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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2402.10764 |
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| _version_ | 1866916128503103488 |
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| author | Barbieri, Santiago Farré, Gerard |
| author_facet | Barbieri, Santiago Farré, Gerard |
| contents | We prove that the solutions of Hölder-differentiable Hamiltonian systems, associated to initial conditions in a small ball of radius $ρ>0$ around a Lagrangian, $(γ,τ)-$Diophantine, quasi-periodic torus, are stable over a time $t^{\text{stab}}\simeq 1/(|ρ|^{1+\frac{\ell-1}{τ+1}}|\ln ρ|^{\ell-1})$, where $\ell>2d+1, \ell \in \mathbb R$, is the regularity, and $d$ is the number of degrees of freedom. In the finitely differentiable case (for integer $\ell$), this result improves the previously known effective stability bounds around Diophantine tori. Moreover, by a previous work based on the Anosov-Katok construction, it is known that for any $\varepsilon>0$ there exists a $C^\ell$-Hamiltonian, with $ \ell\ge 3$, admitting a sequence of solutions starting at distance $ρ_n \to 0$ from a $(γ,τ)$-Diophantine torus that diffuse in a time of order $t^{\text{diff}}_n\simeq 1/(|ρ_n|^{1+\frac{\ell-1}{τ+1}+\varepsilon})$.
Therefore the stability estimates that we show are optimal up to an arbitrarily small polynomial correction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10764 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nearly-optimal effective stability estimates around Diophantine tori of Hölder Hamiltonians Barbieri, Santiago Farré, Gerard Dynamical Systems We prove that the solutions of Hölder-differentiable Hamiltonian systems, associated to initial conditions in a small ball of radius $ρ>0$ around a Lagrangian, $(γ,τ)-$Diophantine, quasi-periodic torus, are stable over a time $t^{\text{stab}}\simeq 1/(|ρ|^{1+\frac{\ell-1}{τ+1}}|\ln ρ|^{\ell-1})$, where $\ell>2d+1, \ell \in \mathbb R$, is the regularity, and $d$ is the number of degrees of freedom. In the finitely differentiable case (for integer $\ell$), this result improves the previously known effective stability bounds around Diophantine tori. Moreover, by a previous work based on the Anosov-Katok construction, it is known that for any $\varepsilon>0$ there exists a $C^\ell$-Hamiltonian, with $ \ell\ge 3$, admitting a sequence of solutions starting at distance $ρ_n \to 0$ from a $(γ,τ)$-Diophantine torus that diffuse in a time of order $t^{\text{diff}}_n\simeq 1/(|ρ_n|^{1+\frac{\ell-1}{τ+1}+\varepsilon})$. Therefore the stability estimates that we show are optimal up to an arbitrarily small polynomial correction. |
| title | Nearly-optimal effective stability estimates around Diophantine tori of Hölder Hamiltonians |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2402.10764 |