it, "strange attractor." From then on this name was that adopted by the scientists to denominate the attractors that display behavior which cannot be predicted in the long term. A Poincaré section of the strange attractor shows a fractal structure, such that the attractor which was bi-dimensional jumped to a dimension greater than 2 but less than 3. Experiments performed in laboratories have confirmed the existence of this strange attractor, which opens a path for formulating the laws of turbulence. Another important aspect of this discovery is it shows that a dynamic system formed from a great number of elements also can arrive at chaotic behavior starting from only three degrees of freedom. Until then it was thought that, as common sense seems to indicate, there can only be chaos when the system has a large number of independent variables.THERMAL CONVECTIONA notable example of a fluid giving rise to a phenomenon of a spontaneous order can be found in thermal convection, or that is the transport of heat caused by a hot fluid when displaced towards a colder zone. H. Benard performed the first studies in 1900. The French physicist discovered that if you heat a thin surface of a fluid (oil), underneath, this can spontaneously organize itself in convection cells of a characteristic size (see figure VI.3), similar to a bees' honeycomb. In each of Benard's cells hot fluid rises through the center and cold fluid descends around the borders. Today we know that this phenomenon of spontaneous organization us very diffused throughout nature: the surface of the Sun is covered with convection cells each one of which has a size on the order of a thousand kilometers, and in general the same cells can be seen in all fluids in which thermal convection produces movements, when they form surfaces much more wide than tall. This is also true for the circulation of air and of the oceanic currents, those that in large part determine the short- and medium-term climate. Assume that we place water or oil in a very large frying pan and uniformly heat it, so that the liquid in contact with the metal has the same temperature in all its parts; on heating it will conduct the heat upwards, where it will dissipate on the surface of the liquid. Thus we have thermal conduction of heat. But if we keep augmenting the heating temperature above a certain value another phenomenon suddenly appears, that of thermal convection from the formation of the cells. What occurs is that the lower, hotter layers of fluid expand, so that they have less density than the colder layers which are above and being lighter, ascend, being replaced by those colder volumes (figure VI.4). These heat as they near the bottom, and simultaneously those that ascend cool, which produces a circular movement. The complexity of these movements is notable: a section of the fluid shows how the sorts of circulation are coordinated in adjacent cells, alternately going clockwise, or counter-clockwise. The correlation of the movements due to processes of thermal convection is that much more noteworthy if one keeps in mind that the size of these Benard cells can reach several millimeters, while the range of action of the forces between molecules is on the order of ten millionth of a millimeter (10^{-7}mm). In each cell there are about 10^{21}molecules. It should be stressed that the experiment is perfectly reproducible, meaning that repeating the same heating conditions, one arrives, starting from the same temperature threshold, at the same formation of cells with similar geometry and rotation velocity. Yet what cannot be controlled is the rotation direction: it is known in advance that with the experience's conditions well controlled, in a certain zone of the surface of the fluid a cell will appear, but it is impossible to predict whether it will rotate in one direction or in the opposite. Repeated tests have demonstrated that the probability that the cell will rotate in one direction or the other is the same for both cases, which is to say that at any point in the fluid, its velocity can be directed as much in one direction as in the contrary. This property can be represented in a graphic like that of figure VI.5, where the velocityVis indicated on the vertical axis, withV^{+}for rotation velocities in one direction andV^{-}for the opposite; on the horizontal axis, the variableRrepresents the conditions of viscosity, density, thickness, and the temperature difference between the lower surface and the top of the cap. On beginning to heatRpasses from zero to growing values until, upon reaching the critical valueR_{c,}a bifurcation of the curve appears: two equal branchesV^{+}andV^{-}, it being impossible to determine in advance which of them the behavior of the dynamic system will traverse, that evidently responds to non-linear conditions, with sensitivity to the initial conditions. If we keep increasing the heating temperature, suddenly the cells are erased, and a random movement begins which is the initiation of turbulence. In 1987, the researchers M. Dubois, P. Atten and P. Bergé performed measurements in the fluid when the Benard cells appear and represented the corresponding thermal oscillations in the space of the phases. In this way they obtained a strange attractor whose Poincaré section has a dimension between 2.8 and 3.1, depending on the heating conditions, which indicates that the behavior of the system can be described with a minimum of four degrees of freedom.CONVECTION IN THE ATMOSPHERE AND METEOROLOGICAL PREDICTIONSAs we know, every day the meteorologists publish their predictions for the weather during the next days, based upon measurement data of the atmosphere, of the terrestrial surface and the oceans, and on satellite observations. Powerful computers process this data at centers in the United States and Europe, through models of the atmosphere like a dynamic system with more than a million independent variables. There are plans to continuously increase the power of those data processing centers and also the number and frequency of measurements throughout the planet. However, no serious meteorologist believes that in some near future they could affirm in a weather forecast something like "Friday the 26th, or that is within 12 days, in greater Buenos Aires the maximum and minimum temperatures will be 19 and 14 degrees, with a humidity of 67% and a 53% probability of rain." Actually, with all the modern instruments, the forecast begins to be uncertain after four or five days. There have been cases (as in England, 1987) when they could not foresee the appearance of a disastrous hurricane 24 hours before its arrival. It is estimated that in the future we will prognosticate up to 14 or 15 days in advance, and only beyond that will be the growing cloud of the uncertain. But we are not referring to an exact prediction for individual points on the planet; if, on the other hand, we consider the climate for an entire region, this indeed is predictable, with considerable more precision than "in June in the province of Buenos Aires the temperature will vary between 18 and 5 degrees." The atmosphere being a fluid, it is studied as a dynamic system formed from an immense number of components, subject to turbulences and convections. The Sun heats the surface of our planet, which in turn elevates the air temperature, and in a fashion similar to what we saw when we examine Benard cells, upon heating from below the air layer that forms the atmosphere, convections appear. The distance between the equator and the poles is about ten thousand kilometers, while the thickness of the atmosphere is some ten kilometers towards the troposphere. In this spherical layer of fluid in rotation, convection cells appear distributed the length of six rings that encircle the planet, three in the Northern hemisphere, and another three in the Southern, as is indicated in figure VI.6. The thick arrows indicate the general direction of the rotation, and the finer ones the additional circulation produced by the rotation of the planet (Coriolis effect). There are further currents of air at very high velocity that circulate in the stratosphere, and which decisively influence long-term weather. They are unstable, mobile cells, with changing limits and sizes. Their dynamic characteristics can vary in amounts that grow by double approximately every couple of days, and in turn act upon the currents of the stratosphere that modify the weather from the temperate regions to the polar. The behavior of the atmosphere thus comprises a non-linear process, with regeneration, highly sensitive to the initial conditions, such that it fulfills the conditions for being a chaotic system. That is to say it does not matter how complex the dynamic models may be, or how precise are measurement data from land, water and air, the laws of physics imposing a limit beyond which it is impossible to make meteorological predictions. Yet even if we cannot foresee more than four or five days for a specific geographical point, could we make a more long-term prediction, which would give the global tendency for a whole region that has relatively large dimensions? The meteorologist Edward Lorenz brought up this problem at the beginning of the decade of the Sixties. At the Massachusetts Institute of Technology, where he worked, the study of non-linear dynamics was developing, and he decided to make a simple atmospheric model of three non-linear equations with only three independent variables in place of the immense quantity there could be for this system. He represented the evolution of the weather in a space of the phases of three dimensions and proved, to his surprise, that the corresponding trajectories gave rise to an attractor of a most curious shape, with two similar links and butterfly wings (see figure VI.7). He had discovered the first strange attractor. In the space of the phases, each one of the wings of the attractor represents a possible state of the atmosphere, for instance a rainy time in the left wing, with dry and stable weather in that of the right. If the initial conditions are those that mark point 1 on the left, the evolution will follow the trajectory which remains in the same wing: the weather will be rainy. But a small perturbation, which changes the initial conditions leading to the atmosphere with the situation represented by point 2, it takes us on trajectory 2, which evolves towards the right wing, and the weather then would be dry and stable. The Lorenz attractor has a fractal structure, with the dimensionD= 2.06. Lorenz published this discovery in 1963, it being the first example of a model calculated with only three independent variable where behavior unpredictable in the long term appeared.THE BUTTERFLY EFFECTLorenz coined the famous expression "butterfly effect" as an example of this extreme sensitivity to initial conditions: when the wings of a butterfly in the Amazon beat today, it could produce an extremely tiny alteration in the state of the atmosphere, which if amplified by doubling every couple of days, will proceed to diverge continually more with respect to what it would have been without the butterfly, such that several weeks later a cyclone might appear in the Caribbean, which, without the insect in question having existed, never would have emerged. The Lorenz model with but three variables only qualitatively describes the chaotic form in which weather evolves, yet it does so in a manner very similar to models with many independent variables. In actuality, models for meteorological prediction have around a million degrees of freedom, and this allows making general forecasts for global regions anticipating up to one month. VII.Far-from-equilibrium systemsTHE EMERGENCE of thermodynamics was a challenge for the physicists of the 19th century, whose schooling was based upon the concepts of Laplace, Lagrange, Hamilton, and their disciples. They came together to study heat equations as universal as those of Newton, and that referred to the global behavior of matter. As we have seen, to apply Newton's laws to dynamic systems, it is necessary to define the positions and velocities of each one of their component elements. But a gas, a liquid or a solid have an immense quantity of components (along the order of 10^{23}molecules in a cubic centimeter in a gas) which is why it was ignored how their global behavior