Arnold diffusion of charged particles in ABC magnetic fields

摘要

We prove the existence of diffusing solutions in the motion of a chargedparticle in the presence of an ABC magnetic field. The equations of motion aremodeled by a 3DOF Hamiltonian system depending on two parameters. For smallvalues of these parameters, we obtain a normally hyperbolic invariant manifoldand we apply the so-called geometric methods for a priori unstable systemsdeveloped by A. Delshams, R. de la Llave, and T.M. Seara. We characterizeexplicitly sufficient conditions for the existence of a transition chain ofinvariant tori having heteroclinic connections, thus obtaining globalinstability (Arnold diffusion). We also check the obtained conditions in acomputer assisted proof. ABC magnetic fields are the simplest force-free typesolutions of the magnetohydrodynamics equations with periodic boundaryconditions, so our results are of potential interest in the study of the motionof plasma charged particles in a tokamak.