Dynamical Systems 1

First part: Chaos

The goal of the second part of the course is to give a Mathematical description of the chaotic behaviour related to the homoclinic intersection. In particular we will prove that a weakly forced pendulum exhibits chaotic orbits.

Stable manifold theorem: existence and regularity of the stable
manifold at a hyperbolic equilibrium point; global stable
manifold. The problem of global behviour of the stable
manifold. Homoclininc intersection. Poincaré's description of chaotic dynamics. Hyperbolic sets and chaotic behviour: Definition and elementary examples (Arnold's cat, Baker's map). Symbolic dynamic, Bernoulli shift. Some examples of equivalence between a dynamical system and coin's toss. Hyperbolicity of the homoclinic intersection set (in the case of transversal intersection). Shadowing Lemma and applications: Statement and proof. Application to the chaotic behaviour in the neighbourhood of a homoclinic intersection point. Melnikov's method for the proof of the existence of transversal
homoclinic intersections. Application to the forced pendulum.

Second part: Poincare' theorem of continuation of periodic orbits. Formulation of the problem in terms of Poincare' map. Streightening theorem, Regularity of Poincare' map Linearization of the Poincare' map, Floquet multipliers, Poincare' Theorem. Applications: Lyapunov center theorem, MacKay and Aubry's Theorem on breathers in lattices of coupled oscillators.

Third part: order.

This part is around Siegel linearization theorem.

Classical theory of linearization of a system in the neighbourhood of an equilibrium point. Normal form, formal theory. The problem: does there exist a coordinate transformation conjugating the system to its linear part? Small denominators. Poincare' domains, Siegel domains, Diophantine type theorems. Siegel theorem.