might be established starting from calculation of the movements of its molecules. With thermodynamics a new focus emerges, where the heat equations, which also are universal, use collective, macroscopic properties as parameters, such as pressure and temperature, and do not seem to require detailed knowledge of what occurs with each of the participant molecules. Thermodynamics appears, then, as a potentially valuable tool for studying processes of global change in dynamic systems formed from a very large quantity of components and, effectively, achieved great advances during the 19th century, especially when, as we have seen in the preceding chapter, it applied the concepts of mechanical statistics to link the pressure and the temperature with the average effects from the movement of a very large number of molecules. In this way the evolution toward thermodynamic equilibrium of isolated structures that appear in physics and chemistry, whose most well-known example is that of the ideal gases, received a satisfactory interpretation via the law of entropy. The second law of thermodynamics establishes that, in general, isolated structures end by decomposing, remaining reduced to a disorganized movement of their elements: the clouds of smoke dissipate, the hot zones and the cold zones in a object fade until arriving at a uniform temperature. The amount of entropy in the Universe--the random, or disordered--can only increase, until reaching the maximum, says the second law. Yet, curiously, in the same era when this law was enunciated the theory of natural selection to explain the evolution of species appeared. Through that theory, Charles Darwin tried to take account of the fact that living organisms constitute continually more organized structures: beginning with the bacteria today we arrive at mammals and at humans. Does this process of growing organization contradict the second law of thermodynamics? Actually, the existence of living beings is not a challenge to this law, which applies to isolated systems.OPEN SYSTEMS CAN ORGANIZE THEMSELVESA living system is open: a person absorbs energy and matter from external sources (the heat of the Sun, the air, meats, vegetables, sources which in turn are structured and, hence, are of low entropy) and expels her waste products, which are of high entropy as a result of the decomposition of organized matter, into other open systems in her environment. While an organism is alive, it remains far from the thermodynamic equilibrium towards which isolated systems tend. We know very well that if someone is completely isolated from the outside environment, in a very short while the inexorable second law is obeyed bringing mortality. Over a considerable time, many scientists thought that the fundamental laws of physics only permitted deducing that systems must reach thermodynamic equilibrium and that, accordingly, the process of biological evolution, with growing complexity of living organisms, was a rare exception. But neither Boltzmann nor Darwin had been able to consider the existence of the phenomena of spontaneous formation of structures in matter, a property that has begun to be studied in these last decades and that appears with surprising frequency in nature. Thus, systems as simple as a layer of liquid or a mix of chemical products can, under certain conditions, exhibit phenomena of coherent spontaneous behavior. An essential condition for this to occur is that it concerns open systems, kept far from thermodynamic equilibrium through sources of energy. It is truly extraordinary that enormous sets of particles, subject only to the blind forces of nature, should be, nevertheless, capable of organizing themselves in configurations of coherent activity. One of the groups that has promoted the study of this type of process--developing in this way the branch of thermodynamics of far-from-equilibrium systems--has been that of Ilya Prigogine and his collaborators, which is why Prigogine received the Nobel prize in chemistry in 1977. These processes also help us understand the mechanisms that lead to oscillations in certain chemical reactions, knowledge which has great transcendence for industrial catalytic processes, without mentioning the importance of this type of reactions for biochemical process in living beings.VERY STRANGE CHEMICAL REACTIONSWe have seen that phenomena of thermal convection can take a liquid or gas that starts initially in a homogeneous state and develop and structure it with time, giving rise to regular forms. An indispensable condition for cells to appear like those of Benard is that the system be open, that is to say have an external source of energy, and that it be remote from equilibrium. This formation of cells through convection is very striking, yet it seems a humble phenomenon if compared with the spectacular effects of chemical oscillations, which have begun to be studied methodically beginning in 1980. In 1958, the Russian biochemist Boris Belousov created a mixture of certain chemical products that usually form a colorless liquid until they react, then turning a pale yellow color. Belousov had mixed the ingredients without worrying what proportions he used, and surprised he observed that the solution changed its color periodically, passing at regular intervals from colorless to pale yellow and back to being colorless, which meant that the reaction retreated and returned to advance as if it could not decide what direction to take. Poor Belousov tried to publish his discovery in the scientific journals, but it was refused. The arbiters who evaluated the work considered the only possible explanation for that phenomenon was an inefficient mix of the reactants, for the laws of thermodynamics opposed the existence of such oscillations. Belousov died without having succeeded in his research being recognized. The skepticism of the chemists of the day should not surprise us too much. In effect, in a typical chemical reaction between two reactants A and B, their molecules are moving at random, and if an A and another B collide they can combine to form a molecule C and another D, which are the so-called products of reaction. This is symbolized as: A + B → C + D The reactants A and B continue progressively disappearing to the extent that the proportion of products C and D increase. Nevertheless, in an isolated system it is observed that the reactants A and B never run out completely and that, over time, the four, A, B, C, and D, co-exist maintaining a fixed proportion of each in the solution. This proportion will no longer vary and we then say that the system is in chemical equilibrium, which corresponds to the equilibrium attained by systems according to the second law of thermodynamics. Yet one can also transform this system into an open one, for example by continually adding more reactants A and B, or retiring part of the C or D that are produced. Here too it can be proved that through an adequate combination of the entrance and exit flows, the proportions keep varying progressively over time until becoming fixed, although with different values from those in the isolated system, a result which, once again, would be expected in accord with the laws of thermodynamics. But what Belousov claimed implied that those combinations instead of progressively growing over time until reaching a stable equilibrium, could retreat towards the initial state, which is equivalent to contradicting the second principle of thermodynamics, and furthermore could do so repeatedly, oscillating in one direction and the other. Phenomena that are periodic or of oscillation are ubiquitous in physics, astronomy and biology, but the chemists thought that reactions were immune from this class of behavior. It is not usually expected that concentrations of the intermediate products of a reaction reach a certain level, then fall to a lower one, then rise and fall repeatedly until at some point stable products resistant to further changes result. In any case, at some laboratories in Moscow they kept treating Belousov's "recipe" as a curiosity of chemistry, and at the beginning of the decade of the Sixties, Anatol Zhabotinskii returned to the theme for his doctoral dissertation. He performed systematic research changing the reactants so as to obtain colors with better contrast. In the reaction now called BZ (Belouysov-Zhabotinskii) the oscillation is manifested by a regular change between red and blue. It is obtained by dissolving in water at a certain temperature certain proportions of sulfuric acid, malonic acid, potassium bromate, and salts of cerium and iron. The resulting oscillations between red and blue have a period of almost a minute and can last for several hours. Oscillations can also be produced in space: if one pours some of that solution on a disk forming a layer of low thickness, some lovely figures appear, with concentric circles or blue spirals upon a red background, which rotate in one direction or the other and keep changing with time. Figure VII.1 shows a successions of photographs of these ripples. Actually the BZ reaction and other similar ones are studied in what is called a continuous flow reactor, continually introducing the reactants and removing the excess products to maintain a constant volume. The reaction can so be rigorously controlled and maintained like an open chemical system, far from equilibrium.FEEDBACK AND CATALYSISToday all the stages of this type of reaction have been studied in detail, establishing the corresponding equations, and with them the oscillations have been simulated by computer. It is now known that the conditions for these to appear are that, in addition to dealing with a far-from-equilibrium system, there must be a feedback, which is that some of the products that appear at one stage of the process be capable of influence upon its own speed of formation. It chemistry this is called "autocatalysis." A catalyzer influences the speed with which the substances present react chemically and remains unchanged during this process. In autocatalysis, if the substance being produced acts on its own velocity of production by augmenting it, increasing its concentration and, at the same time, producing a higher quantity of that substance, one has positive feedback, which responds to a non- linear equation. Varying the entrance flow of reactants in the continuous flow reactor, the chemical system can pass into either of two different stable states, each one of which having its own oscillation period, which is to say that bifurcations appear for the possible evolution of the system, which recall those found in the dynamic of fluid flow. For some ranges of flow in the reactor a more complex behavior appears, with a mixture of various oscillation frequencies of diverse amplitudes. The current explanation is that, as opposed to a "normal" chemical reaction, where the reactants and products continue to be uniformly distributed in the solution, here a very tiny heterogeneity or difference at one point is incremented through the effect of autocatalysis, such that in a region a specific chemical substance can dominate the reaction while in the neighboring one, its concentration exhausts itself. Thus complex oscillations are activated in the reaction system. A chemical oscillator, for example, can initially be of a uniform red color and, as the reaction continues, white spots appear that become concentric blue rings, that are destroyed by colliding with each other. One has then the situation of sensitivity to the initial conditions that, as we have seen, characterizes non-linear dynamic systems and, in fact, representing the evolution of these chemical systems in the space of the phases reveals strange attractors. It must be remembered that catalyzers are employed in many important processes, like those linked to petroleum for example, such that detailed comprehension of the dynamic of chemical reactions which are important for industry can have large economic consequences. Moreover, these chemical clocks suggest the natural rhythms that appear in living organisms. If one keeps in mind that feedback also appears in biochemical reactions, produced through the catalytic effect of enzymes, and that living beings are far-from-equilibrium systems, it can be hoped that this new focus might help understanding the behavior of many biological mechanisms. VIII.How do we define