Melnikov's persistence for completely degenerate Hamiltonian systems

2022/12/31 by Du, Jiayin, Ji, Shuguan, Li, Yong
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2301.00206

Abstract

In this paper, we study the Melnikov's persistence for completely degenerate Hamiltonian systems with the following Hamiltonian H(x,y,u,v)=h(y)+g(u,v)+ε P(x,y,u,v),~~~(x,y,u,v)∈ \mathbbTⁿ×G× ℝ^d× ℝ^d, where n≥2 and d≥1 are positive integers, G⊂ℝⁿ, g=o(|u|²+|v|²) admits complete degeneracy and certain transversality, and ε 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.

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