Dynamical Systems 1

The program illustrated in the following is a rather long list of topics and it is unlikley that all of them will be addressed. We nevertheless expect to be able to deal with most of the first two parts.

PERIODIC ORBITS
* Poincaré-Bendixson theory; Poincaré local sections; limit sets and their properties. Existence of limit cycles in Liénard systems.
* Fixed point methods; Brouwer theorem.
* Continuation of periodic orbits: the Poincarè theorem. The Poincarè map and its linearization and eigenvalues. Floquet multipliers.
* The Lyapunov center theorem (Periodic orbits near equilibrium points). Applications to the three body problem.
* Poincaré-Birkhoff fixed point theorem; existence of fixed points and periodic orbits for area-preserving twist homeomorphisms of the annulus. Billiard dynamics.
* Dispersion and existence of breathers solutions in infinite non linear chain of particles.
* Periodic orbits in partial differential equations

CHAOS
* Stable and unstable manifolds. Stable manifolds Theorem.
* Homoclinic intersections, Chaotic Dynamics and hyperbolic sets.
* Shadowing Lemma and chaos in the vicinity of homoclinic intersections.
* Chaotic dynamics for the pendulum with a periodic external force. The Melnikov's method

NORMAL FORM THEORY AND SIEGEL THEOREM
* Formal theory. Problem of existence of a change of variables reducing and ordinary differential equation to its linear part. Resonances. Poincarè theorem on the existence of a "formal normal form transformation" eliminating the nonlinear part of the equation.
* Small divisors and Siegel Theorem. Poincare' and Siegel domains. Non-resonance, diophantine conditions. Measure estimates for the small divisors satisfying diophantine conditions. Proof of the Siegel theorem and quadratic scheme