complexity?UNTIL now we have used the term complexity for a state where many different factors interact among themselves. Yet we must give greater precision to this concept, since the complexity of a system should not be confused with that of a system that is merely complicated. In reality one should talk of complex behavior of a system since, as we have seen, a dynamic system can be very simple but under certain conditions exhibit unexpected behavior of very complex characteristic which we call chaotic.BETWEEN ORDER AND CHAOS: COMPLEXITYA quartz crystal is an ordered system, with its atoms vibrating around positions ultimately fixed in the crystalline net; a virus has characteristics of order in its structure, similar to an organic crystal, yet when it infects a cell, it rapidly commences to replicate genes like a live organism: it is complex; the movement of the molecules of a gas in thermal equilibrium is truly chaotic. Complexity thus covers a vast territory that is between order and chaos. There is not actual agreement concerning the meaning of "complexity." We all know that human beings have great complexity, and that the fishes have enough; mushrooms are somewhat complex, and a virus, less so. The galaxies are complex, but in a different sense. We all agree about that, but could we come to agreement on using a more scientific definition, which would specify quantitatively how much more complexity a plant has than a bacteria? In the physical sciences, a law is not a law if it cannot be expressed in mathematical language; no scientist would enthuse over measures of complexity that sounded like this: "a fish is considerably more complex than a virus, but a little less than a mammal." It might seem that this difficulty in discovering a good definition of complexity disappears if we ignore living organisms and apply it only to dynamic systems formed from inert matter, where it is easier to know whether or not they are complicated. Yet it is not exactly so either. According to the conditions of the system, these can have order or complexity or even chaos, whether they treat of only three components or with an enormous number of them. It is the territory of complexity, which is situated between order and chaos, that comprises the new challenge for science. In any case, it is difficult to be able to predict that that search for a precise definition of complexity will end with the discovery of a unique magnitude that would give us a number to use in the physics equations, such as occurs with velocity, with pressure or with mass. In the final analysis, the complexity of a galaxy implies a relation among its component stars qualitatively different from that which exists between the cells that form a sheep. It does not seem that this difference could be reduced to a unique magnitude similar to the temperature. Possibly the best solution to the problem is that proposed by computation specialists. Actually the majority of scientists dedicated to trying to define complexity pertain to that area, whose tradition is to see practically everything as reducible to information and, accordingly, quantifiable in terms of bits and bytes.HOW TO MEASURE COMPLEXITYIf this viewpoint is adopted, then we could measure complexity as a function of, for example, the time that a computer requires to execute a program which simulates a complex physical process. In principle there is no limit to the types of processes that can be simulated with a program, such that this would be a good starting point for the definition we seek. Thereby, complex phenomena include not only clouds of symbols (numbers, computer programs, words) but also physical processes and living organisms. Such a focus originated with the publication in 1948 of a work by Claude Shannon, of Bell Telephones, on the mathematical theory of communication, which led to a search for an equivalent to the second law of thermodynamics but for information. As a consequence of that presentation, scientists have become accustomed to viewing physical entities--bicycles or oil flows--and asking, how much information is required to describe this system? Moreover the phenomenal advance in the capacity of computers has propelled the computational focus for processes in physical systems, combining observations of such systems with the construction of models for the computer. A process is simulated this way, so as to obtain results that traditionally only were achieved by modifying the physical conditions of the system and measuring the effect. This becomes especially useful for systems like those studied in cosmology, where one cannot alter for instance the structure of a planet, or in the social sciences, where we cannot change the economic conditions of a nation in order to determine which variables are important. Given that many processes in nature are expressible through models which permit simulating their evolution with a computer, one could try to measure the complexity of a system by the difficulty of representing it in an algorithm, that is, with a computer program. How is that difficulty measured? Different formulas have been proposed, like that of measuring the minimum time necessary for a machine to execute the program, or alternatively measure the minimum memory capacity the computer should have to run that program. But since these magnitudes depend on the type of machine utilized, it is necessary to refer them to some ideal computer which acts as normalizing governor. This abstract machine was conceived by the English mathematician Alan Turing in 1936. It can be considered as a mechanical artifact with a printer, through which the memory passes, consisting of a paper ribbon as long as necessary and upon which are marked a sequence of spaces; each one of the spaces can be blank or marked with a line, which is equivalent to the binary notation 0 and 1 respectively (see figure VIII.1). The machine can perform one of four operations each time it uses the memory, namely when it traverses the ribbon's spaces and reads them: move a space forwards, or move a space backwards, or erase a line, or print a line. Since the ribbon is as long as needed, it is understood that the machine has an unlimited storage capacity for data, also that no fixed limit of time is given for completing its operations. The machine begins to operate when the program is introduced, and continues until finishing by printing the result. In this manner it is capable of performing any program expressed in binary code, or that is, in a mathematical language formed from two unique signs 0 and 1. What characterizes Turing's universal machine is that, given an adequate input program, it can simulate the behavior of any other digital computer, even the most complex. Obviously it is much slower, so that no one has tried to construct one, despite its simplicity. We now have the instrument for precisely measuring the quantity of information that we send in a message: the fundamental unit of information is the bit, which is defined as the smallest information capable of indicating a selection between two equally possible things. In the binary notation, a bit equals a digit, and can be 0 or 1. Thus, for example, the number 29 is written 11101 in the binary code and, hence, contains five bits of information. We have here, then, the sought for normalization, that permits measuring the complexity independently of the type of computer employed.THERE ARE NUMBERS THAT ARE COMPUTABLE AND OTHERS WHICH ARE NOTWe shall attempt now to define complexity with greater precision. It seems sensible to try to do so beginning with abstract objects such as numbers are, since the relations among them are, obviously, quantitative. Turing distinguished two classes of numbers: those computable and those non-computable. The computable are those for which an algorithm, or computer program, exists, that when run on the machine delivers us the number, no matter how big it may be, which includes even being infinite. To clarify this concept let us suppose that we have a computer connected via satellite with another used by a friend who lives in Japan, to whom we need to transmit certain numbers. As we know that the more seconds of transmission our message requires, the greater will be the bill which the company presents, it befits us to compress it to the maximum. For that it is advisable to see whether it has some characteristic that assists towards that end. Suppose that we desire to send a number like, for example: 1234567891011121314151617181920212223242526272829303132 Upon examination we find that its digits are formed by writing the first 32 whole numbers in order. Its structure follows a perfectly determinate law, since I know that the digit 7 is followed by 8, with 9 coming next, et cetera. Accordingly, it will be sufficient to transmit a message with the instructions for constructing that number. The corresponding algorithm to transmit is a very short program, which basically executes the following instruction: "Print the first 32 whole numbers in order" Our friend's computer will generate the number, which can be as large as we want changing very little the length of the program (for example, "Print the first 1,000 whole numbers in order"). By contrast, a non-computable number is one whose only possible algorithm turns out to be the same number written into the program. We shall easily understand this if we wish to transmit to our friend in Japan a number that we have constructed by rolling a die many times and recording the figures from 1 to 6 which we successively obtain. Thus after throwing the die 25 times we obtained this number: 3546221356431142652142663 In this instance, since each digit was generated at random, it has no relation whatever with that which follows or precedes it, such that the only algorithm that can reproduce the number is the program which simply copies it: "Print 3546221356431142652142663" In the same fashion, if I wanted to transmit the number obtained when I kept throwing the die until completing a thousand throws, I have no other possibility than to revert to the program: "Print 3456221356431142642142663..." Here the three dots signify the other 975 digits from 1 to 6. I should resign myself to the fact that the message with the program will continue to be as large as the number. In summary, for a computable number it is possible to write a relatively short computer program that will calculate it, though the number may be infinitely large. However, for a number generated at random, not being computable, the program that calculates it must contain the same number and will be at least as large as it is. Therefore, the complexity of a number could be measured by the minimum amount of instructions for a Turing machine program--that is, its minimum length--capable of reproducing the number. In consequence, a number generated at random corresponds to a high complexity--of chaos--while a computable number is ordered, having low complexity. In the middle will be the complicated numbers, generated through a combination of chance and order. Yet, what relation can there be between these valid considerations for mathematical entities and the systems of objects in nature whose complexity we intend to estimate? A pathway for seeing the relation is given in the fact that the binary code which the computer uses to express and number can also express any other information. The 28 letters of the alphabet can be made to correspond with 26 different sequences of ones and zeros, along with the punctuation signs, et cetera; this permits us, for example, to consider the complexity of the construction of words in a language. Something ordered would be a string of letters likeaaaaa,because it can be concisely written in a program as "5 ×a." However, a random sequence of letters, likedcflksivgdhglkjthlakijgueernsedgmk,is chaotic, not having a program that can print it which is more concise than the sequence itself. Between both extremes is the complexity that appears in the way letters are grouped to form words and sentences. This method also permits coding the information that corresponds to physical objects. Thus, an ordered object, such as a chunk of crystal formed of carbon atoms (a diamond) can be formulated in an algorithm which specifies the quantity of carbon atoms that for it as, for example, "10^{27}× carbon." In the same