There is a quantity called entropy, which measures the amount of disorder in a system, and entropy must continually increase. The arrow of time is the direction of increase of entropy. So how did the chicken create the ordered egg from disordered chicken feed? Do living systems somehow borrow a decrease of entropy from their environment?

Do they push the entropy into their environment, making it even more disordered than it would otherwise have been, and use the spare negative entropy to build an egg? Chickenkind has been borrowing an awful lot of negative entropy over the millennia. Entropy as it is currently conceived may not make a good arrow. The law arises from a particular thought experiment: the interaction of two previously independent systems. Imagine two cats, one black and one white. The white cat has its own set of white fleas; the black cat has black fleas. This remains constant--until the cats meet.

The fleas can then hop from one to the other. Now both cats have a mixture of black and white, creating a kind of gray cloud of fleas. Just as one party with ten children is far more chaotic than two parties each with five, there is more disorder among the gray fleas because there are more fleas! The relentless-increase-of-entropy rule does not apply to a single free-running system and so does not conflict with the time- reversibility of dynamics.

If you ran the cats backward, each would depart with a set of fleas, and you would then define the white fleas to be those on the white cat. Where originally there was just one system--a pair of cats with a shared pool of fleas--there are now two distinct subsystems, each comprising one cat and its set of fleas.

Thus it seems that when previously integrated systems become isolated, entropy decreases. Nothing relentless about that.

## Fractal Geometry Nature

Of course scientists knew that the cats have to interact with their surroundings, but the interaction seemed so small that they could safely ignore it. That was in the days before chaos. Imagine time-reversing some almost-isolated subsystem of the universe. For a short period it really will seem to undo its previous behavior, proceeding backward. However, it continues to interact--albeit very weakly--with the unreversed portion of the universe. That interaction, we have good reason to believe, is chaotic. And one of the basic features of chaos is the butterfly effect: very tiny changes become amplified to produce major changes in the observed motion.

On the strange attractor that represents the weather, two points that are infinitesimally close to each other quickly diverge over time on different paths. So, in the time-reversed portion of the universe, the butterfly effect comes into play, and soon that subsystem is no longer following the intended time-reversed motion. This has nothing to do with random motion: the chaotic fleas bouncing around on the cats are performing their own predetermined dance. The laws of physics imply that the dance can run backward. But suppose we manage to reverse the dance of the fleas on the black cat, while leaving those on the white cat running in the original direction.

Only if the black cat were truly isolated from the rest of the universe could the laws of physics allow it to keep running backward forever. Given the butterfly effect, even the tiniest degree of nonisolation is fatal to the argument. Say we remove the white cat a billion light-years from the black one. When the white cat thinks about having a scratch, the molecules in its brain will shift slightly.

That in turn will change their gravitational attraction.

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Only an absolutely isolated system is time-reversible. Strictly speaking, there is only one of these: the universe as a whole. The answer seems to be related to the external conditions under which the transition takes place. Molecules that escaped into the air during the cooking return; the long protein strands repair themselves; the yolk and white separate.

Going in this direction, the egg has to hope that the conditions surrounding it are precisely right. Running forward, systems are apparently immune to cats and other interferences. The history of the entire universe has to fit together consistently, and the butterfly effect destroys any attempts to reverse small bits of it. While chaos may run the universe on its greatest scale, it may also be at work on its smallest.

On the level of subatomic particles, Lady Luck seems to rule. Radioactive atoms decay at random, their only regularities being statistical. A large quantity of radioactive atoms has a well-defined half-life, a period of time during which half the atoms will decay. Then how does the atom know what to do? Might the apparent randomness of quantum mechanics be fraudulent?

Underneath the confusion, is chaos really at work? Perhaps it would be useful to think of an atom as some kind of vibrating droplet of cosmic fluid. Radioactive atoms would vibrate very energetically, and every so often a smaller drop could split off--what we would perceive as decay. The vibrations would be too rapid for us to measure in detail, so that we could only measure averaged quantities such as energy levels.

Now, classical mechanics tells us that a drop of fluid can vibrate chaotically. Its motion is deterministic--in other words, it is ruled by simple, comprehensible laws--but it is unpredictable. Occasionally, seemingly at random, the vibrations conspire to split off a tiny droplet. The butterfly effect makes such an event unpredictable; but it has well-defined statistics, a half- life.

