Melnikov's persistence for completely degenerate Hamiltonian systems

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 $n\ge 2$ and $d\ge 1$ are positive integers, $G\subset {\mathbb{R}}^n$, $g=o(|u{|}^2+|v{|}^2)$ admits complete degeneracy and certain transversality, and $\epsilon P$ is the small perturbation. This is a try in studying lower-dimensional invariant tori in the normal complete degeneracy. Under Rüssmann-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.