manner, plants, persons and bacteria will be similar, under this focus, to words, sentences and paragraphs: mixtures of order and randomness. A related approach is found in the algorithmic theory of complexity formulated by Andrei Kolmogorov, Gregory Chaitin and Ray Solomonov. This theory considers computation of the physical magnitudeQspecified as a sequence of digitsS,and establishes thatS,and henceQ,is random if the minimum computing program required to obtainSis the program where all that is done is to copyS:"PrintS" Computational complexity can also be defined by this method, as the amount of time a computer needs to resolve a certain problem and which therefore measures its difficulty. It is shown that there are two basic types of dynamic systems whose movements can be computed: ordered systems and chaotic systems. The first, as is the case with the harmonic oscillator, have orbits that require a quantity of input information, or that is a sequence of ones and zeros, whose length is relatively short, yet the corresponding computational process produces much much greater output information as a result. The input information is of short length because the initially nearby trajectories of the system separate slowly with the passage of time, so that it is not necessary to know all the digits of initial information to enable prediction through computation of future states. From the viewpoint of computational complexity, the time required to compute a trajectory is proportional to the logarithm of the system's time. Thus, if one calculates the trajectory for the timet= 15 seconds, she would require a computation time of 1.17, and fort= 150,000 seconds, a time of 5.18, always much less than that which the system requires to traverse that trajectory. Instead, for chaotic systems, the quantity of input information is the same as the output, since the initially close trajectories diverge exponentially over time, and this has as a consequence that every second of time the system loses a decimal digit of precision in the output information. To maintain the degree of exactitude in the output information it is necessary, then, to compensate for that loss by adding an additional digit of precision to the input information and, hence, if one wants to integrate a chaotic trajectory with exactitude, they must introduce as much information as what they extract. If we examine this from the point of view of computational complexity instead of algorithmic complexity, we see that the time the computer requires to calculate a chaotic trajectory is proportional to what that system requires to traverse it, in such a manner that the computer needs as much time to calculate as the chaotic system itself to execute the process, and accordingly, it is not possible to make a prediction. All these methods allow quantifying the complexity of the extreme processes of order and chaos, yet they are not very satisfactory for complexity that is between both. In effect, the complexity of a dynamic system like that of a living organism is, according to the foregoing definitions, less than that of a gas in thermodynamic equilibrium, since in the latter entropy has reached its maximum and the movement of its atoms is totally random. This would seem to imply that complexity and randomness are equivalent, which takes us nowhere if the intention is to understand the complexity of systems that display organization, such as are abundantly found in nature. The idea of complexity does not coincide with that of entropy, for there are systems which can have equal order, or equal disorder, and differ in complexity. C. Bennet, of IBM Research, and other researchers have proposed a different focus, that offers as a result a distinction by which the complexity of a fish will be greater than that of a crystal or than that of a gas. The evolution of life on Earth shows us ever more organized organisms, which is to say that something we can call complexity appears to be increasing, despite that the quantity of order in the Universe is diminishing. Bennet's proposal is to measure that process of organization, especially for the self-organized systems that appear in nature. What characterizes them is that, in addition to being complex, begin initially from simple systems (a layer of fluid which is heated, a solution of chemical substances, a fertilized egg). The idea of organization or complexity of a system will thus be strictly linked to the process that can go from that simple initial system to the totally developed complex system. Complexity will then be measured by the time it will take a computer to simulate the entire development of that system until arriving at its definitive organization, and counting the number of "logical steps" for the length of the process. It has for an input algorithm the basic rules of its growth, which include, for example, genetic data, if we are dealing with human beings. Thus, considering natural selection as a theory of the origin of the human species, the number of logical steps will be the estimated number of times that our genetic material has been modified starting from when it was contained in an initial bacteria. The complexity will be measured then by the time a computer will require to simulate that evolution passing through all those logical steps. Yet none of the mentioned definitions of complexity that is between order and chaos have been accepted by the scientific community as totally satisfactory, there are new proposals and the debate still continues. Moreover, the lack of definition is not an impediment because the theme of complexity in itself is registering important advances, in a process that resembles that which accompanied comprehension of the nature of heat and its relation to energy. The advances in the science of heat were realized from the beginning of the 18th century, but by the middle of the 19th century it could be defined with precision basing itself on the kinetic theory. It can be hoped that the advances achieved will permit defining complexity in a much shorter time. IX.Applications in biology and economicsTHERE EXISTS a mathematical equation especially adequate for examining these properties common to various systems and given its importance we turn to describing it.THE LOGISTIC EQUATIONIt concerns the logistic equation, which is a simple equation, very fruitful in a number of applications in many fields of study of complex systems: ecologists, biologists, economists, et cetera. It is an equation that operates upon one number and transforms it into another upon which it operates, and so on repeatedly, in an iterative process. Its special characteristics are only in evidence when the number of iterations is large, so that to apply it at least a pocket calculator is required, though a computer is the most adequate tool. The logistic equation produces two opposite effects from any number:1)it is incremented, producing another greater number which, in turn, is returned to be incremented by the equation and so on repeatedly;2)it keeps reducing those resulting numbers as they grow, such that we have here a process with a controlled feedback. What would happen when the equation had operated a good number of times? Common sense would tell us there should finally result some intermediate number, neither too large nor too small. Yet here the grand surprise comes: this can be totally mistaken, so mistaken that we can also run into some unsuspected behavior, of a chaotic nature.APPLICATIONS IN BIOLOGYThe log equation was proposed in 1845 by the sociologist and mathematician Pierre Verhulst; its surprising properties were made manifest by the physicist and biologist Robert May in the decade of 1970, when he applied it to the study of the population dynamic of plants or animals. In such populations there is feedback in every natural cycle due to reproduction controlled by the negative effect of predators or by the increasing scarcity of food, which thus impedes those populations from growing explosively. The equation allows calculation, starting from the characteristics of the population at a given moment, of how these will vary over time. We wish to know how the number of individuals in a population will vary annually, from what we know, that in the initial year there are 1,000 and that it increases at a constant rhythm of 10% per year. If control by predators or by the availability of nourishment did not exist, we could make the following table:And so on successively. This procedure can be expressed mathematically with the equation:

YearNumber of individualsTotal0 1 2 3 1,000 1,000 + 100 1,100 + 110 1,210 + 121 1,000 1,100 1,210 1,331 X+ 1 =_{t}K × X_{t}Wheretindicates the quantity of years elapsed starting from the initial year which ist= 0, andXis the variable that symbolizes the number of individuals. Thus,Xis that quantity in the year_{t}t,andX+ 1 in the following year. The parameter_{t}Kindicates the annual rate of increase ofX;in the previous example it isK= 1.1. Accordingly, what this expression tells us is that if we know what the population is in the yeart,it is enough to multiply the corresponding number by the rate of increaseKto determine the population there will be in the following year. To facilitate the calculations we work with a normalizedX,meaning that it can only vary between 0 and 1. Thus,X= 1 corresponds to the maximum possible for that population, or that is 100%, and_{t}X= 0.5 at 50%, and it does not matter whether we are considering 15,950 rabbits or 12 million trees; all that interests us is to calculate the annual variation of the population in relation to the previous or subsequent values. Returning to the example, evidently that increment of 10% annually will lead us to an impossible situation: if we begin with 1,000 rabbits in the year zero, 200 years later there will be 19 billion rabbits covering the surface of the planet; it is then required to add to the equation a term that reflects the real situation, where food is not going to reach in feeding the growing population, and furthermore there will continually be more foxes and other predators which feed on rabbits. The logistic equation is, definitively, the following:_{t}XHere two opposite actions are executed; the more factor_{t + 1}= K × X_{t}(1 - X_{t})Xgrows, the more the factor (1 -X) diminishes, reducing the final result. If, for example,_{t}Kis 1.1, for a smallXof 0.1, with the first equation it resulted:_{t}X= 1.1 × 0.1 = 0.11 But since the reducing factor is here (1 - 0.1) = 0.9 the new equation is:_{t + 1}X= 0.11 × 0.9 = 0.099 Or, the reducing factor affects the result very little. When_{t + 1}Xgrows greatly, for instance to 0.8 at a maximum, this factor becomes 0.2, and one then has instead:_{t}X= 1.1 × 0.8 = 0.88 The result corrected by the reducing factor of 0.2:_{t + 1}X= 1.1 × 0.8 × 0.2 = 0.176 This diminishes the population by a fifth. We have then a mathematical expression that permits unambiguously calculating the value of_{t + 1}Xin other words we are dealing with dynamic systems where the future depends in a deterministic manner upon the past, without uncertainties. All the information about the system will be found contained in the logistic equation, and applying it we can know how it will vary over time. Thus, if_{t;}Xis the initial value of the variable whose evolution over time we wish to know, it will be_{0}Xwhen it arrives at time_{1}t= 1: (a)XAnd for_{1}= K × X_{0}× (1 - X_{0})Xwe will have: (b)_{2}XWhere we can substitute_{2}= K × X_{1}× (1 - X_{1})Xfor equation (a) resulting in the following:_{1}XAnd similarly for_{2}= K(K × X_{0}(1 - X_{0})(1 - K × X_{0}(1 - X_{0})))t= 3:XAs can be seen, the equation progressively and rapidly converts to an ever more complicated formula; if we were to attempt to know the future for_{3}= K(K(KX_{0}(1 - X_{0})(1 - KX_{0}(1 - X_{0}))) × (1 - (KX_{0}(1 - X_{0})(1 - KX_{0}(1 - X_{0})))))t= 20 (20 years, or 20 generations, or whatever we use as a measure of time) we would need some 300 pages for nothing more than expressingX_{20,}and to arrive at a timet= 50, which is not unrealistically far in the future, it would not fit in the size of a library. In the computer era, it is meaningless to follow this path; for if there is something that those artifacts know how to do, it is to reiterate at great velocity the same operation as often as desired, and additionally they do so