Whitney Smooth Families of Invariant Tori within the Reversible Context 2 of KAM Theory

TL;DR

This paper establishes a general theorem on the persistence of smooth families of invariant tori in a specific reversible context of KAM theory, extending understanding of stability in reversible dynamical systems.

Contribution

It introduces a new theorem proving the persistence of Whitney smooth families of invariant tori in the reversible context 2 of KAM theory, building on prior results for systems with singular matrices.

Findings

Proves persistence of Whitney smooth families of invariant tori in reversible context 2

Extends KAM theory to a new class of reversible systems

Builds on previous work on quasi-periodic stability with singular matrices

Abstract

We prove a general theorem on the persistence of Whitney infinitely smooth families of invariant tori in the reversible context 2 of KAM theory. This context refers to the situation where dim Fix G < (codim T)/2 where Fix G is the fixed point manifold of the reversing involution G and T is the invariant torus in question. Our result is obtained as a corollary of the theorem by H.W.Broer, M.-C.Ciocci, H.Hanssmann, and A.Vanderbauwhede of 2009 concerning quasi-periodic stability of invariant tori with singular "normal" matrices in reversible systems.