对偶Lindstedt级数与KAM定理
Dual Lindstedt series and KAM theorem
摘要：
我们证明了存在一个Lindstedt系列，当一个哈密顿量是由一个扰动驱动到无穷大。这个级数似乎是对偶的标准Lindstedt级数，因为它可以通过交换扰动和未扰动系统的作用得到。这个对偶序列的存在意味着一个对偶KAM定理成立，并且当一个前导阶哈密顿量存在时，它是非退化的，环面改造的效果可以观察到一个系统从规则运动到充分发展的混沌，再回到规则运动，再现不变环面。我们将这些结果应用到一个扰动谐振子证明数字环面改造的外观。Tori重整作为一种将混沌行为限制在有限参数空间范围内的效应出现在某些哈密顿系统中。对偶KAM定理，如这里证明的，适用时的扰动，结合一个动力学项，再次提供了一个可积系统。
英文摘要：
We prove that exists a Lindstedt series that holds when a Hamiltonian is driven by a perturbation going to infinity. This series appears to be dual to a standard Lindstedt series as it can be obtained by interchanging the role of the perturbation and the unperturbed system. The existence of this dual series implies that a dual KAM theorem holds and, when a leading order Hamiltonian exists that is non degenerate, the effect of tori reforming can be observed with a system passing from regular motion to fully developed chaos and back to regular motion with the reappearance of invariant tori. We apply these results to a perturbed harmonic oscillator proving numerically the appearance of tori reforming. Tori reforming appears as an effect limiting chaotic behavior to a finite range of parameter space of some Hamiltonian systems. Dual KAM theorem, as proved here, applies when the perturbation, combined with a kinetic term, provides again an integrable system.
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