See also: Chaos

The theory of chaos rigorously treats dynamic systems deterministic, but which present a fundamental phenomenon of instability called “Sensibilité to the initial conditions” which, modulating an additional property of recurrence , makes them nonprédictibles in practice on “the long” term.

Introduction

Heuristic definition of a chaotic system

A dynamic Système is known as chaotic if a “significant” portion of sound Espace of the phases shows simultaneously the two following characteristics:

• the phenomenon of Sensitivity to the initial conditions.

• strong a recurrence.

The presence of these two properties involves an extremely disordered behavior qualified rightly the “chaotic one”. The chaotic systems are opposed in particular to the integrable systems of the traditional Mécanique, which were a long time the symbols of a very powerful regularity in Theoretical physics. Dynamics quasi-periodical of an integrable system seemed itself to find its illustration perfect in the majestic movements of planets of our Solar system around the Sun; let us remember that Voltaire, itself introducer of the mechanical of Newton in France at the 18th century, spoke about God like “Large Clock and watch maker”…

What the “theory of chaos”?

During its history, the theoretical physics had already been confronted with the description of complex systems macroscopic, like a volume of gas or liquid, but the difficulty of describing such systems seemed to rise from the very great number of degrees of internal freedom of the system on a microscopic scale (atoms, molecules). The Mécanique statistics had in this case allowed to return account in a satisfactory way of the macroscopic properties of these systems to balance. It was thus a great surprise when one realized at the end of the 19th century that a dynamics of large a Complexité could result from a simple system having a very small number of degrees of freedom , provided that it has this property of sensitivity to the initial conditions.

Was the theory of chaos born in the years 1970?

The answer to this question is: yes AND not.

• Not, because the phenomenon of sensitivity to the initial conditions was discovered as of the end of the 19th century by Henri Poincaré in work concerning the Problème with NR body in Celestial mechanics, then by Hadamard with a mathematical model abstracted now baptized “geodetic Flot on a surface with negative Courbure”. This discovery involved a great number of important work, mainly in the field of mathematics. This work is evoked in the paragraph historical Développements located further.

• Yes, because it is truly only in years 1970 that the theory of chaos gradually took a lead in the front of the scientific scene, operating an epistemological rupture strong. The suggestive term of “chaos” was introduced besides only in 1975 by the two mathematicians Li and Yorke. The late character of this change of paradigm is explained easily: the theory of chaos indeed owes its popularization with fulgurating progress of the Informatique as from the years 1960-70. This new science indeed made available to the not-mathematicians direct visualization from the incredible complexity of these dynamic systems, before reserved to the only “initiates” able to absorb the suitable mathematical formalism.

As illustration, the figure opposite is a typical example of images produced by the theory of chaos; it is here about a geometrical object discovered by Lorenz in 1963, and baptized since “attractile strange”. (This object will be commented on low, in the paragraph: Lorenz & meteorology .)

But the theory of the chaos is not a discipline the contemplative ones, only filled with wonder by the vision at the beautiful images colors generated by their preferred computer! It is true a scientific Théorie, which knew to find order hidden under the apparent disorder. But this new order is very different from the old order: the determinism relentless of an integrable dynamics quasi-periodical succeeded a description of nature basically probabilistic, characterized by the existence of invariants taking the form of measurements of probabilities, of Attracteur S, Dimension S Fractale S… All the Sciences, including social , are concerned with this change of paradigm, in order to answer the intuition of Plato (reformulated by James Gleick):

“ (…) behind the visible and particular shapes of the matter must hide ghostly forms which are used to them as invisible model ”.

Determinism, of Laplace with Poincaré (Prehistory)

See also: Determinism

The stability of the solar system

See also: Problem with NR body

The starting point of the theory of chaos is the problem with “3 bodies” which consist in studying the movement of three bodies in gravitational interaction, such as for example the system: {Sun - Ground - the Moon}, presumedly isolated from the remainder of the universe. The aim of this research is to determine if the Solar system is “stable” on the long run, or if one of the bodies is likely one day to strike another body, or to be ejected solar system towards the infinite one.

