Tracing KAM tori in presymplectic dynamical systems

TL;DR

This paper develops a KAM theorem for presymplectic systems, enabling the precise identification of invariant tori under certain conditions, with implications for understanding complex dynamical behaviors.

Contribution

It introduces an a posteriori KAM theorem tailored for presymplectic systems, including explicit non-degeneracy conditions and parameter space reduction for exact systems.

Findings

Established a KAM theorem for presymplectic systems.

Demonstrated the existence of invariant tori under specified conditions.

Reduced parameter space dimension for exact presymplectic systems.

Abstract

We present a KAM theorem for presymplectic dynamical systems. The theorem has a " a posteriori " format. We show that given a Diophantine frequency $\omega$ and a family of presymplectic mappings, if we find an embedded torus which is approximately invariant with rotation $\omega$ such that the torus and the family of mappings satisfy some explicit non-degeneracy condition, then we can find an embedded torus and a value of the parameter close to to the original ones so that the torus is invariant under the map associated to the value of the parameter. Furthermore, we show that the dimension of the parameter space is reduced if we assume that the systems are exact.