Applied Dynamical Systems Theory

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:

1. 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
2. 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
3. Probabilistic properties of one-dimensional dynamics: Frobenius-Perron equation and -operator, partitions, invariant measure, absolute continuity, SRB measure
4. 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