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

There are several reasons to investigate nonlinear maps. Maps dynamics is very complicated and computers make quickly 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 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 x_n+1 = F(x_n) is determined uniquely by initial coordinate x_o. 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 δx_n between close points increases exponentially with time. This instability may be detected by the positive Lyapunov exponent
Λ = lim_n→∞ L_n , L_n = 1/n log|δx_n/δx_o|.
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 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 the picture 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 determinate with absolute precision the finite state too we shall 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 "-/+" buttons to trace mixing process.

E.g. for the considered above quadratic map probability distribution function of an orbit points (invariant measure) is equal for almost all initial x_o. 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.

Smale horseshoe and homoclinic structures

Smale horseshoe map has 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

Complicated set of infinit number of unstable orbits may be repelling (see strange cantor repeller). Then close orbits wander chaotically in its vicinity for a long time before to come to a regular attractor. This phenomenon is called chaotic transient.

For the quadratic map parameter C values for regions with regular dynamics are dense in [-2, 1/4]. Therefore arbitrary close to a C_ch value with chaotic dynamics there is region with regular attracting cycle. But close to C_ch period of this cycle 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).

Contents Next: Sawtooth map & Bernoulli shifts
updated 12 July 2007