Introduction

This introductory chapter provides an overview of Arnold diffusion. The famous question called the ergodic hypothesis, formulated by Maxwell and Boltzmann, suggests that for a typical Hamiltonian on a typical energy surface, all but a set of initial conditions of zero measure have trajectories dense in this energy surface. However, Kolmogorov-Arnold-Moser (KAM) theory showed that for an open set of (nearly integrable) Hamiltonian systems, there is a set of initial conditions of positive measure with almost periodic trajectories. This disproved the ergodic hypothesis and forced reconsideration of the problem. For autonomous nearly integrable systems of two degrees or time-periodic systems of one and a half degrees of freedom, the KAM invariant tori divide the phase space. These invariant tori forbid large scale instability. When the degrees of freedoms are larger than two, large scale instability is indeed possible, as evidenced by the examples given by Vladimir Arnold. The chapter explains that the book answers the question of the typicality of these instabilities in the two and a half degrees of freedom case.

STABILITY OF NEARLY-INTEGRABLE SYSTEMS WITH DISSIPATION

We study the stability of a vector field associated to a nearly-integrable Hamiltonian dynamical system to which a dissipation is added. Such a system is governed by two parameters, namely the perturbing and dissipative parameters, and it depends on a drift function. Assuming that the frequency of motion satisfies some resonance assumption, we investigate the stability of the dynamics, and precisely the variation of the action variables associated to the conservative model. According to the structure of the vector field, one can find linear and long-term stability times, which are established under smallness conditions of the parameters. We also provide some applications to concrete examples, which exhibit a linear or long-term stability behavior.