Special relativity effects on chaos and periodic orbits - a semianalytic approach

In the study of nonlinear dynamics and chaos, analytically solvable physical models are rare. We investigate the classical motion of a trapped relativistic particle, subject to an electric and a magnetic field. When the directions of the electric and magnetic fields coincide, the dynamics are integrable. As the magnetic field rotates away from the direction of the electric field, a transition to chaos occurs. Based on the approach in a previous work in the nonrelativistic regime [Narimanov , ], we introduce a particular Lorentz transformation to extend the framework to the relativistic regime, which enables us to derive the Poincaré map and obtain semianalytic formulas for periodic orbits of low periods. The transformation also makes it possible to assess, semianalytically, the role of the degree of the relativistic motion of the particle in its dynamics, revealing that the special relativity effects can enhance chaos. As numerical simulations have played a dominant role in investigating nonlinear physical systems exhibiting chaos and in many cases are the only feasible tool to probe into the system dynamics, our semianalytic approach is unique and leads to additional insights into the interplay between special relativity and chaos.