Dual Lindstedt series and Kolmogorov–Arnol’d–Moser theorem

Abstract

We prove that there 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 Kolmogorov–Arnol'd–Moser (KAM) theorem holds, and when a leading order Hamiltonian exists, which is nondegenerate, 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 that limits 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.