Dynamic Systems

The dynamic systems usually indicate the branch of active research of the Mathématiques, at the border of the Topologie, the analyzes, of the Géométrie, the Théorie of measurement and the Probabilités, and which endeavors to study the properties of a dynamic system. The nature of this study differs according to the studied dynamic system, nature which depends on the tools used (analytical, geometrical or probabilistic).

Historically, the first questions concerned with the dynamic systems related to the Mécanique at one time when it was included in the Enseignement of mathematics. One of the major questions and which justified mathematical research is the problem of the stability of the Solar system. Work of Lagrange on the subject consisted in interpreting the influence of the bodies other than the Sun on a planet like a succession of infinitesimal shocks: this work finds echoes in the Théorème KAM (Kolmogorov - Arnold - Moser).

The dynamic systems developed and specialized during the 19th century. They initially related to the iteration of the continuous applications and the stability of the differential equations. But gradually, progressively of the diversification of mathematics, the dynamic systems widened considerably. They include/understand today the study of the continuous actions of groups, whose interest lies in its applications in Géométrie, and the ergodic Théorie, born from the advent of the Théorie of measurement and who finds his echoes in Probabilité S.

The diversity of study

Differential equations

The study of the behavior of the differential equations are at the origin of the dynamic systems. The questions which occurrent are several natures. They concern:

the behavior limits solutions;
the stability of the solutions compared to the initial conditions;
the stability of the solutions compared to disturbances relating to the differential equation.

A relatively simple case seemingly is the study of the homogeneous differential equations of order 1. These differential equations are summarized with the data of a field of vectors on open of a real vector space, which one seeks the curves known as integral. The study of the fields of vectors has an impact in differential Géométrie.

Iteration of continuous applications

The iteration consists in considering made up successive the F ⁿ of a certain application continues F . The powers are in general positive, but can be negative when the reverse of F is defined. The study relates primarily to its limiting properties.

The interest carried to the iteration of continuous applications came following work from Poincaré on the stability of the Solar system. It is on this occasion that the application of first return of Poincaré was introduced. The application of return of Poincaré and its known opposite operation under the name of suspension make it possible to pass from a field of vectors to the problem of the iteration of an application.

One usually distinguishes:

a continuous dynamic System: the data of a separate topological space X and of an application continues $f: X \ rightarrow X$ , often a homeomorphism;
a differentiable dynamic System: the data of a real differential variety X and of a differentiable application $f: X \ rightarrow X$ , often a diffeomorphism (it is often of use to impose a degree of regularity);
a dynamic Système complexes: the data of a holomorphic variety X and of a holomorphic application $f: X \ rightarrow X$ , often a biholomorphism.

If necessary, one joins the compact adjective to announce that X is supposed to be compact. The study of a differentiable dynamic system and its characteristics is called the differentiable Dynamique and the study of a holomorphic dynamic system the holomorphic Dynamique.

The Homéomorphismes of the circle wrongly seem to be a simple case of study, but show with the contaire the complexity of the questions which arise. The classification of the C^1-diffeomorphisms whose derivative is with limited variation is determined by an invariant called the Nombre of rotation. A current research tends to generalize these results in higher dimension.

The ergodic theory

The study of the continuous actions of groups

See also: Action continues of a topological group

The continuous actions of a topological group on a topological space can be seen like a generalization of the continuous floods. In this language, a continuous flood is a continuous action of the additive group R .

For the topological groups locally compact, the existence of a measurement of Haar makes it possible to include methods of the theory of measurement. In particular concepts of moyennability were formalized.

In differential geometry, the interest relates to the study of the differentiable actions of the groups of Dregs on the differential varieties.

The unicity of the dynamic systems

In spite of the diversification of its sphere of activity, this mathematical field keeps a unit as well in the questions put as in the description of the studied objects.

Space phases

Whatever its nature (topological, differentiable, measurable, measured,…), a dynamic system is the joint data:

Of a space of the phases,
Of a time, which can be discrete (natural entireties, relative entireties, discrete group), continuous (realities positive, real, related group of Dregs),…
Of a law of evolution.

Asymptotic behavior

Stability

Popular culture

A certain number of results of the dynamic systems had an impact considerable in the popular culture. In particular, the theory of chaos for the continuous dynamic systems and the whole of Mandelbrot in holomorphic dynamics had an impact in the artistic fields.