without mistakes or boredom. It follows that, instead of pursuing the classical method of writing an equation that will be valid for every possiblet, Xand_{0}K,so as to later introduce into it the numbers corresponding to the case of interest, what we shall do is give the computer the initial dataX_{0}andK,the number of iterations that we need, and the instructions for it to execute the basic calculation which as we have seen is very elemental, consisting of performing a subtraction and two multiplications. With the program thus prepared any personal computer permits us to obtain values such asX_{100,000}or any other in a very short while, and also to provide experiences like seeing step by step howX_{t}varies on incrementingt,and what happens whenKis changed. Whoever desires to feel the power of iterative calculation can do so without the need of using a computer: a simple pocket calculator is sufficient, and a good dose of patience. If both are available, I propose to perform a very simple iterative calculation, which consists in raising a number by squaring, the result returning to be raised by squaring again, and so on successively up to ten times in all. We shall input the number "0.9999" into the calculator, and we press the "x^{2}" key, or if this is unavailable, that of multiplication, "×" followed by "=". We repeat the same operation ten times. A calculator of eight digits gives as the result:X_{0}= 0.9999X_{10}= 0.9026637 We now see what happens when the initial conditions is quickly varied. What will happen if we use asX_{0}a value that differs by only 0.1% from the preceding? For:X_{0}= 0.9999 It now is:X_{10}= 0.3589714 Thereby the final result is 40% of the preceding, having changed the initial value by only 0.1%! We have here an example of sensitivity to the initial conditions which, as we have seen, is a necessary condition for the appearance of chaotic phenomena and of complexity in the dynamic systems.HISTORIES OF FISHES AND CRUSTACEANSIn these iterative mathematical operations other unexpected characteristics appear, which manifest themselves beginning with some dozens of cycles of calculation, and that we turn to describe. If someone wishes to do them themselves, it is preferable to use a computer, the less you enjoy doing long accounts with a calculator. The program with instructions for the computer has a general form in the BASIC language: INPUT K X = 0.6 FOR n = 1 to 100 X = K * X * (1 - X) PRINT X NEXT n STOP WhereKis the rate of increase, and we have set the number of iterations at 100 and the initialXto 0.6. To demonstrate the surprising behavior of the logistic equation, we shall begin by comparing it with the results of the mathematical studies performed by Vito Volterra in the decade of 1920 to explain periodic fluctuations in the fish population of the Mediterranean. Let us consider a population of crustaceans, and of fishes that feed on them, and we shall assume that the crustaceans have a low rate of reproduction,K= 1.01, and that their initial population isX_{0}= 0.6. Making the calculation results in the population decreasing every year such that, at the end of a time, the colony will disappear:But what happens when

tXtX0 1 0.6000 0.2424 5 10 0.1147 0.0732 Kis greater, for exampleK= 2?The population remains stabilized at 0.5. If the reproduction rate increases to 2.7, the equation shows an annual fluctuation, varying between 0.61 and 0.64 due to the opposition between growth and the action of the fishes, but finally, after about 15 cycles, it stabilizes at approximately

tXtX0 1 2 0.6000 0.4800 0.4992 3 4 5 0.5000 0.5000 0.5000 X= 0.63, a value which therefore makes it an attractor for this behavior. Yet it is starting with a rate of reproductionKgreater than 3.0 that something new happens: the system strongly fluctuates at the outset, and finally the population of the colony oscillates between two stable values, indicating that the attractor has bifurcated into two. Thus, forK= 3.3:The stable values here are 0.47943 and 0.82360, and each of them repeats every two years. Upon representing on one graph the annual variation in the population of crustaceans (see figure IX.1) we see that once the fluctuations have stablized, if in one year they increase to 0.83360, this turns out to be a veritable feast for the fishes, which, in turn, increase so much that they lower the quantity of crustaceans to the lower level of 0.47943 in the following season. This will diminish the fish population due to the scarcity of food, allowing the crustaceans in the following year to abound, and so on cyclically. The process described can also be represented as was done by Volterra, in a space of the phases, which permits visualizing the cyclical behavior of the fishes-crustaceans system in figure IX.2, where the vertical axis indicates to us the crustacean population and the horizontal that of the fishes. If we start the colony in

tXtX0 1 2 3 4 5 0.60000 0.79200 0.54363 0.81872 0.48978 0.82466 14 15 97 98 99 100 0.47941 0.82360 0.47943 0.82360 0.47943 0.82360 AwithX_{0}= 0.6 crustaceans will be increasing along curve 1, which simultaneously marks the increase in predators until reaching levelB.Starting from there the population of fishes dominates, taking the system around curve 2 with diminishment of crustaceans, which drags the fish populations toward a minimum, and this repeats itself cyclically as a period, in this case of two seasons. The logistic equation adjusts itself very well to other natural cycles like those of insects and bacteria, which were studied by R. May in the decade of 1970. Let us continue exploring the equation through the computer, which fulfills in this age at the end of the 20th century a fundamental role as a tool of science, analogous to that of the microscope and of the telescope in previous centuries. If we increaseKto over 3.45, the two values that repeat every two years resume being unstable, and each one bifurcates to produce a population which oscillates around four different values that are repeated every four years, so that the period has doubled. ForK= 3.45:The stable values are found around 0.42820, 0.45290, 0.84410, and 0.85320. With

tXtX0 72 73 74 75 0.6 0.42688 0.84405 0.45412 0.85524 76 97 98 99 100 0.42713 0.84470 0.45258 0.85474 0.42835 K= 3.56 a new instability is produced, with bifurcations that produce eight fixed values, with periods of double the preceding, and this occurs again withK= 3.596, giving 16 values; later more and more bifurcations keep appearing, until finally withK= 3.56999 it arrives at a chaotic state, with myriad values ofXfor the colony which oscillate in an unpredictable way between 1 and 0.ROUTES TO CHAOSWe see in this example that a dynamic system can begin from an ordered state and under certain conditions evolve towards a chaotic state. Actually there exist various routes among regular, ordered behavior and the chaotic, unpredictable state. over which dynamic systems can move; also at some stages in the routes complex behaviors appear that can give them surprising properties. Such routes can move in both senses: dynamic ordered systems can pass into completely complex behavior or reach a chaotic one, as occurs with the pendulums and other oscillators, or pass from chaos to an organized complexity, like in chemical clocks or the cells of Benard. We have seen that for these behaviors to appear in systems with a quantity of components that can be small or very large, an indispensable requirement is that the number of variables which intervent in the system's dynamics be limited. If the number of independent variables, or degrees of freedom, is less than three, the behaviors analyzed do not appear; moreover, if that number is very large, it will be impossible to distinguish by these methods whether their evolution is due to the phenomena we are studying of obeys fundamentally random factors. The routes between order and chaos can be classified into three principal types according to the different modalities under which those transitions are produced:1) Almost periodicity.The system is represented in the space of the phases with an almost periodic attractor inscribed on a torus and the transition transforms it into a strange attractor.2) Sub-harmonic cascades.The system displays oscillations of a certain periodT,and at the start of the transition a bifurcation is produced, others appearing with a double period 2T,later 8Tand so on successively in a cascade. This is observed among others in phenomena of thermal convection (chapter VI) and in the BZ reaction (chapter VII).3) Intermittencies.The system sporadically produces fluctuations of great amplitude. These transitions usually appear in hydrodyamic processes and also in the oscillations of electronic circuits, where it is manifested as a low-frequency noise which appears occasionally. With a basis in the foregoing classification, in the case of the logistic equation, the variation of theKparameter implies a route to chaos through duplication of the periods, that is, the subharmonic cascade. As we see, the role ofKis to define the complexity of the behavior, and it becomes convenient to visualize its effect on the system's evolution through a graphic like that of figure IX.3, where we shall represent population on the vertical axis and on the horizontal the value ofK.WHERE CHAOS APPEARSThe figure shows the panorama that R. May found with his model based upon the logistic equation, whereKvaried with variation in the provision of food. Again, the figure represents the population on the vertical axis and the parameterKon the horizontal axis. He thus discovered the successive bifurcations that indicate the increase in the oscillations in the insect population, and which later, after various bifurcations, entered into a chaotic zone indicated in the figure, where the population in the model fluctuates erratically, such as occurs in reality with that of the insects. We have depicted in the frames of figure IX.3 some of these situations, which respond to a specific value ofK.Thus for that ofK= 1.01, the populationPthat initially was at 0.6, diminishes over timeT,represented on the horizontal axis, until reaching zero: it is extinguished. The frame forK= 2 indicates to us that for that value, after some cycles with fluctuations the population becomes stable. ForK= 3.3, the population oscillates between the two values 0.47943 and 0.82360, taking one of those values in one cycle, and the other in the following cycle. These two values are further represented in the principal graphic: if atK= 3.3 we trace a vertical line, it will cut the two branches of the bifurcated curve, at the values mentioned forP.In the same manner, forK= 3.4 there are four values that repeat in successive cycles, values which correspond on the principal curve to the intersections of a vertical line with the four branches. WithK= 3.7, there are so many distinct values ofPthat it is not possible to discover any relation between one cycle and the next; with the principal curve, a vertical line will pass through thousands of intersections, marked as points on the vertical stripes. It is in the zone of deterministic chaos. Finally, forK= 4, one has a chaos with infinite possible values for each cycle. As for today, one of the questions posed by the ecologists is that of determining whether the behavior predicted by this model occurs in the real populations. Opinions among biologists concerning these theories are divided: from those who think that the unpredictability inherent in deterministic chaos is a very important factor to explain the evolution of species, to those who consider real populations not to have chaotic dynamics because they will be extinguished, and that the researchers in laboratories and through simulation with computers are too removed from cases as they appear in nature. This is not a simple task, because in ecological systems it turns out to be very difficult to separate out the multiple environmental factors, nor can parameters like the reproduction rate be varied in order to see how the effect varies. In any case experiments have been made on animal or vegetable populations isolated in the laboratory, where indeed some conditions can be altered, varying, for instance, the surrounding temperature to accelerate the metabolism of horse flies or of protozoa. The studies of these experimental populations