Integrability via Reversibility

TL;DR

This paper introduces a class of reversible, volume-preserving systems that exhibit robust integrability with invariant tori, and also contains systems with positive Lyapunov exponents, demonstrating complex dynamical behavior.

Contribution

It presents a new class of reversible systems with proven robust integrability and explores their properties, including coexistence of chaos and regular motion.

Findings

Invariant tori fill an open dense subset of the phase space.

Robust integrability persists under perturbations within the class.

Existence of systems with positive Lyapunov exponents coexisting with invariant tori.

Abstract

A class of left-invariant second order reversible systems with functional parameter is introduced which exhibits the phenomenon of robust integrability: an open and dense subset of the phase space is filled with invariant tori carrying quasi-periodic motions, and this behavior persists under perturbations within the class. Real-analytic volume preserving systems are found in this class which have positive Lyapunov exponents on an open subset, and the complement filled with invariant tori.