The problem with 3 bodies is as old as Newtonian mechanics; indeed, as of the birth of this theory, its founder was interested in the problem with three bodies with an aim of predicting the movement of the Moon. All the astronomers with his continuation tackled this problem, of which Laplace, which believed to have proven the stability of the solar system by using the Théorie of the disturbances to the first order. Unfortunately, the perturbatif development with the first order is insufficient to conclude definitively. One century after Laplace, Henri Poincaré thus seized the problem. One examines below the evolution of the ideas which distinguishes the thought from Laplace of that of Poincaré.

Concept of conservative differential dynamic system

See also: dynamic System

For a system having N degrees of freedom, the Espace of the phases $\backslash \; Gamma$ of the system has N dimensions, so that the complete state $x\; (T)\; \backslash \; in\; \backslash \; Gamma$ of the system at the moment T is in general a Vecteur with component N . One then considers typically a first order differential connection of the type:

where the function F defines the studied dynamic system (it is in general also a vector with N dimensions, i.e. a whole of N scalar functions). This physical system, presumedly conservative, is deterministic if and only if the dynamics of the system associates with each initial condition $x\_0$ one and only one final state $x\; (T)$. It is necessary for that there exists a bijective application $\backslash \; phi\_t:\; \backslash \; Gamma\; \backslash \; to\; \backslash \; Gamma$ of the space of the phases on itself such as:

When time T varies, this bijection generates a flood on $\backslash \; Gamma$, i.e. a continuous Groupe with a parameter $\backslash \; phi\_t$. This mathematical modeling corresponds for example to the Hamiltonian flood of the traditional Mécanique, like with the geodetic Flot.

Laplace, or triumphing determinism

Extremely the successes obtained in Celestial mechanics, Laplace write in 1814 in the introduction of philosophical sound Essai on the probabilities :

“ We must thus consider the state present of the universe like the effect of its former state, and as the cause of that which will follow. An intelligence which for a given moment knows all the forces whose nature is animated and the respective situation of the beings which compose it, if besides it were enough vast to subject its data to the analysis, would embrace in the same formula the movements of the more large body of the universe and those of the lightest atom: nothing would be dubious for it, and the future like the past would be present at its eyes.

the human spirit offers, in the perfection which it knew to give to Astronomy, a weak draft of this intelligence. Its discoveries in Mechanics and Geometry, united with that of universal gravity, put it at range to include/understand in the same analytical expressions the last and future states of the system of the world. While applying the same method to some other objects of its knowledge, he managed to bring back to general laws the phenomena observed, and to envisage those which circumstances given must do to hatch. All these efforts in the research of the truth tend to unceasingly bring it closer to the intelligence which we have just conceived, but from which there will remain always infinitely distant. This tendency specific to mankind is what makes it higher than the animals, and its progress in this kind distinguishes the nations and the centuries and makes their true glory. ”

This text today celebrates is actually largely prophetic, with direction where Laplace does not have the general theorem of existence and unicity of the solution of a differential equation, which will be shown later on, and is the subject of the following paragraph.

The theorem of Cauchy-Lipschitz

See also: Theorem of Cauchy-Lipschitz

It is the mathematician Cauchy who states in 1820 the general theorem of existence and unicity of the solution of a differential equation. Lipschitz will give him its final form in 1868.

Poincaré, or wavering determinism

Approximately a century after Laplace, Poincaré written in the introduction of sound Probability theory a text from which the tonality is extremely different from that of its famous predecessor. It is between 1880 and 1910, that Poincaré, which seeks to prove the stability of the solar system, discovers a new continent resulting from the equations of Newton and hitherto unexplored.

“ How to dare to speak about the laws of fate? Isn't the chance the antithesis of any law? Thus Rerirand is expressed, at the beginning of sound Probability theory . The probability is opposed to the certainty; it is thus what one is unaware of and, consequently it seems, which one could not calculate. There is an at least apparent contradiction and on which one wrote already much.