effectively revealed the transitions that correspond to the first bifurcations, but without a duplication of periods as clear and vivid as that which appears in physical systems. What unequivocally appears is the transition to chaos. The other powerful method that is being used for the study of the dynamic of populations is computer simulation, in which the researchers create a model that generates series of data representing the size of a population, over the course of many generations. This is an imaginary world, where the researcher specifies at will the factors which govern the system, and later analyzes them using methods that are applied to real living systems. This mathematical method permits constructing a space of the phases with as many dimensions as there are independent variables in play, and seeking strange attractors, which, if they appear, are evidence that you are before a dynamic non-linear, deterministis system. One of the difficulties that presents itself is that each species usually interacts with many others, and for each a variable or dimension should be added, so that one must work with multidimensional spaces of the phases, where it is very easy to confuse statistical fluctuations with the presence of attractors. The method has been applied to instances like that of the measles epidemics in New York appearing over a period of 40 years, and this revealed the presence of a tri-dimensional attractor in a space of four dimensions.RHYTHMS IN LIVING ORGANISMSMathematical methods have also been applied to the search for indications of deterministic chaos in the dynamic of individual organisms, that is, in physiologically and neurobiologically rhythmic processes, which permits them being studied as sets of mutually influencing oscillators that have feedback cycles with a non-linear dynamic. A highly studied case is that of cardiac rhythms, which can throw light on arrhythmias and upon the interpreation of electrocardiograms before and after a cardiac attack. The biophysicist R. Cohen performed a computer simulation of cardiac rhythms and proved that in the prelude to a heart attack there appears a bifurcation of the heartbeat period. This can be explained considering that the electrical pulses that force the muscular fibers of the ventricles to contract and thus pump the blood, work as an oscillator system with a regular period. If for some pathology there appears an alteration in the oscillation period of the electrical pulses in a group of fibers, these two different oscillations combined can enter into the situation we examined for two coupled pendulums, with a cascade of bifurcations of period 2T,4T...until paralyzing the heart. It is clear that here we deal with an effect which appears in a computer simulation and that it is not easy to obtain experimental data for a heart attack, for doctors and patients are sufficiently occupied with trying to overcome the crisis. So it is that the closest to the real phenomenon is to do measurements in a laboratory. In 1980, the physiologist L. Glass initiated a series of investigations with heart cells from chicken embryos. These, in a cultivating medium, keep spontaneously pulsing at a rhythm of from 60 to 120 beats per minute, and thus are a natural oscillator. If a microelectrode is introduced into that mass of cells periodic electrical shocks can be applied to it, so that now one has an oscillating system with two coupled rhythms, one intrinsic and the other forced. This latter can vary, so we observe how the cardiac heartbeat varies with periods 2Tappearing, later 4T,and also totally irregular, chaotic situations which suggest fibrillation. Another application appears in neurophysiology, where the dynamic complexity has been analyzed of electroencephalograms (EEG) which register the cerebral activity of human beings performing diverse tasks such as, for example, counting backwards from 700 by sevens. The fractal dimension was computed of the attractor that appears when the set of oscillations registered by the EEG is represented in the space of the phases, discovering that it moves above the basic value of 2.3 to a value of around 2.9 when the subject is making the effort of counting backwards. The conclusion to which he arrived is that forms of the EEG with a higher fractal dimension, or that is being more complex, correspond to a more alert mental state. Yet all these physiological and neurological studies expose the same weak flank as that with animal studies: how difficult it is to apply these theories to the actual cases that appear in nature, and which are much more complex than the models that can be established through a computer. The difficulties in applying these methods of study are recognized by R. May, who nevertheless has promoted them for twenty years based on the fact that populations as well as biological processes are governed by non-linear mechanisms, and that hence they should display chaotic behaviors in addition to the stable cycles. It still seems premature to pronounce upon the final result. Evidently it is necessary to keep perfecting the application techniques of a method that, in the last analysis, is very recent, before being able to determine to what degree these theories on the transitions between order and chaos comprise an explanation of the dynamic of living being.ECONOMIC CYCLESThe field of the social sciences as well is working intensely on the application of these novel concepts. In particular it becomes attractive for the economists to try to document their capacity of prediction in the financial markets. In the final analysis, the financial markets and the nations' economists are dynamic systems with mechanisms of feedback and self- regulation. We know that if one raises the price of a product in an excessive manner, demand diminishes and the price should fall. And so here also one can try to create a model based upon the logistic equation:P1_{t + 1}= A × P_{t}(- PWhere we call_{t})Pa price on the day, month or moment_{t}t, Athe rate of increase, andPthe price in the following period. As we have seen previously, the properties of this mathematical equation indicate that for an_{t + 1}Aless than 3, after a certain time the pricePwill be stable, and that ifAis more than 3, the price would have periodic fluctuations, with double cycles which oscillate between two values at the beginning, and later pass to four, and so on successively until for an even greaterA,the value ofPcan have unusual behaviors, such that never repeat, that is to say are chaotic. In such a situation, a graphic of the values ofPcan be easily confused with a series of numbers generated at random. In recent years various research groups confronted this problem of applying the focus used by physics on non-linear dynamic systems to the field of the economy. The most well known is that begun in the Santa Fe Institute, in New Mexico, where in 1987 three winners of the Nobel prize (the economist Kenneth Arrow and the physicists Murray Gellman and Philip Anderson) brought together economists, physicists, biologists, and specialists in computation to begin a program of research on the economy considered as a complex dynamic system. They have managed to construct computer models of the economy where the investors are, in turn, computer programs capable of recognizing rules of variation in the prices and of acting in consequence, and which furthermore learn from experience. The result of this computer simulation begins to approach that which is observed in the actual economies. Yet the same as what we saw for the applications of an equation of the same type to ecological systems and to live organisms, economic reality is even much more complex, affected by constant changes in society and the price of every product is linked to that of many others. Thus, it is possible that the effects of deterministic chaos that appear in those mathematical models may not be present in a real economy, and that it may be equally valid to keep describing the economy through linear processes with data which have an important component of "noise," or that is, of chance fluctuations. In order to decide which focus is more adequate, the real cases must be used, which are very difficult to apply because they require very large sets of data, and those that count are in general scarce, in addition to having a strong noise component. But in recent years new techniques have been invented for statistical analysis that are capable of distinguishing between fluctuations due to chance and those that might exhibit regularity is they are adequately examined. The basic idea is that complex systems can reveal their structure if the data are translated to a space of the phases with an appropriate number of dimensions. These methods have been applied with encouraging results to studies concerning populations and other biological systems, on turbulent fluids, and now also for these economic models. Among the most powerful are the algorithms that use a single magnitude (population, temperature, price, or whatever which will yield a sufficiently large series of data) taken at regular intervals of time. In its most elementary form, this temporal series is a list of codes which represent the experimental data, for example, a population of bacterias measured every hour: 0.453; 0.671; 0.632; 0.661; 0.702; 0.799; 0.530; 0.501... We wish to determine whether an attractor appears in a space of the phases that has three dimensions at a minimum. From everything that we have seen here, that then requires representing the measurements of three independent variables, yet the data available to us only gives us information about one variable. Is there a solution for this situation? The mathematicians D. Ruelle and N. Packard discovered a trick that resolves the problem, and F. Takens succeeded in demonstrating that this artifice if mathematically correct. The method consists in fabricating another two series with the same measurement values, but displaced over time. Calling the successive values obtainedX_{1,}X_{2,}X_{3...}the pseudoseries are constructed:YOr let us say that, instead of a single temporal series, we have the original_{1}= X_{2,}Y_{2}= X_{3,}Y_{3}= X_{4...}Z_{1}= X_{3,}Z_{2}= X_{4,}Z_{3}= X_{5...}Xand two copies displaced one and two places in time,YandZ:Thereby for the timet= 1 we represent in the tri-dimensional space of the phases the pointX= 0.453,Y= 0.671,Z= 0.632; fort= 2, the pointX= 0.671,Y= 0.632,Z= 0.661, and so on successively. If the result is that the point remain grouped in a limited region forming an attractor, this is an indicator there is a regularity, some periodic behavior. If, however, they remain spaced more or less uniformly throughout the entire space of the phases, we might be dealing with a random phenomenon, or it may be that a space with more dimensions is required for an attractor to appear. It is clear that the higher the number of necessary dimensions, the more doubtful becomes the presence of an attractor. One of the difficulties in the application of these methods lies in that, even for dynamic systems like turbulent fluids it is not easy to discern which are the pertinent variables. This is even truer for systems which are studied in fields far from physics, where it is even more difficult to determine the dimension of the space of the phases. X.Order, chaos and complexity in mathematicsWE HAVE seen some of the surprising qualities of the logistic equation discovered when R. May applied it to the field of biology and that caused many mathematicians to begin to study it in detail performing large calculations with their computers. It establishes a new method of investigating the laws of mathematics that differs from the approaches where all knowledge is attained through logical steps within an abstract framework and not through experiments. Thanks to computers, mathematical experimentation is done today and this is particularly so in that it is applied to the dynamic by the role played in it by iterative processes.THE LOGISTIC EQUATION REVEALS A VERY COMPLEX WORLDA simple equation like the logistic reveals aspects unsuspected until various millions of calculations are made that indicate a very intimate relation between the dynamic of complex systems and the structure of number systems. From the moment when the iterations for this type of equation reveal a