Dual Lindstedt series and KAM theorem

TL;DR

This paper introduces a dual Lindstedt series for Hamiltonians with large perturbations, leading to a dual KAM theorem that explains the reformation of invariant tori and the transition between regular and chaotic dynamics.

Contribution

It establishes the existence of a dual Lindstedt series and a dual KAM theorem applicable to strongly perturbed Hamiltonian systems, with numerical validation on a harmonic oscillator.

Findings

Tori reforming limits chaos in certain Hamiltonian systems.

Dual Lindstedt series can be derived by interchanging roles of perturbation and unperturbed system.

Numerical evidence of tori reformation in a perturbed harmonic oscillator.

Abstract

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.