Regular motion and symmetry in the relativistic standard map

In nonlinear dynamical systems, area-preserving maps were investigated extensively as a useful method for characterizing the non-integrable Hamiltonian systems. Especially, the nonrelativistic acceleration of charged particles by an infinite sequence of constant amplitude longitudinal waves with equally spaced phase velocities is represented by the standard map. This map exhibits regular and chaotic motion and was studied in various fields of physics. The central problem of the standard map is a transport process under the coexistence of regular motion and chaos. Recently, Chernikov et al., introduced the relativistic generalization of the standard map. The chaotic motion is restricted to the vicinity of the fixed points and the breakup of the last KAM torus occurs at higher wave amplitude than that for the standard map. The purpose is to clarify the relativistic effects on the nonlinear motion of particles by varying the wave phase velocity in a wide range and to discuss properties of the regular motion by constructing the families of symmetry lines.