sensitivity to the initial values it is shown that not only the systems of the physical world can be deterministic and unpredictable at the same time, but also this is so in those systems of the mathematical world. That impossibility of predicting a result in the long term will only be eliminated were one to measure the physical world with infinite precision, or if in the mathematical world one were to calculate utilizing all the infinite digits of which the majority of numbers are formed. Let us now examine the surprising results that derive from making calculations for the logistic equation for aKgreater than 3.5 where complex behavior appears (see figure X.1). We see that the periodTbifurcates in an ever more rapid cascade upon increasingK,and that the distance between the corresponding values ofXare becoming ever less. In a zone beginning withK= 3.57 we have for each growing value ofKrespectively 1,024, 2,048, 4,096... periods, and it would take a microscope to distinguish the structure formed by the points. Another notable property is that of "renormalization," discovered by M. Feigenbaum in the decade of 1970: for sufficiently high bifurcations, for instance with 2,048 periods, if those bifurcate again to 4,096 they repeat the structure of the 2,048, if and when represented by a very precise increase in the scales; the increase inKshould be 4.66920166... and the increase in X should be 2.502908... These numbers of Feigenbaum's are universal, like π, because the same cascading structure of bifurcations and the same Feigenbaum numbers also appear in other equations if and when they are continuous functions ofXand having only one maximum. Cascading bifurcations and the Feigenbaum numbers appear not only in the calculations done by mathematicians with a computer, but also when many behaviors in nature are mathematically represented. In a more detailed graphic we depict the variation ofXwith respect toK(see figure X.2). We shall see that beyond the bifurcations, for aKbetween 3.55 and 4 vertical stripes appear covered with spots that correspond to the myriad places where the system could be for a value ofK,so that if the computer generates a value corresponding to a pointX= 0.57739, for example, the subsequent will be at some place between the upper and lower extremes of the dark stripe, and that is all we can anticipate until the computer reveals to us the result of the new calculation. Further thanX= 4 the results show no structure whatsoever betweenX= 0 andX= 1. It is such that the points are, in this zone, indistinguishable from those that could have been marked by randomly generating data, despite that we are using a perfectly deterministic mathematical equation. We cannot attribute this impossibility of predicting to the existence of unknown factors, because the equation has no ambiguity. The reason is that as we have seen, we can neither measure nor represent a present state with infinite precision, and even if this limitation is not important in many situations, we now perceive that non-linear behaviors abound, where insignificant causes that we thought would not affect a dynamic system produce such disproportionately large effects that they change its behavior in an unpredictable fashion. We can consider the region betweenK= 3.55 andK= 4 to correspond to complexity and, starting fromK= 4, to chaos with points which fill the entire space.WINDOWS LIKE ISLANDS IN A CHAOTIC SEAIn the zone of complexity more obscure curves can be observed, which demonstrate a greater concentration of points or, that is, greater probability of the system's presence, and also something especially significant: observe that there are white vertical bands interspersed, crossed by a few lines. These receive the name of windows, and indicate the return to regular cyclical behavior, where two or three different periods appear. Such intervals with low oscillation frequencies are termed intermittencies. If one examines a window of three periods like that ofK= 3.83 with higher definition, that is to say, with more computer calculations, and represented in the central zone where we have enclosed boxAin figure 10.2, at an amplified scale, we encounter another great surprise: the period repeats its bifurcation, and thus initiates a cascade with successive duplications of 6, 12, 24, 48... repeating the original scheme in miniature, with windows which in turn demonstrate repeated cascading bifurcations (boxB) where there are other windows with their own cascades and so on as many times as it is desired to explore this unknown universe. Here we have the same phenomenon of self-similarity that we had found in fractal figures.MANDELBROT SETS AND COMPLEX PLANESLet us return then to the concept of generating fractals through the process of iteration. When the mathematician B. Mandelbrot reviewed the work of Gaston Julia, a disciple of H. Poincaré, on iterative calculations with complex numbers, he decided that they suggested a path for constructing fractal figures starting from mathematical equations. It is necessary here to review the concept of a complex number. As we know there are various classes of numbers, calling the most basic and elementary the natural numbers: 0, 1, 2, 3... These can be added or multiplied to produce other natural numbers. But if we want to get to a systematic method, it is convenient to introduce the negative numbers -1, -2, -3... Up to here we deal only with whole numbers, and this can create problems when we try to divide one whole by another, so that we need fractions or rational numbers 1/2, -1/2, 1/3, -1/3, 3/2, -3/2... and the irrational numbers, that require an infinite number of digits to be expressed, such as π, and2 . All those different types of numbers form the system of real numbers. But there remains one limitation: if you want to obtain the square root of a number, it can only be done with those that are positive; thus, the root of 9 is 3, which multiplied by itself regenerates the 9, but the root of -9 does not exist, since -3 × -3 is also 9. The square roots of negative numbers were denominated, consequently, "imaginary" numbers to differentiate them from "real" ones, yet since it turns out to be supremely convenient to use them for calculations, the difficulty was resolved by "inventing" a square root calledifor the negative number -1. It then results thati^{2}= -1 and the mathematical nature ofidoes not worry us. Like any negative number,-acan be written as -1 ×a,and its square root,can be written as -a-1 ×or that is -aiWith this subterfuge these numbers can thus be considered as real as the "real," so achieving major versatility for the numerical system. Finally, if we combine "real" numbers with "imaginaries" a number results that we call complex, not because it is complicated, but because it has various components. A complex number is of the form:a .Z = X × iYWhereXandYare ordinary real numbers, andiis the square root of -1. Similarly to the real numbers, the complex numbers can be visualized representing them through a graphic in a system of coordinates, but in a complex plane, where the vertical axis isiY,and the horizontal isX.The pointZis located at the intersection of lines parallel to the two axes that share the value ofXand that ofiY(figure X.3). Complex numbers have their own arithmetic, algebra and analysis, and on graphing the result of performing mathematical operations on them, aspects of great importance and strange beauty are discovered. Mandelbrot started from a very simple iterative operation: assign an initial value to the complex variableZ,raise it to the square and add a constant number that we shall callC:Z = ZThe^{2}× CZso obtained repeats being raised to the square and added to the sameC,and so on successively, an operation that requires a computer to discover its properties, for those are placed in relief when the graphical result in the complex plane appears on the screen (figure X.4). The heart-shaped figure, called a "Mandelbrot set," represents all the values ofZthat will not come to have an infinite value even if infinite iterations were done. Surrounding the figure are all the points that do tend to an infinite value when one performs ever more iterations. The frontier between both zones has properties of self-similarity and a complexity so great that it can only be captured when the computer is instructed to shade them in different tones of gray, or better yet in distinct colors, according to the different velocities with whichZgrows upon iteration. There is a classic book for admiring these figures,The Beauty of Fractals,by H. Peitgen and P. Richter, and computer programs like Fractint, which have divulged the beauty of such forms, where each Mandelbrot set has very small similar figures attached which seen with the amplification of the computer as if it were a microscope repeat and repeat in a self-similar form. If we keep increasing the amplification, other characteristic forms appear, which resemble dendrites, whirlpools, seahorse tails, that in turn repeat ever more microscopic details, an endless process, since it could be iterated an infinite number of times and, additionally, varying the constantCcan produce and infinite variety of forms (see figure X.5). Even the most experienced observer is filled with admiration before this process, through which beginning from a simple equation we arrive through iteration at a result with structures so extraordinarily complex that they recall the results of repeating the logistic equation. In reality there are some similar aspects between both mathematical expressions, which lead one to consider that the Mandelbrot equation is the version for complex variables of the logistic equation, where now, instead of the variableX, Zis applied, and in place of the growth factorKthe constantC.Thus by graphically representing the variation ofXupon increasingKthe surprising forms of figure X.2 were obtained, while the representation ofZwhile continually varyingCproduces the Mandelbrot figures. In both cases one has an "ordered" region with stable values, which is the region forKless than 3 in the logistic equation, corresponding to the interior of the Mandelbrot set. The zone of "chaos," for aKgreater than 4 in the first case can be compared to the most remote region in the heart figure of the second, where the values ofZgo rapidly to infinity, and an intermediate region with a very complex structure, plus properties of self-similarity. Just as there are windows for the logistic equation, for instance forK= 3.84, with a stable cycle in which there appear replicas in miniature of the original figure, also with the Mandelbrot set for the position with imaginary partiY= 0 and real partX= -1.75, is a small island in the form of a heart (figure X.4) which, seen in detail, becomes a new continent similar to the principal Mandelbrot set with "buds" that, examined with the microscope turn out to be diminutive replicas of the initial set, which in turn exhibit other more minuscule buds. The Mandelbrot figure and each attached circular disk correspond to a particular periodic orbit: that in the form of a heart to period 1, the largest disk to period 2, followed by disks of periods 8, 16... Such complexity shows us that, just like what we observed with many phenomena in nature, complex behavior can appear even with simple laws. One of the most important results of these mathematical investigations is that, upon their basis, a new method is emerging for confronting study of dynamic systems: the complex dynamic, which does not mean complicated, but instead based on complex numbers. XI.Can chaos be tamed?IT IS now more than a century since the mathematicians established the basis for studying non-linear systems. For three decades it has been applied to research the phenomena called chaotic which appear in nature. The attitude before chaos has been equivalent to that one has with diseases: she investigates their causes to avoid their appearance, since what one wants are predictable processes, behaviors without surprises, perfectly controllable domesticated systems. Chaos implies catastrophes, and one tries to keep it at a prudent distance. Yet this vision of chaos is changing during recent years, before the evidence that it can be controlled and even reach doing useful things.CHAOS CAN BE USEFULIn the First Conference on Experimental Chaos occurring