The course aims to give an introduction to the theory of dynamical systems, with particular emphasis on the coexistence between ordered and chaotic motions in the same system. The simplest methods of perturbation theory and of ergodic theory will be described, and the connections with the dynamical foundations of statistical mechanics illustrated. The course comprises a laboratory for the numerical study of simple models.
Expected learning outcomes
At the end of the course the student will
1) get the notion of the coexistence between ordered and chaotic motions, having observed it in numerical simulations 2) know the notions of hyperbolic fixed point, stable and unstable manifolds, homoclinic points and homoclinic orbits. 3) be able to find, by numerical computations, the stable and unstable manifold for the hyperbolic fixed point of the standard map, and computed the homoclinic orbits. 4) know the statement of the stable manifold theorem. 5) know the model of the torus rotation, understanding the difference between the ration and irrational cases, and know the Arnold's cat map. 6) know the basic notion about the ergodic system, and be able to prove the ergodicity of the irrational torus rotation, and to prove the mixing character of the Arnold's cat map. 7) know the basic notion about the Hamiltonian perturbation theory, and the statement of KAM theorem. 8) be able to prove the stable manifold theorem.
Lesson period: Second semester
(In case of multiple editions, please check the period, as it may vary)