Could the apparently random decay of radioactive atoms be something similar, but on a microcosmic scale? After all, why are there any statistical regularities at all? The review is focused on those strictly deterministic dynamic Systems that present the peculiarity of being sensitive to initial conditions, and, when they have a propriety of recurrence, cannot be predicted over the long term. Chaos theory has a few applications for modeling endogenous biological rhythms such as heart rate, brain functioning, and biological docks.

Johannes Kepler published the three laws of planetary motion in his two books of 1 and , 2 and Galileo Galilei wrote, in , 3 :. Philosophy Is written In this vast book, which continuously lies open before our eyes I mean the universe. But It cannot be understood unless you have first learned to understand the language and recognize the characters in which it is written.

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It is written in the language of mathematics, and the characters are triangles, circles, and other geometrical figures. For example, this principle is accepted a priori in physics. In Isaac Newton then consolidated the causality principle by asserting that the two concepts of initial conditions and law of motion had to be considered separately. Newton, having developed differential calculus and written the gravitational law, can be seen as the researcher who launched the development of classical science, ie, physics up to the beginning of the 20th century before relativity and quantum mechanics.

Newton wrote his manuscript on differential calculus in Gottfried Wilhelm von Leibniz had another point of view on this theme, and he published his book in Yet, later it was Leibniz' ideas that were used. These steps in the acquisition of human knowledge are described in Arthur Koestler's books on astronomy 8 and mentioned in science dictionaries. Determinism is predictability based on scientific causality Table I. One distinguishes schematically between local and universal determinism. Local determinism concerns a finite number of elements.

A good illustration would be ballistics, where the trajectory and the site of impact of a projectile can be precisely predicted on the basis of the propulsive force of the powder, the angle of shooting, the projectile mass, and the air resistance. Local determinism raises no particular problem. Obviously, one cannot.

In a whirlwind of dust, raised by elemental force, confused as it appears to our eyes, in the most frightful tempest excited by contrary winds, when the waves roll high as mountains, there is not a single particle of dust, or drop of water, that has been placed by chance, that has not a cause for occupying the place where it is found; that does not, in the most rigorous sense of the word, act after the manner in which it ought to act; that is, according to its own peculiar essence, and that of the beings from whom it receives this communicated force.

A geometrician exactly knew the different energies acting in each case, with the properties of the particles moved, could demonstrate that after the causes given, each particle acted precisely as it ought to act, and that It could not have acted otherwise than It did.

However, It was the mathematician and astronomer Pierre-Simon Laplace who most clearly stated the concept of universal determinism shortly after d'Holbach, In 12 :. We ought then to regard the present state of the universe as the effect of Its anterior state and as the cause of the one which is to follow. Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose It - an Intelligence sufficiently vast to submit these data to analysis-- it would embrace in the same formula the motions of the greatest bodies of the universe and those of the lightest atom; for It nothing would be uncertain and the future, as the past, would be present to Its eyes.

The creativity of Laplace was tremendous. He demonstrated that the totality of celestial body motions at his time, the sun and the planets could be explained by the law of Newton, reducing the study of planets to a series of differential equations. Urbain Jean Joseph Le Verrier discovered the planet Neptune in , only through calculation and not through astronomical observation.

He then developed further Laplace's methods by, for example, approximating solutions to equations of degree 7 and concluded 14 :. It therefore seems impossible to use the method of successive approximations to assert, by virtue of the terms of the second approximation, whether the system comprising Mercury, Venus, Earth, and Mars will be stable Indefinitely. It is to be hoped that geometricians, by integrating the differential equations, will find a way to overcome this difficulty, which may well just depend on form.

In the middle of the 19th century, it became clear that the motion of gases was far more complex to calculate than that of planets. One of their main postulates was the following: an isolated system in equilibrium is to be found in all its accessible microstates with equal probability.

In , Maxwell described the viscosity of gases as a function of the distance between two collisions of molecules and he formulated a law of distribution of velocities. Boltzmann assumed that matter was formed of particles molecules, atoms an unproven assumption at his time, although Democrites had already suggested this more than years previously. He postulated that these particles were in perpetual random motion. It is from these considerations that Boltzmann gave a mathematical expression to entropy.