in the United States in October in 1991 reports were presented which demonstrate that, and which refer to research underway to stabilize cardiac rhythms, control the oscillations in chemical clocks, increase the potency of laser beams, synchronize the output of electronic circuits. In every case, the results of applying chaos and controlling it have been very encouraging. Many of these applications have been initiated by the physicists William L. Ditto, of the Georgia Institute of Technology, and Louis M. Pecora, of the U.S. Naval Research Laboratory, who see two fundamental reasons for chaos to be of utility. In the first place, the deterministic chaos in dynamic systems is, in reality, a complex structure of many ordered states, none of which predominate over the others, as opposed to an ordered system that has a unique behavior. The researchers have demonstrated that if the chaotic system is adequately perturbed, it can be stimulated to adopt one among those ordered behaviors. The great advantage with respect to the classical ordered systems is their superior flexibility, because they can jump rapidly from one behavior to others out of a wide collection. The other reason is that if indeed one cannot predict its behavior, it is determined, such that if two practically identical chaotic systems of an adequate type are guided or moved by the same chaotic signal, both will have the same behavior even though it cannot be predicted in advance. Chaotic synchronization if the best demonstration that here we deal with deterministic chaos, since if we dealt with pure chance the behaviors would have no reason to coincide. Two isolated chaotic systems cannot be synchronized, because although they may be practically identical and begin to function at the same time, immediately their minuscule differences will be amplified and will cause them to diverge more and more. But if they are guided by a single chaotic signal both will have identical chaotic behavior, with the condition that they be stable, or that is if you perturb them a little, their trajectory in the space of the phases change only a little. If this condition is fulfilled, both systems will suppress whatever difference lies between them and will act in a chaotic--that is, unpredictable--yet synchronized manner. This is being applied in communication systems and signal processing.CHAOS IS CONTROLLABLEThe applications of controlling chaos are based on the OGY method (developed by Ott, Grebogi and Yorke, of the University of Maryland) who succeeded in having a system that displayed an entire collection of periodic oscillations move to having only one. The method consists in obtaining in the space of the phases the trajectory information as it passes through the Poincaré section of the strange attractor, and wait until the trajectory returns to pass through the section. If this happens in the vicinity of a desired periodic orbit, at that precise moment the dynamic system is perturbed modifying the parameter sufficiently for it to remain in that orbit. What they proved is that since chaotic dynamic systems are so sensitive to initial conditions, reacting very rapidly and with great versatility to this control, therefore their use can prove very advantageous. This has been applied to the control of chaotic fluctuations in the intensity of laser systems, attaining greater flexibility and stability, with potential output increased by a factor of 15. Another application of importance is the control of chaos in a biological system: utilizing a piece of the heart of a rabbit an arrhythmia was provoked, and in that situation it was stimulated with electrical signals produced by the OGY method. These were sufficient to re-establish a normal cardiac rhythm, which has encouraged the researchers to seek control of arrhythmias in human hearts and to project developing pacemakers and defibrillators based on this controlled chaos.ConclusionsWE HAVE reviewed the multiple aspects presented in the behavior of non-linear dynamic systems and the diverse methods that are used for its study. The applications of concepts such as self-similarity, fractals, the strange attractors, have awakened great enthusiasm among researchers in the most diverse disciplines, founded upon their notable results for non-linear physical systems. That is giving a great impetus to statistical mechanics, to the study of phase transitions, of spin windows, of turbulence, themes which also open new perspectives on other scientific fields. It might be, in effect, that those concepts are equally valid for other complex phenomena such as the fluctuations in insect or animal populations, in the economy or the behavior of the brain, et cetera. But to determine that we must know how to characterize the chaos in the system under study, and that becomes much more difficult the greater the number of independent variables in play. Even for physical systems like turbulent fluids with many degrees of freedom it is not entirely understood what is the role of deterministic chaos in the different transitions between order and chaos. When there are few degrees of freedom in play, one looks for a strange attractor, but this is made more difficult when applied to the search for attractors for the brain or the financial market, which then requires prudence, because expectations generated in this field have fomented a sort of fashion, where there are cases in which the same scientific works that a few decades ago were presented with results obtained by applying the classical tools are now done deriving them from fractal dimensions, without an apparent conceptual advantage for the interpretation of the phenomena studied. In any event, this is a very young discipline, in constant evolution. The outlook is very good for it is understood that, instead of avoiding non-linearity and complexity, they can be employed to provide more flexible, rapid systems with unexpected behavior which presents a wide gamut of possibilities. Yet the implications of deterministic chaos for the realm of knowledge extend beyond its utility for studying the different sorts of dynamic systems, and the new techniques being deployed as its applications grow. Another significant aspect refers to the way in which changes in our concept of the world are conditioned by our beliefs, a theme studied by Thomas S. Kuhn inThe Structure of Scientific Revolutions.When a new paradigm is produced, the scientists see new and different things in the world of research, even with instruments and in places already known. A notorious example is the way that the falling trajectory of bodies was perceived before and after the studies of Galileo. As the historian of science Pierre Thuillier shows us in his bookFrom Archimedes to Einstein,wise persons before Galileo were educated in the physics of Aristotle, according to which a body only moves when it is subjected to a force. Accordingly when a rock is thrown, or one shoots an arrow, or when a cannon fires a projectile, the body moves in a straight line from the impetus received, until this is exhausted due to air resistance, the moment when it falls vertically to the ground, to the place which corresponds to it according to the natural order. It is for that reason that the drawings in artillery manuals written by the experts around the year 1500 show trajectories in the form of an inverted L. One might suppose that an expert in artillery must have observed sometime how a stone or an arrow fall when shot toward a target, or the shape waterfalls have or wine poured from a jug. But only after the diffusion of Galileo's teachings, who demonstrated that these trajectories have the form of a parabolic curve, did the fall of projectiles, waterfalls, et cetera begin to be correctly represented. It is as if projectiles, which before Galileo fell in a straight line, would now begin to move following a curve! In a similar fashion, an investigator of the stature of Leonardo da Vinci, impassioned observer of nature, made drawings of what he saw in dissections of cadavers of animals and human beings. Yet when he drew the heart, he represented it with two ventricles and one auricle, for he saw it that way, imbued with what Galen, the maximum medical authority of that era, taught in that respect. In this century something similar occurs, with a change of paradigm that begins with the appearance of quantum and relativist physics, and which continues with the transformation that invokes the real aspect of non-linear phenomena. Until the 19th century, only phenomena were studied that obeyed integrable equations, linear in particular. If there was interest in a non- linear phenomenon, it was transformed through a linear approximation. It seemed as if in nature the truly important was the family of linear phenomena and that the others were an exception, a rare species undesirable for the difficulty of their treatment. In witness to this are the physics and mathematics books written until only a few decades ago, where non-linear systems are barely mentioned. Now, however, it is perceived that the immense majority of natural phenomena are non-linear, and that the others are the exception and not the rule. If indeed much meaningful physics exists which can be applied to linear phenomena, increasingly more institutions and scientists are dedicated to the dynamics of non-linear processes. Before there were no non-linear phenomena worthy of study, and today they are legion. Once again it is proved that the perception of what each of us has before ourselves is conditioned by our theories and beliefs concerning the world, to a very often unsuspected degree. As we have seen, the incorporation of the theme of chaos implies a change in focus, a different vision from what was common to scientists until the start of the 20th century: an "absolute" determinism like that which Laplace would so clearly formulate is no longer considered valid, and thus it is accepted that systems which obey deterministic laws can have unpredictable behavior, which makes inevitable their description through probabilities, and furthermore that this applies for the majority of them, including cases as apparently simple as the movement of every object that is subjected to the action of more than two forces. Accordingly, the study of chaos has revealed that the unpredictability of this complex world can be reconciled with the existence of simple and ordered natural laws. The mathematics corresponding to this field also require a different focus, now that the fundamental application of the computer permits visualization, through images which often are of rare beauty, the global, qualitative behavior of the equations of chaotic systems. Here too we see how very simple equations, applied through iteration, and with characteristics of self-similarity, give birth to fractal forms astonishingly similar to those that appear in nature: trees, mountains, clouds, which it was never dreamed could be represented mathematically. The mathematics of chaotic dynamic systems are non-linear, and this, due to the consequent sensitivity to the initial conditions, implies that, as opposed to those which obey integrable equations, only the global behavior of those systems can be known. The example of Poincaré's study of the movement of the planets illustrates this: the only linear, integrable system is that studied by Kepler, namely that which forms the Sun and the Earth. To manage to approximate to a linear system the other eight bodies comprising the solar system were excluded. In this way the elliptical trajectory of our planet can be obtained, but if you want to predict its long-term position with even minimum exactitude the Moon must be added, which influences terrestrial movement, and the remaining planets plus the gravitational interactions among them, which gives us a non-integrable, chaotic system, and where even the comets that come into the confines of the system or disappear for various reasons can, thanks to the extreme sensitivity to such perturbations, totally alter the whole. So it is not possible to analyze it as one would a linear system, through separation of the factors that affect the behavior from those which practically do not: every factor, however infinitesimal it may appear initially, can unleash a drastic change in the complete set. 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