Aubry-Mather Theory for Conformally Symplectic Systems

TL;DR

This paper extends Aubry-Mather theory to conformally symplectic systems, revealing invariant sets that influence the systems' long-term behavior and their attracting or repelling properties.

Contribution

It introduces an Aubry-Mather framework for dissipative conformally symplectic systems, establishing the existence and properties of invariant sets analogous to the conservative case.

Findings

Existence of Aubry and Mather sets in conformally symplectic systems

Analysis of the attracting and repelling nature of these sets

Role of invariant sets in the asymptotic dynamics of dissipative systems

Abstract

In this article we develop an analogue of Aubry-Mather theory for a class of dissipative systems, namely conformally symplectic systems, and prove the existence of interesting invariant sets, which, in analogy to the conservative case, will be called the Aubry and the Mather sets. Besides describing their structure and their dynamical significance, we shall analyze their attracting/repelling properties, as well as their noteworthy role in driving the asymptotic dynamics of the system.