Homoclinic orbits to invariant tori in Hamiltonian systems
Homoclinic orbits to invariant tori in Hamiltonian systems
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发表时间：
1998
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通讯作者：
A. Valdés;P. G. Serrés
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作者：
A. Valdés;P. G. Serrés
我们考虑具有不变量环面和重合须的可积哈密顿系统（如一些旋转体和钟摆）的扰动。我们的目标是测量扰动晶须之间的分裂距离，重点是检测它们的交叉点，这些交叉点会产生到扰动环面的同斜轨道。提出了一种考虑晶须的拉格朗日性质的几何方法。这样，分裂距离就是分裂势的梯度。在常规情况下（也称为优先级不稳定：须状环面的李雅普诺夫指数保持固定），分裂势很好地近似于梅尔尼科夫势。该方法被设计为奇异情况研究的第一步（也称为优先稳定：当扰动趋于零时，须状环面的Lyapunov指数趋于零）。
We consider a perturbation of an integrable Hamiltonian system which possesses invariant tori with coincident whiskers (like some rotators and a pendulum). Our goal is to measure the splitting distance between the perturbed whiskers, putting emphasis on the detection of their intersections, which give rise to homoclinic orbits to the perturbed tori. A geometric method is presented which takes into account the Lagrangian properties of the whiskers. In this way, the splitting distance is the gradient of a splitting potential. In the regular case (also known as a priori-unstable: the Lyapunov exponents of the whiskered tori remain fixed), the splitting potential is well- approximated by a Melnikov potential. This method is designed as a first step in the study of the singular case (also known as a priori-stable: the Lyapunov exponents of the whiskered tori approach to zero when the perturbation tends to zero).