A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold
A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold
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DOI：
10.3934/dcds.2020166
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
M. Gidea;Rafael de la Llave;T. M. Seara
中科院分区：
文献类型：
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作者：
M. Gidea;Rafael de la Llave;T. M. Seara
我们描述了一种显示哈密顿系统不稳定性的最新方法。该方法的主要假设是满足一些显式的横向条件——可以通过有限计算在具体系统中验证这些条件。特别是，对于可积哈密顿系统的几种类型的扰动，只需检查某些梅尔尼科夫型积分是否具有非简并零点即可验证该假设。这适用于 \begin{document}$ C^r $\end{document} 拓扑中的 Baire 通用扰动集，对于 \begin{document}$ r \in [3, \infty) \cup \{\omega\} $\end{document} 。我们的方法不要求未扰动的哈密顿系统是凸的，或者扰动是多项式的，这些都是非通用属性。只要验证了横向条件，就可以得出结论，存在轨道，这些轨道会改变作用坐标，其量与扰动的大小无关。事实上，我们可以获得遵循动作空间中任何路径的轨道，直到误差随着扰动的大小而减小。
We describe a recent method to show instability in Hamiltonian systems. The main hypothesis of the method is that some explicit transversality conditions – which can be verified in concrete systems by finite calculations – are satisfied. In particular, for several types of perturbations of integrable Hamiltonian systems, the hypothesis can be verified by just checking that some Melnikov-type integrals have non-degenerate zeros. This holds for Baire generic sets of perturbations in the \begin{document}$ C^r $\end{document} -topology, for \begin{document}$ r \in [3, \infty) \cup \{\omega\} $\end{document} . Our method does not require that the unperturbed Hamiltonian system is convex, or that the perturbation is polynomial, which are non-generic properties. Provided that the transversality conditions are verified, one concludes the existence of orbits which change the action coordinate by a quantity independent of the size of the perturbation. In fact, one can obtain orbits that follow any path in action space, up to an error decreasing with the size of the perturbation.