Upper bounds on Arnold diffusion times via Mather theory

Abstract

This paper considers five examples of hamiltonian systems for which the existence of Arnold's mechanism for diffusion has been shown. These systems have in common that each of them is the perturbation that couples a number of rotators to a pendulum. The main result is that, for all systems considered and for all suffiently small values of the perturbation paramenter, there are orbits whose action variables have a drift of order one in a time which is inversely proportional to the splitting of the homoclinic whiskers. The paper reviews the necessary results concerning the Hamiltonian and uses them to apply Mather theory to the estimate on the drift. These results are important in veryfying the optimality of Nekhoroshev-type theorems

Measure of the exponential splitting of the homoclinic tangle in four-dimensional symplectic mappings

“…The general mechanisms which produce chaos and diffusion in the Arnold web is still the subject of numerical and theoretical studies (Chierchia and Gallavotti 1998;Bessi 1996Bessi , 1997Bessi et al 2001;Berti and Bolle 2002;Berti et al 2003;Lega et al 2003;Guzzo et al 2005;Froeschlé et al 2005;Todorovic et al 2008). In this article we study one of the most popular mechanisms for the production of chaos and diffusion in quasi-integrable systems, which is related to the so called homoclinic tangle of the hyperbolic orbits in the single resonances.…”

Hamiltonian Systems: Stable or Unstable?

“…From this paper, one can see that the Lagrangian action of orbits attains its local minimum along the diffusion orbits constructed by Arnold. One can see the more clear statement in their following work [12].…”

An example of Arnold diffusion for near-integrable Hamiltonians

“…7 In what follows we only consider the case when z-components of θ 0 and θ 1 differ by 2. From now on, we choose c = (c 1 , c 2 , 0) with |c| = 1.…”