Mechanisms of Instability in Nearly Integrable Hamiltonian Systems

There are many systems that appear in applications that have negligible friction, like in celestial mechanics and Astrodynamics, motion of charged particles in magnetic fields, chemical reactions, etc.

A general model for this kind of systems is to consider time periodic perturbations of integrable Hamiltonian systems with $2$ or more degrees of freedom.

One problem that has attracted attention for a long time is whether the effect of perturbations accumulate over time and lead to large effects (instability) or whether these effects average out (stability).

The first result in proving instability for this model was presented by Arnold in 1964, but the perturbation considered was not general enough (there do not appear "large gaps" in the resonant tori).

In this talk we present some mechanisms that cause instabilities for general perturbations if the unperturbed systems considered are a priori-unstable, that is, they can be written as a product of $d$ rotators and $n$ penduli, $d\ge 1$, $n\ge 1$.

The main technique is to develop a toolkit to study, in a unified way, tori of different topologies and their invariant manifolds, their intersections as well as shadowing properties of these bi-asymptotic orbits. Part of this toolkit is to unify standard techniques (normally hyperbolic manifolds, KAM theory, averaging theory) so that they can work together. A fundamental tool used here is the scattering map of normally hyperbolic invariant manifolds.

The conditions needed are explicit. This makes it possible to show that this phenomenon occurs in concrete examples, as the restricted three body problem in celestial mechanics.

Tere M. Seara, Universitat Politecnica de Catalunya, Spain

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