Mixed Dynamics in a Parabolic Standard Map

TL;DR

This paper investigates a family of smooth, symplectic maps on the torus that exhibit both chaotic and regular dynamics, using numerical and analytical methods to explore their mixed Lyapunov behavior.

Contribution

It introduces a new family of diffeomorphisms with mixed dynamical behavior, expanding understanding of coexistence of chaos and regularity in symplectic maps.

Findings

Existence of positive measure sets with positive Lyapunov exponents

Existence of positive measure sets with zero Lyapunov exponents

Demonstration of mixed dynamics through analytical and numerical evidence

Abstract

We use numerical and analytical tools to demonstrate arguments in favor of the existence of a family of smooth, symplectic diffeomorphisms of the two-dimensional torus that have both a positive measure set with positive Lyapunov exponent and a positive measure set with zero Lyapunov exponent. The family we study is the unfolding of an almost-hyperbolic diffeomorphism on the boundary of the set of Anosov diffeomorphisms, proposed by Lewowicz.