KAM algorithms for Lagrangian tori in time-quasi-periodic Hamiltonian systems

Abstract

In this paper, we present an algorithm to compute (fiberwise) Lagrangian tori in periodic and quasi-periodic Hamiltonian systems, whose convergence is proved in [1], and an algorithm to continue them with respect to parameters. We exhibit the algorithm with two models. The first is a tokamak model [2,3], which proposes a control method to create barriers to the diffusion of magnetic field lines through a small modification in the magnetic perturbation. In this context, we additionally estimate the breakdown threshold of the barrier in the tokamak model under study. The second model is related to the vorticity defect model in fluid dynamics and to the single wave model in plasma physics [4], both of which arise from successive reductions of vorticity-velocity interactions.