Journal of Nonlinear Science

, Volume 27, Issue 3, pp 721–774 | Cite as

Arnold Diffusion of Charged Particles in ABC Magnetic Fields



We prove the existence of diffusing solutions in the motion of a charged particle in the presence of ABC magnetic fields. The equations of motion are modeled by a 3DOF Hamiltonian system depending on two parameters. For small values of these parameters, we obtain a normally hyperbolic invariant manifold and we apply the so-called geometric methods for a priori unstable systems developed by A. Delshams, R. de la Llave and T.M. Seara. We characterize explicitly sufficient conditions for the existence of a transition chain of invariant tori having heteroclinic connections, thus obtaining global instability (Arnold diffusion). We also check the obtained conditions in a computer-assisted proof. ABC magnetic fields are the simplest force-free-type solutions of the magnetohydrodynamics equations with periodic boundary conditions, and can be considered as an elementary model for the motion of plasma-charged particles in a tokamak.


Motion of charges in magnetic fields Hamiltonian dynamical systems Arnold diffusion Global instability Heteroclinic connections 

Mathematics Subject Classification

34C20 37C29 37D05 37J40 



The authors are very grateful to A. Delshams, M. Guardia, A. Haro, G. Huguet, R. de la Llave and T.M. Seara for useful discussions and suggestions. We especially want to thank T.M. Seara for her patience and kindness answering several questions on the papers (Delshams et al. 2006, 2008, 2016). The authors are supported by the ERC Starting Grant 335079. This work is supported in part by the ICMAT–Severo Ochoa Grant SEV-2015-0554 (MINECO) and the Grants MTM2012-3254 (A.L.) and 2014SGR1145 (A.L.).


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Consejo Superior de Investigaciones CientíficasInstituto de Ciencias MatemáticasMadridSpain

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