The Poincare'-Nekhoroshev map

TL;DR

This paper introduces a generalized Poincaré-Nekhoroshev map, extending the classical Poincaré map, to analyze the persistence and bifurcation of invariant tori in Hamiltonian systems, providing new insights into their stability and behavior.

Contribution

It presents a novel generalization of the Poincaré map inspired by Nekhoroshev's work, with applications to invariant tori in Hamiltonian dynamics.

Findings

The generalized map helps analyze invariant tori persistence.

It offers new methods for studying bifurcations.

Properties of the map enhance understanding of Hamiltonian stability.

Abstract

We study a generalization of the familiar Poincar\'e map, first implicitely introduced by N.N. Nekhoroshev in his study of persistence of invariant tori in hamiltonian systems, and discuss some of its properties and applications. In particular, we apply it to study persistence and bifurcation of invariant tori.