### Page Content

### to Navigation

## Course Outline

This course introduces to fundamental properties of chaotic dynamical systems. It is based on rigorous mathematical concepts applied to time-discrete one-dimensional maps. These models are easy to understand and analytically tractable.

**Topics:**

**Simple examples of dynamical systems:**driven nonlinear pendulum, bouncing ball, billiards, kicked rotor, standard map, Bernoulli shift, tent map, logistic map, rotation on the circle, piecewise linear maps**Topological properties of one-dimensional maps:**cobweb plots, periodic points, periodic orbits, stability analysis, bifurcations, expanding/contracting maps, hyperbolicity, topological transitivity, sensitive dependence on initial conditions**Probabilistic properties of one-dimensional dynamics:**Frobenius-Perron equation and -operator, partitions, invariant measure, absolute continuity, SRB measure**Assessing chaos, and related properties:**Poincaré-Bendixson theorem, Poincaré surface of section, rigorous definitions of chaos, Lyapunov exponents, ergodicity, mixing, entropies, Pesin’s theorem, escape rates, Cantor set, fractals