We used to get simple solutions of 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 orbits of the
standard map 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).
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 between close points increases exponentially with time.
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. Therefore instability of bounded
orbits and impossibility of exact measurements lead to probabilistic
description of nonlinear dynamical systems. This phenomenon is called
dynamical chaos.
Blue square mixing by the standard map. N is number of iterations.
Press "-" button to trace the mixing process.
You see below one of chaotic orbits of the quadratic map
for c = -2.
As since for unstable orbits coordinate variation δxk
grows under iterations therefore divergence of close orbits can be detected by
positive Lyapunov exponent
Λ = limn → ∞ Ln ,
Ln = 1/n
log|δxn /δxo|.
The Lyapunov exponents L calculated for orbits below are
positive for almost all initial points xo.
Symbolic dynamics and Chaos
Sometimes 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
orbits. Dynamical systems with such complicated orbits are considered
chaotic.
Nonlinear maps
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.
Simple 1D sawtooth and tent
maps demonstarte exponential divergence of close orbits, mixing,
dense set of unstable periodic orbit, ergodicity and invariant
distributions (measures).
More complicated quadratic map has very rich dynamics. For this map
regions of regular and chaotic dynamics are entangled in an intricate manner
(see the bifurcation diagram to the left) and scenarios of transition
to chaos are common for many other dynamical systems.
Dissipative Henon map can have strange attractors
(the right picture). Conservative maps (baker's and standard map) are similar
to Hamiltonian systems.
Contents
Next: Sawtooth map & Bernoulli shifts updated 11 July 2006