KAM theorem on modulus of continuity about parameter
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2022
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| _version_ | 1866909317557387264 |
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| author | Tong, Zhicheng Du, Jiayin Li, Yong |
| author_facet | Tong, Zhicheng Du, Jiayin Li, Yong |
| contents | In this paper, we study the Hamiltonian systems $ H\left( {y,x,ξ,\varepsilon } \right) = \left\langle {ω\left( ξ\right),y} \right\rangle + \varepsilon P\left( {y,x,ξ,\varepsilon } \right) $, where $ ω$ and $ P $ are continuous about $ ξ$. We prove that persistent invariant tori possess the same frequency as the unperturbed tori, under certain transversality condition and weak convexity condition for the frequency mapping $ ω$. As a direct application, we prove a KAM theorem when the perturbation $P$ holds arbitrary Hölder continuity with respect to parameter $ ξ$. The infinite dimensional case is also considered. To our knowledge, this is the first approach to the systems with the only continuity in parameter beyond Hölder's type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_04383 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | KAM theorem on modulus of continuity about parameter Tong, Zhicheng Du, Jiayin Li, Yong Dynamical Systems 37J40 (Primary), 58F27 (Secondary) In this paper, we study the Hamiltonian systems $ H\left( {y,x,ξ,\varepsilon } \right) = \left\langle {ω\left( ξ\right),y} \right\rangle + \varepsilon P\left( {y,x,ξ,\varepsilon } \right) $, where $ ω$ and $ P $ are continuous about $ ξ$. We prove that persistent invariant tori possess the same frequency as the unperturbed tori, under certain transversality condition and weak convexity condition for the frequency mapping $ ω$. As a direct application, we prove a KAM theorem when the perturbation $P$ holds arbitrary Hölder continuity with respect to parameter $ ξ$. The infinite dimensional case is also considered. To our knowledge, this is the first approach to the systems with the only continuity in parameter beyond Hölder's type. |
| title | KAM theorem on modulus of continuity about parameter |
| topic | Dynamical Systems 37J40 (Primary), 58F27 (Secondary) |
| url | https://arxiv.org/abs/2210.04383 |