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.
Click mouse to get a new orbit marked by the red color
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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
Λ = limn→∞ Ln ,
Ln = 1/n
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.
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.
The blue square mixing by the standard map. N is number of iterations.
Press "-" button to trace the mixing process.
Symbolic dynamics, Smale horseshoe and
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
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.
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).
Next: Sawtooth map & Bernoulli shifts
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updated 12 July 07