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Global Melnikov Theory in Hamiltonian Systems with General Time-dependent Perturbations. (arXiv:1710.01849v1 [math.DS])
来源于:arXiv
We consider a mechanical system consisting of $n$ penduli and a
$d$-dimensional generalized rotator subject to a time-dependent perturbation.
The perturbation is not assumed to be either Hamiltonian, or periodic or
quasi-periodic. The strength of the perturbation is given by a parameter
$\epsilon\in\mathbb{R}$. For all $|\epsilon|$ sufficiently small, the augmented
flow has a $(2d + 1)$-dimensional normally hyperbolic locally invariant
manifold $\tilde\Lambda_\epsilon$.
We define a Melnikov vector, which gives the first order expansion of the
displacement of the stable and unstable manifolds of $\tilde\Lambda_0$ under
the perturbation. We provide an explicit formula for the Melnikov vector in
terms of convergent improper integrals of the perturbation along homoclinic
orbits of the unperturbed system.
We show that if the perturbation satisfies some explicit non-degeneracy
conditions, then the stable and unstable manifolds of $\tilde\Lambda_\epsilon$,
$W^s(\tilde\Lambda_\epsilon)$ and $W 查看全文>>