And initially what chance? The old ones distinguished the phenomena which seemed to obey harmonious laws, established once and for all, and those that they allotted randomly; these was those that one could not provide because they were rebellious with any law. In each field, the precise laws did not decide a whole, they traced only the limits between which it was randomly allowed to be driven.

to find a better definition of the chance, it is necessary for us to examine some of the facts that one agrees to look like fortuitous, and to which the probability theory appears to apply; we will seek then which are their common characters. The first example that we will choose is that of unstable balance; if a cone rests on its point, we know well that it will fall, but we do not know a which side; it seems to us that the chance alone will decide some. If the cone were perfectly symmetrical, if its axis were perfectly vertical, if it were subjected to no other force which gravity, it would not fall at all. But the least defect of symmetry will make it slightly lean side or other, and as soon as it leans, as little as it is, it will fall completely on this side. So even symmetry is perfect, a very light trepidation, an air blast will be able to make it incline few seconds of arc; it will be enough to determine its fall and even the direction of its fall which will be that of the initial slope. ”

Sensitivity to the initial conditions

In the preceding framed paragraph, Poincaré puts forward the phenomenon known today under the denomination of sensitivity to the initial conditions : for a chaotic system, a very small error on the knowledge of the initial state x₀ in the space of the phases will be (almost always) quickly amplified.

Quantitatively, the growth of the error is locally exponential for the strongly chaotic systems, baptized according to the ergodic Théorie K-systems (the K is for Kolmogorov), like for the systems very strongly chaotic, known as B-systems (the B is for Bernoulli). This amplification of the errors quickly makes completely inoperative the predictive capacity which rises from the unicity of the solution, ensured by Cauchy-Lipschitz.

Typically, for a chaotic system, the errors grow locally according to a law of the type:

where $\backslash \; tau$ is a time characteristic of the chaotic system, called sometimes “ horizon of Lyapounov ”. The prédictible character of the evolution of the system remains only for the moments $T\; \backslash \; L\; \backslash \; tau$, for which the exponential one is worth 1 roughly, and thus such as the error its initial size keeps. On the other hand, for $T\; \backslash \; gg\; \backslash \; tau$, any prediction becomes practically impossible, although the theorem of Cauchy-Lipschitz remains true.

Historical developments

August 1st

Poincaré & the stability of the solar system

See also: Problem with NR body

One century after Laplace, Henri Poincaré was harnessed with the problem of the stability of the solar system. Between 1880 and 1886, it starts by publishing a series of memories entitled: “ On the curves defined by a differential equation ” which gives rise to the qualitative analysis differential equations. Poincaré in particular introduced there the capital concept of Portrait of phase, which geometrically summarizes the aspect of the solutions in the Espace of the phases of the system. Then, in 1890, it publishes the famous report entitled: “ On the problem of the three bodies and the equations of the dynamics ”, which will be worth to him the price of king Oscar, king de Norvège and of Sweden and impassioned mathematics. The history is famous: the report prize winner comprised an error detected by the young Phrägmen mathematician whereas it prepares the manuscript for the printer. This error will oblige Poincaré to proceed to deep rehandlings in its report, and to as refund the expenses of impression of the first report, a sum higher of some thousand crowns than the price as it had received. But this error was fertile, because instead of the stability of the solar system, Poincaré discovered the potential chaos hidden in the equations of dynamics.

More recently, of the numerical calculations carried out by the astronomer Jacques Laskar in 1989-1990, then confirmed by Sussman & Wisdom in 1992, showed that the solar system is chaotic , with a horizon of Lyapounov of about 200 million years.

The Russian school of the years 1890-1950

Lyapounov & the stability of the movement

October 12th, 1892, Lyapounov supports at the University of Moscow a thesis of doctorate entitled: the general problem of the stability of the movement . It introduced the idea there to measure the possible divergence between two orbits resulting from close initial conditions. When this divergence believes exponentially with time for almost all the initial conditions close to a given point, one has the phenomenon of Sensibilité to the initial conditions, idea to which are attached the exhibitors of Lyapounov, which give a quantitative measurement of this exponential divergence local .

The school of Gorki: 1930-1940

• Andronov

• Pontryagin
• Lefschetz

The oscillator of Van DER pol.

See also: Oscillating of Van DER pol.

Emergence & development of the ergodic theory

See also: ergodic Theory

• Birkhoff

• Von Neumann
• Koopman
• Hopf
• Hedlung

Predictibility & calculability

Norbert Wiener and John von Neumann was worried however about the possibility of predicting by calculation a future situation starting from a state present. If Wiener considered the task difficult, even impossible since “small causes” that one would necessarily omit to include in the model can produce “great effects” (it gave the indication of the “snowflake starting an avalanche”), Von Neumann saw there an exceptional opportunity for the new apparatuses which one had not baptized yet Ordinateur S: “If a snowflake can start an avalanche”, answered it Wiener, “then the prediction by calculation will tell us very exactly what a precise snowflake to intercept so that the avalanche does not occur! ” Wiener showed skeptic: a hypercritical state remained a hypercritical state, and to remove this particular flake would not make in its opinion that “make it possible another to replace it in this function”. According to him, nothing would thus be solved (point of view admitted today). The two men did not push more before this disagreement.

