Topology and Computations in Dynamics
Jean-Philippe Lessard,
(Basque Center of Applied Mathematics, BCAM)
Martes 3 de mayo 2011
Studying and proving the existence of solutions of nonlinear dynamical systems using standard analytic techniques is a challenging problem. In particular, this problem is even more challenging for PDEs, variational problems or functional delay equations which are naturally defined on infinite dimensional function spaces. In the last twenty years, several methods combining the strengths of topology, analysis, and rigorous computations have been introduced to prove the existence of solutions of such nonlinear equations.
Following the same directions, I will first present some of my recent work concerning the development of rigorous computational techniques to prove existence of steady states, time periodic solutions, traveling waves and connecting orbits of dynamical systems. Then, once these dynamical objects are rigorously computed, I will show that it is possible to use topology to organize the information from the above mentioned rigorous computations. The philosophy behind this organization lies in the fact that topological methods like Morse theory can be extremely powerful book-keeping tools. More precisely, the power of Morse theory is that it is defined in terms of low dimensional local objets which are in essence computable (zero-dimensional critical points and one-dimensional connecting orbits), and can be used to characterize global properties of nonlinear problems. Such global properties include for instance the global homology of manifolds, the existence of chaotic dynamics, lower bounds on the number of periodic solutions of a Hamiltonian system or the topology of maximal invariant sets of dynamical systems.