Arnold diffusion in a model of dissipative system

TL;DR

This paper investigates Arnold diffusion in a dissipative rotator-pendulum system, demonstrating conditions under which energy growth occurs despite dissipation, extending classical diffusion results to conformally symplectic systems.

Contribution

It introduces explicit conditions on dissipation for Arnold diffusion to occur in a non-symplectic, dissipative system, expanding the understanding of energy diffusion beyond conservative models.

Findings

Energy growth persists with small dissipation under certain conditions.

Explicit dissipation parameters are identified for diffusion.

The mechanism can be adapted to more general couplings.

Abstract

For a mechanical system consisting of a rotator and a pendulum coupled via a small, time-periodic Hamiltonian perturbation, the Arnold diffusion problem asserts the existence of `diffusing orbits' along which the energy of the rotator grows by an amount independent of the size of the coupling parameter, for all sufficiently small values of the coupling parameter. There is a vast literature on establishing Arnold diffusion for such systems. In this work, we consider the case when an additional, dissipative perturbation is added to the rotator-pendulum system with coupling. Therefore, the system obtained is not symplectic but conformally symplectic. We provide explicit conditions on the dissipation parameter, so that the resulting system still exhibits energy growth. The fact that Arnold diffusion may play a role in systems with small dissipation was conjectured by Chirikov. In this work, the coupling is carefully chosen, however the mechanism we present can be adapted to general couplings and we will deal with the general case in future work.