Global Melnikov Theory in Hamiltonian Systems with General Time-Dependent Perturbations
Global Melnikov Theory in Hamiltonian Systems with General Time-Dependent Perturbations
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DOI：
10.1007/s00332-018-9461-2
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发表时间：
2017-10
影响因子：
3
通讯作者：
M. Gidea;R. Llave
中科院分区：
文献类型：
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作者：
M. Gidea;R. Llave
本文考虑一个由n个自由度的转子和n个自由度的转子组成的机械系统。旋转体的相空间定义了一个正常双曲不变流形。我们应用一个依赖于时间的扰动，这是不假设是哈密尔顿，或周期，或准周期，因为我们允许相当一般的时间依赖性。扰动的强度由参数给出。对于充分小的，增广流-通过使时间成为一个新的变量-有一个正常的双曲局部不变流形。为了，。我们定义了一个Melnikov型向量，它给出了扰动下稳定流形和不稳定流形的位移的一阶展开式。我们提供了一个明确的公式Melnikov向量的收敛不当积分的扰动沿着同宿轨道的未扰动系统。我们表明，如果扰动满足一些明确的非退化条件，那么稳定和不稳定的流形的，和，分别，相交沿着一个横同宿流形，而且，分裂的和可以明确计算，到一阶，在Melnikov型向量。这意味着，沿着一些同宿轨线的行程产生一个非平凡的增加阶的转子的作用变量，为所有足够小的扰动。我们得到的公式是独立的无扰运动，并给出，在同一时间，对周期，准周期，或一般型轨道的影响。当微扰是哈密顿的，我们表示的影响的微扰，到一阶，在Melnikov潜在的。此外，如果扰动是周期的，我们得到了Melnikov势的非简并条件是一般的。
We consider a mechanical system consisting ofn-penduli and ad-degree-of-freedom rotator. The phase space of the rotator defines a normally hyperbolic invariant manifold. We apply a time-dependent perturbation, which is not assumed to be either Hamiltonian, or periodic, or quasi-periodic, as we allow for rather general time dependence. The strength of the perturbation is given by a parameter. For allsufficiently small, the augmented flow—obtained by making the time into a new variable—has a normally hyperbolic locally invariant manifold. For,. We define a Melnikov-type vector, which gives the first-order expansion of the displacement of the stable and unstable manifolds ofunder 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,and, respectively, intersect along a transverse homoclinic manifold, and, moreover, the splitting ofandcan be explicitly computed, up to the first order, in terms of the Melnikov-type vector. This implies that the excursions along some homoclinic trajectories yield a non-trivial increase of orderin the action variables of the rotator, for all sufficiently small perturbations. The formulas that we obtain are independent of the unperturbed motions in, and give, at the same time, the effects on periodic, quasi-periodic, or general-type orbits. When the perturbation is Hamiltonian, we express the effects of the perturbation, up to the first order, in terms of a Melnikov potential. In addition, if the perturbation is periodic, we obtain that the non-degeneracy conditions on the Melnikov potential are generic.