A course on Dynamical Systems

A course on Dynamical Systems

Date

Tue-Thu 15h-17h, from September 26, 2013

Registration

–

Location

Room 005, Facultat de Matemàtiques i Estadística, UPC

The lectures will be chosen among the following list of topics, according to the interests and backgrounds of the students and adapting them to the list of short courses on nearby topics given in the Math centers of the Barcelona area.

Date

Tue-Thu 15h-17h, from September 26, 2013

Registration

–

Location

Room 005, Facultat de Matemàtiques i Estadística, UPC

The lectures will be chosen among the following list of topics, according to the interests and backgrounds of the students and adapting them to the list of short courses on nearby topics given in the Math centers of the Barcelona area.

Lecturers

Amadeu Delshams (UPC)

Carles Simó (UB)

Contents
  • Preliminary statements about discrete and continuous dynamical systems. The skeleton of a system: invariant objects.
  • Some paradigmatic examples: Henon, standard and separatrix maps, Duffing equations, Rossler model, etc.
  • Fixed points/stationary states. Local analysis of stability. A first approach to Normal Forms. Bifurcations and unfoldings.
  • Hyperbolicity: stable and unstable manifolds. Normally hyperbolic invariant manifolds (NHIM). Homoclinic and heteroclinic phenomena.
  • Chaos: symbolic dynamics, topological entropy, invariant Cantorian sets.
  • Introduction to invariant measures and to ergodic theory.
  • Averaging and normal forms. The problem of small divisors and the Diophantine conditions. Slow-fast systems.
  • Hamiltonian systems. Integrability versus non-integrability. KAM theory. Destruction of invariant tori. Aubry-Mather invariant sets. Diffusion mechanisms. The role of the splitting of the invariant manifolds of NHIM.
  • Some topics in Celestial Mechanics and Astrodynamics. Invariant objects of the N-body problem, and applications to Astronomy and Space Science.
  • Dissipative systems. Invariant measures, and strange attractors.
  •  Evolution PDE analyzed using dynamical systems tools.
Bibliography
  • V.I. Arnold and A. Avez. Ergodic problems of classical mechanics. W. A. Benjamin, Inc., New York-Amsterdam, 1968.
  • V.I. Arnold, V.V. Kozlov, and A.I. Neishtadt. Mathematical aspects of classical and celestial mechanics.  Springer-Verlag, Berlin, 2006.
  • B. Hasselblatt and A. Katok. A first course in dynamics.
    Cambridge University Press, New York, 2003.
  • K.R. Meyer, G.R. Hall, and D. Offin.Introduction to Hamiltonian dynamical systems and the N-body problem.  Springer, NY, 2009.
Share This