In physical terms, entropy is the measure of the uniformity of the distribution of energy, also viewed as the quantification of randomness in a system. Since the particle motion in gases is unpredictable, a probabilistic description is justified. Changes over time within a system can be modelized using the a priori of a continuous time and differential equation s , while the a priori of a discontinuous time is often easier to solve mathematically, but the interesting idea of discontinuous time is far from being accepted today.

One can define the state of the system at a given moment, and the set of these system states is named phase space see Table I. A very small cause, which eludes us, determines a considerable effect that we cannot fail to see, and so we say that this effect Is due to chance. If we knew exactly the laws of nature and the state of the universe at the initial moment, we could accurately predict the state of the same universe at a subsequent moment.

But even If the natural laws no longer held any secrets for us, we could still only know the state approximately. If this enables us to predict the succeeding state to the same approximation, that is all we require, and we say that the phenomenon has been predicted, that It Is governed by laws. But this is not always so, and small differences in the initial conditions may generate very large differences in the final phenomena. A small error in the former will lead to an enormous error In the latter. Prediction then becomes impossible, and we have a random phenomenon. Moser In , and Vladimimlr Igorevltch Arnold In , he showed further that a quasiperiodic regular motion can persist in an Integrable system Table I even when a slight perturbation Is Introduced Into the system.

The theorem also describes a progressive transition towards chaos: within an Integrable system, all trajectories are regular, quaslperlodlc; introducing a slight perturbation one still has a probability of 1 to observe a quasiperiodic behavior within a point chosen arbitrarily in the phase space.

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When a more significant perturbation is introduced, the probability of a quasiperiodic behavior decreases and an increasing proportion of trajectories becomes chaotic, until a completely chaotic behavior is reached. In terms of physics, in complete chaos, the remaining constant of motion is only energy and the motion is called ergodic. Kolmogorov led the Russian school of mathematics towards research on the statistics of dynamical complex system called the ergodic theory. In a linear system Table I , the sum of causes produces a corresponding sum of effects and it suffices to add the behavior of each component to deduce the behavior of the whole system.

Phenomena such as a ball trajectory, the growth of a flower, or the efficiency of an engine can be described according to linear equations. In such cases, small modifications lead to small effects, while Important modifications lead to large effects a necessary condition for reductionism. The nonlinear equations concern specifically discontinuous phenomena such as explosions, sudden breaks In materials, or tornados.

Although they share some universal characteristics, nonlinear solutions tend to be individual and peculiar. In contrast to regular curves from linear equations, the graphic representation of nonlinear equations shows breaks, loops, recursions all kinds of turbulences. Using nonlinear models, on can identify critical points in the system at which a minute modification can have a disproportionate effect a sufficient condition for holism. The above observations from the field of physics have been applied in other fields, in the following manner: in the terms of reductionism, the whole can be analyzed by studying each of its constituents, while in holism, the whole is more than the sum of its constituents, and therefore cannot be deduced from its parts.

When should one analyze rhythmic phenomena with reductionist versus holistic models? This is a question that one can ask in the field of chronobiology. He first observed the phenomenon as early as and, as a matter of irony, he discovered by chance what would be called later the chaos theory, in , 18 while making calculations with uncontrolled approximations aiming at predicting the weather. The anecdote is of interest: making the same calculation rounding with 3-digit rather than 6-digit numbers did not provide the same solutions; indeed, in nonlinear systems, multiplications during iterative processes amplify differences in an exponential manner.

By the way, this occurs when using computers, due to the limitation of these machines which truncate numbers, and therefore the accuracy of calculations. Lorenz considered, as did many mathematicians of his time, that a small variation at the start of a calculation would Induce a small difference In the result, of the order of magnitude of the initial variation. This was obviously not the case, and all scientists are now familiar with this fact. Yorke, in The figure that appeared was his second discovery: the attractors. The Belgian physicist David Ruelle studied this figure and he coined the term strange attractors in It is also Ruelle who developed the thermodynamic formalism.

There are four types of attractors. Figure 1 describes these types: fixed point, limit-cycle, limit-torus, and strange attractor. According to Newton's laws, we can describe perfectly the future trajectories of our planet. However, these laws may be wrong at the dimension of the universe, because they concern only the solar system and exclude all other astronomical parameters. Then, while the earth is indeed to be found repetitively at similar locations in relation to the sun, these locations will ultimately describe a figure, ie, the strange attractor of the solar system.

