Variational Structures for Infinite Transition Orbits of Monotone Twist Maps

TL;DR

This paper investigates the chaotic dynamics of area-preserving monotone twist maps, establishing the existence of infinite transition orbits using variational methods, which oscillate infinitely between fixed points.

Contribution

It introduces a variational framework for monotone twist maps and proves the existence of infinite transition orbits, advancing understanding of complex dynamical behaviors.

Findings

Existence of infinite transition orbits proven

Variational methods applied to twist maps

Oscillating trajectories between fixed points established

Abstract

In this paper, we consider chaotic dynamics and variational structures of area-preserving maps. There is a lot of study on the dynamics of their maps and the works of Poincare and Birkhoff are well-known. To consider variational structures of area-preserving maps, we define a special class of area-preserving maps called monotone twist maps. Variational structures determined from twist maps can be used for constructing characteristic trajectories of twist maps. Our goal is to prove the existence of an infinite transition orbit, which represents an oscillating orbit between fixed points infinite times, through minimizing methods.