# 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
*d***x**/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