Lorenz & meteorology

See also: dynamic System of Lorenz

Presentation

In 1963, the meteorologist Lorenz highlighted the probably chaotic character of meteorology.

Mathematically, the coupling of the atmosphere with the Océan is described by the system of coupled partial derivative equations Navier-Stokes of the Mécanique of the fluids. This system of equations was too much complicated to solve numerically for the first computers existing at the time of Lorenz. This one had thus the idea to seek a very simplified model of these equations to study a particular physical situation: the phenomenon of Convection of Rayleigh-Bénard. It leads then to a differential dynamic system having only three degrees of freedom, much simpler to integrate numerically than the starting equations. It then observed, by mere chance, that a tiny modification of the initial data (about for thousand) involved very different results. Lorenz had just put forward the Sensibilité to the initial conditions (already observed in numerical Analyze in solutions of differential equations on computer, inter alia by Marion Créhange at the University of Nancy).

The metaphor of the butterfly

In 1972, Lorenz makes a conference with the American Association for the Advancement off Science entitled: “ Predictability: Does the Flap off have Butterfly' S Wings in Brazil Set off has Tornado in Texas? ”, which is translated into French by:

“ Prédictibilité: does the beat of wings of a butterfly in Brazil cause a tornado in Texas? ”.

This metaphor, become emblematic of the phenomenon of sensitivity to the initial conditions, is often interpreted wrongly in a causal way: it would be the beat of wing of the butterfly which would start the storm. It of it is nothing; Lorenz writes indeed:

One will be able to read interesting it article of Nicolas Witkowski: hunting for the effect butterfly , Alloy 22 (1995), 46-53.

Stephen Smale: topology & structural stability

August 1st

The Russian school of the years 1950-1980

See also: System of Anosov

• Anososv

See also: Theorem KAM

• the Sinai

• Arnold

Transition from a regular dynamics towards chaos

That is to say a dynamic system dependant on a parameter $\backslash \; driven$:

It happens that dynamics changes behavior when the parameter $\backslash \; driven$ varies. One could highlight three great scenarios of passage of a regular dynamics at a chaotic dynamics during the variation of a parameter.

Scenario of Feigenbaum

Feigenbaum proposed a scenario known as: “ by doubling of period ” to describe the transition from a regular dynamics towards chaos. This scenario finds its origin in the behavior of the logistic Suite, which is defined by recurrence by an application of segment 1 in itself:

where N = 0,1,… indicates discrete time, X the single variable dynamic, and $0\; \backslash \; the\; \backslash \; driven\; \backslash \; the\; 4$ a parameter. The dynamics of this application presents a behavior very different according to the value from the parameter $\backslash \; mu$:

• For , the system has a gravitational fixed point, which becomes unstable when $\backslash \; driven\; =\; 3$.

• For , the application has a Attracteur which is a periodic orbit, of period $2^n$ where N is an entirety which tends towards the infinite one when $\backslash \; mu$ tends towards 3.57…

• When $\backslash \; driven\; =\; 3.57\dots$, the application has attractile of Feigenbaum Fractal (but nonstrange) discovered by the biologist May (1976).

• the case $\backslash \; driven\; =\; 4$ had been studied since 1947 by Ulam and von Neumann. It should be noted that one can in this precise case establish the exact expression of invariant measurement ergodic.

When the parameter $\backslash \; mu$ increases, one thus obtains a succession of junctions of the regularity towards chaos, summarized on the figure opposite.

Scenario of Lane-Takens

By quasi-periodicity…

August 1st

Scenario of Pommel-Manneville

Intermittently…

August 1st

Some examples

• Transformation of the baker. The transformation of the baker has many alternatives, which all have as a common point “to make go back” very quickly to the macroscopic level of negligible microscopic differences, weaker than the Résolution of the instrument used.

• Transformation of the photomaton.

Dependant articles

• Determinism

• Mechanical Hamiltonian
• integrable System
• Theorem KAM
• Problem with NR body
• ergodic Theory
• dynamic System
• quantum Chaos

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