Chaotic Dynamics of Charged Particles in Magnetic Fields with Singular Forces

Abstract

We study charged-particle dynamics under a non-Lipschitz velocity-dependent force with magnetic and damping effects, focusing on continuation through zero velocity and chaos in the velocity direction. For a time-independent directional force and fixed parameters, we introduce a $C^1$ regularization. When an unregularized trajectory reaches zero velocity in finite time, inner rescaling yields convergence to a reference orbit within the regularization ball, while the regularized trajectories converge locally uniformly outside it to a unique continuation. For the directional dynamics, stereographic projection yields a parameterized Euclidean system. By choosing the skew-symmetric magnetic part, we obtain a nonautonomous parameterized center manifold reduction with a prescribed finite Taylor jet. Under an additional factorization assumption, the reduced dynamics admits Shilnikov homoclinic orbits and associated chaos.