Chaos in simple maps
There are several reasons to investigate nonlinear maps
- PC makes very quickly amazing fractal pictures
- Maps dynamics is very rich and complecated
See e.g. tangled orbits in the standard map dynamics
and mixing
- One can study flows dynamics (determined by systems of nonlinear
differential equations) by Poincare maps and classical unstable periodic
orbits determine quantum system spectrum in the quasiclassical limit.
Surprisingly, very simple maps will turn out to yield rather good
qualitative models for behavior in ordinary and partial differential
equations.
Dynamical Chaos
Of cause nonlinear system dynamics is deterministic and is determined
completely by initial conditions. But usually initial coordinates are
known only approximately. Therefore due to exponential divergence of
orbits after some iterations close orbits are dispersed and mixed in
the phase space and it is impossible to predict position of the system.
This phenomenon is called dynamical chaos.
It is supposed that chaotic systems have
- Exponential divergence of close orbits (unstability of bounded orbits
and mixing)
- Dense set of unstable periodic orbit
- Measures. Ergodicity. Decay of correlations (mixing) or positive entropy
All these phenomena can be observed in nonlinear maps (and PC display :)
Types of nonlinear maps
We consider
- noninvertible 1D maps (e.g. quadratic maps) are very simple but
have main features of chaotic systems
- dissipative maps (the Henon map) with strange attractors
- conservative maps (the standard map) similar to Hamiltonian
systems
Contents
Next: Sawtooth map & Bernoulli shifts
updated 14 June 2005