Melnikov's persistence for completely degenerate Hamiltonian systems

TL;DR

This paper investigates the persistence of lower-dimensional invariant tori in completely degenerate Hamiltonian systems using KAM theory and topological degree methods, extending Melnikov's theory to degenerate cases.

Contribution

It introduces a novel approach combining homotopy invariance and quasi-linear KAM iteration to prove invariant tori persistence in degenerate Hamiltonian systems.

Findings

Established conditions for the persistence of invariant tori.

Applied topological degree to handle degeneracy.

Extended Melnikov's theory to new degenerate settings.

Abstract

In this paper, we study the Melnikov's persistence for completely degenerate Hamiltonian systems with the following Hamiltonian \begin{equation*} H(x,y,u,v)=h(y)+g(u,v)+\varepsilon P(x,y,u,v),~~~(x,y,u,v)\in \mathbb{T}^n\times{G}\times \mathbb{R}^d\times \mathbb{R}^d, \end{equation*} where $n\geq2$ and $d\geq1$ are positive integers, $G\subset\mathbb{R}^n$, $g=o(|u|^2+|v|^2)$ admits complete degeneracy and certain transversality, and $\varepsilon P$ is the small perturbation. This is a try in studying lower-dimensional invariant tori in the normal complete degeneracy. Under R\"{u}ssmann-like non-degenerate condition and transversality condition, we apply the homotopy invariance of topological degree to remove the first order terms about $u$ and $v$ and employ the quasi-linear KAM iterative procedure to derive the persistence of lower-dimensional invariant tori.