Constructive Methods in KAM Theory for Quasi-Periodic Time-Dependent Hamiltonian Systems and Applications

Abstract

[eng] In this work, we prove a KAM theorem and present an algorithm formulated in an a-posteriori format, using the parameterization method to look invariant tori in non-autonomous Hamiltonian systems with n degrees of freedom that depend periodically or quasi-periodically (QP) on time, with f external frequencies. Such a system is described by a Hamiltonian function in the 2n-dimensional phase space, M, that depends also on f angles, ϕ ∈ TR. We take advantage of the fibbered structure of the extended phase space M × TR. As a result of our approach, the parameterization of tori requires the last f variables, to be precise ϕ, while the first 2n components are determined by an invariance equation. This reduction decreases the dimension of the problem where the unknown is a parameterization from 2(n + f) to 2n. We employ a quasi-Newton method, in order to prove the KAM theorem. This iterative method begins with an initial parameterization of an approximately invariant torus, meaning it approximately satisfies the invariance equation. The approximation is refined by applying corrections that reduce quadratically the invariance equation error. This process converges to a torus in a complex strip of size ρ∞, provided suitable Diophantine (γ, τ ) conditions and a non-degeneracy condition on the torsion are met. Given the nature of the proof, this provides a numerical method that can be effectively implemented on a computer, We exhibit the algorithm with two models. The first is a Tokamak model [CVC+05, VL21], which proposes a control method to create barriers to the diffusion of magnetic field lines through a small modification in the magnetic perturbation. The second model [dCN00], known as the vorticity defect model, describes the nonlinear evolution of localized vorticity perturbations in a constant vorticity flow. This model was originally derived in the context of plasma physics and fluid dynamics.