Introduction to Chaotic Dynamical Systems

Course content

Mappings and Orbits. Periodic orbits. Linear mappings. Differentiable mappings. Fixed points and periodic points. Eventually periodic points. Family of mappings: Bifurcations. The quadratic map. Symbolic dynamics. Ingredients of chaos: Sensitivity. Dense orbits and transitivity. Schwarzian derivatives. Wiggly iterates. Cantor set and chaos. The logistic map. Period-three points and chaos. Period-doubling bifurcation: method of bifurcation diagrams. Positive Lyapunov exponents. The Two-dimensional chaos: the Henon map, the horseshoe map. Alpha unpredictability. Ultra Poincaré chaos. Alpha labeling. Abstract similarity map. Abstract similarity sets. Chaos in stochastic processes. Homoclinic chaos. Hyberbolicity and chaos.