Computer-assisted applications of KAM theory

Abstract: Stability of Hamiltonian systems is relevant to problems from celestial mechanics, particle accelerators, plasma confinement, quasigeostrophic flows, etc. Motivated by such applications, one is interested in detecting mechanisms of stability in concrete models, and in providing quantitative information on the stable trajectories. KAM theory concerns the existence of quasi-periodic solutions, that are geometrically described as orbits lying inside invariant tori.

In this lecture we will overview a methodology for rigorously detecting quasi-periodic orbits in Hamiltonian systems. The methodology involves analytical, geometrical and computational methods and covers from pen and paper rigorous results to computer-assisted rigorous results, passing through algorithms (and the study of their convergence) and implementations. We will present some ideas for performing computer assisted proofs in this context. In particular, we will see FFT-methods for representing rigorously real-analytic periodic functions, that are used to parameterize tori in phase space. We will see some applications in this context. We will finish the lecture with some other applications and further topics.

analysis of PDEsclassical analysis and ODEsfluid dynamics

Audience: researchers in the topic

Online Northeast PDE and Analysis Seminar