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 $ C^r $-topology, for $ r \in [3, \infty) \cup \{\omega\} $. 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.

中文翻译：

---

哈密​​顿系统中不稳定性的一般机制：沿常双曲不变流形跳过

我们描述了一种新方法来显示哈密顿系统中的不稳定性。该方法的主要假设是，可以满足某些明确的横向条件（可以通过有限计算在混凝土系统中进行验证）。特别是对于可微积分哈密顿系统的几种类型的摄动，可以通过仅检查某些条件来验证该假设。 Melnikov型积分具有非简并零。对于$ C ^ r $拓扑中的Baire通用扰动集，在[3，\ infty）\ cup \ {\ omega \} $中，$ r \ in成立。我们的方法不需要非扰动的哈密顿系统是凸的，也不需要扰动是多项式的，这是非一般性质。只要证明了横向条件，一个结论是存在轨道，该轨道改变作用坐标的量与扰动的大小无关。实际上，人们可以获得在动作空间中遵循任何路径的轨道，误差随着扰动的大小而减小。