Melnikov's persistence for completely degenerate Hamiltonian systems
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916401612062720 |
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| author | Du, Jiayin Ji, Shuguan Li, Yong |
| author_facet | Du, Jiayin Ji, Shuguan Li, Yong |
| contents | 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 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2301_00206 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Melnikov's persistence for completely degenerate Hamiltonian systems Du, Jiayin Ji, Shuguan Li, Yong Dynamical Systems 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 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. |
| title | Melnikov's persistence for completely degenerate Hamiltonian systems |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2301.00206 |