A chaotic system leads to amplification of initial distances in the phase space; two trajectories that are initially at a distance D will be at a distance of 10 times the value of D after a delay of once the value of characteristic Lyapunov time Table I. If the characteristic Lyapunov time of a system is short, then the system will amplify its changes rapidly and be more chaotic.

However, this amplification of distances is restricted by the limits of the universe; from a given state, the amplification of the system has to come to an end. It is within the amplification of small distances that certain mathematicians, physicists, or philosophers consider that one can find randomness.

## Chaotic Dynamics and Fractals - 1st Edition

The solar system characteristic Lyapunov time is evaluated to be in the order of 10 years. The terms of negative and positive feedback Table I concern interactions that are respectively regulations and amplifications. An example of negative feedback is the regulation of heat in houses, through interactions of heating apparatus and a thermostat. Biology created negative feedback long ago, and the domain of endocrinology is replete with such interactions.

An example of positive feedback would be the Larsen effect, when a microphone is placed to close to a loud-speaker. In biology, positive feedbacks are operative, although seemingly less frequent, and they can convey a risk of amplification. Negative and positive feedback mechanisms are ubiquitous in living systems, in ecology, in daily life psychology, as well as in mathematics. A feedback does not greatly influence a linear system, while it can induce major changes in a nonlinear system.

Thus, feedback participates in the frontiers between order and chaos. Mitchell Jay Feigenbaum proposed the scenario called period doubling to describe the transition between a regular dynamics and chaos. His proposal was based on the logistic map introduced by the biologist Robert M. May in The logistic map is a function of the segment [0,1] within itself defined by:. The dynamic of this function presents very different behaviors depending on the value of the parameter r:.

This function of a simple beauty, in the eyes of mathematicians I should add, has numerous applications, for example, for the calculation of populations taking into account only the initial number of subjects and their growth parameter r as birth rate.

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When food is abundant, the population increases, but then the quantity of food for each individual decreases and the long-term situation cannot easily be predicted. He described his theory in a book, 27 where he presented what is now known as the Mandelbrot set. A characteristic of fractals is the repetition of similar forms at different levels of observation theoretically at all levels of observation. Thus, a part of a cloud looks like the complete cloud, or a rock looks like a mountain.

Fractal forms in living species are for example, a cauliflower or the bronchial tree, where the parts are the image of the whole. A simple mathematical example of a fractal is the so-called Koch curve, or Koch snowflake. This is then repeated for each of the smaller segments obtained. Fractal objects have the following fundamental property: the finite in the case of the Koch snowflake, a portion of the surface can be associated with the infinite the length of the line. A second fundamental property of fractal objects, clearly found in snowflakes, is that of self similarity, meaning that parts are identical to the whole, at each scaling step.

A few years later, Mandelbrot discovered fractal geometry and found that Lorenz's attractor was a fractal figure, as are the majority of strange attractors. He defined fractal dimension Table I.

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Mandelbrot quotes, as illustration of this new sort of randomness, the French coast of Brittany; its length depends on the scale at which it is measured, and has a fractal dimension between 1 and 2. This coast is neither a one-dimensional nor a two-dimensional object. For comparison the dimension of Koch snowflake is 1. Catastrophe theory is interesting in that it places much emphasis on explanation rather than measurement.

## Fractal Links/References

Thom was at the origin of a renewed debate on the issue of determinism. I'd like to say straight away that this fascination with randomness above all bears witness to an unscientific attitude. It is also to a large degree the result of a certain mental confusion, which is forgivable in authors with a literary training, but hard to excuse in scientists experienced in the rigors of rational enquiry. What in fact is randomness? Only a purely negative definition can be given: a random process cannot be simulated by any mechanism or described by any formalism.

Ilya Prigogine, author of a theory of dissipative structures in thermodynamics, considers that the universe is neither totally deterministic nor totally stochastic. Initial conditions can no longer be assimilated to a point in the phase space, but they correspond to a region described by a probability distribution. It is a nonlocal description, a new paradigm. Thank you. Is this the German version?

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