Deterministic Chaos

Chaos in simple maps

We used to get simple solutions for simple equations (e.g. oscillator, Keplerian orbits, limit cycles of the Van der Pol generator). But very often simple nonlinear systems have extremely complicated orbits which look completely chaotic. For example you see the standard map orbits below. Ellipses correspond to regular (integrable) motion but "grey" regions are filled by tangled chaotic orbits.

Controls: Click mouse to get a new orbit marked by the red color
(sorry if your browser doesn't support Java).

There are several reasons to investigate nonlinear maps. Maps dynamics is very complicated and PC makes quickly enough amazing fractal pictures (integration of differential equations is much more boring). One can study flows dynamics by Poincare maps too. Surprisingly, very simple maps turn out to yield rather good qualitative models for behavior in ordinary and partial differential equations.

Unstable orbits and deterministic chaos

Any orbit of a dynamical system defined by differential equation dx/dt = F(x) or by discrete map xn+1 = F(xn) is determined uniquely by initial coordinate xo. Chaos is associated with unpredictable random motion, therefore ("by definition") orbits of deterministic dynamical systems can not be chaotic. But very often nonlinear systems have unstable orbits. In that case distance δxk between close points increases exponentially with time. This instability may be detected by the positive Lyapunov exponent
    Λ = limn→∞ Ln ,     Ln = 1/n log|δxn /δxo|.
You see below one of chaotic orbits of the quadratic map with positive Lyapunov exponents L calculated for shown segment

For real physical systems it is impossible to determine initial coordinates with absolute accuracy. It is possible to set only probability distribution function to find system in a small (but finite) region of the phase space. For a short time all orbits from this region move together and this "packet" is similar to a particle. But due to instability small initial region is stretched and mixed in the phase space (see applet below). It is similar to ink-drop spreading in water under mixing. For bounded motion after a time close orbits are dispersed and mixed in the phase space. As since we can not to determinate with absolute precision the finite state too we shell average this picture on a small scale. After that one can predict only probability to find system in a point (precisely - in a small region) of the phase space.
The blue square mixing by the standard map. N is number of iterations. Press "-" button to trace the mixing process.
Thus for the considered quadratic map probability distribution function of an orbit points (invariant measure) is equal for almost all initial xo. Therefore for large time it is natural to use statistical description of this deterministic system and replace time averaging by averaging with the invariant measure. In that way instability of bounded orbits leads to probabilistic description of nonlinear dynamical systems. This phenomenon is called dynamical chaos.

Symbolic dynamics, Smale horseshoe and homoclinic structures

Often it is possible to partition the phase space of a system in such a way that its orbits are determined uniquely by the sequence of passing of these partitioned regions (symbolic sequence). For example symbolic dynamics is constructed if dynamical system is reduced to the Smale horseshoe map. By means of symbolic dynamics it is possible to show that system have unstable periodic orbit with any period and continuum of non-periodic chaotic orbits. Horseshoe map exists if there is a homoclinic point i.e. an intersection of stable and unstable manifolds of a saddle point.

Strange repellers and chaotic transient

This complicated set may be repelling (see strange cantor repeller). Close orbits wander chaotically in its vicinity for a long time before come to a regular attractor. This phenomenon is called chaotic transient.
For the quadratic map parameter C regions with regular dynamics are dense in [-2, 1/4]. Therefore arbitrary close to Cch with chaotic dynamics there is a region with regular attracting cycle. But close to Cch period of this orbit grows, chaotic transient duration tend to infinity and it is impossible practically to destinguish regular and chaotic dynamics (see intricate entangling of regular and chaotic dynamics regions on the bifurcation diagram to the left).
Strange attractor (to the right) and intricate entangling of regular and chaotic dynamics regions (to the left) for the dissipative 2D Henon map are shown below.

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updated 12 July 07