Kolmogorov's Theorem for Degenerate Hamiltonian Systems with Continuous Parameters
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910582903406592 |
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| author | Du, Jiayin Li, Yong Zhang, Hongkun |
| author_facet | Du, Jiayin Li, Yong Zhang, Hongkun |
| contents | In this paper, we investigate Kolmogorov type theorems for small perturbations of degenerate Hamiltonian systems. These systems are index by a parameter $ξ$ as \( H(y,x,ξ) = \langleω(ξ),y\rangle + \varepsilon P(y,x,ξ,\varepsilon) \) where $\varepsilon>0$. We assume that the frequency map, $ω$, is continuous with respect to $ξ$. Additionally, the perturbation function, $P(y,x,\cdot, \varepsilon)$, maintains Hölder continuity about $ξ$. We prove that persistent invariant tori retain the same frequency as those of the unperturbed tori, under certain topological degree conditions and a weak convexity condition for the frequency mapping. Notably, this paper presents, to our understanding, pioneering results on the KAM theorem under such conditions-with only assumption of continuous dependence of frequency mapping $ω$ on the parameter. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2206_05461 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Kolmogorov's Theorem for Degenerate Hamiltonian Systems with Continuous Parameters Du, Jiayin Li, Yong Zhang, Hongkun Dynamical Systems In this paper, we investigate Kolmogorov type theorems for small perturbations of degenerate Hamiltonian systems. These systems are index by a parameter $ξ$ as \( H(y,x,ξ) = \langleω(ξ),y\rangle + \varepsilon P(y,x,ξ,\varepsilon) \) where $\varepsilon>0$. We assume that the frequency map, $ω$, is continuous with respect to $ξ$. Additionally, the perturbation function, $P(y,x,\cdot, \varepsilon)$, maintains Hölder continuity about $ξ$. We prove that persistent invariant tori retain the same frequency as those of the unperturbed tori, under certain topological degree conditions and a weak convexity condition for the frequency mapping. Notably, this paper presents, to our understanding, pioneering results on the KAM theorem under such conditions-with only assumption of continuous dependence of frequency mapping $ω$ on the parameter. |
| title | Kolmogorov's Theorem for Degenerate Hamiltonian Systems with Continuous Parameters |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2206.05461 |