- 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.

- 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

- 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

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