Study on the physical meaning for generalized Mandelbrot-Julia sets based on the Langevin problem

Abstract

Based on a dynamics research of the typical Langevin problem, i.e., a moving charged particle under the continuous influence of a constant impulse in a double-well potential and a time-dependent magnetic field, using the stroboscopic sampling, we propose complex difference equations which can describe the change rule of particle's velocity. By selecting appropriate magnetic intensity and time intervals (sampling period), we reduce the difference equations to complex mapping which is used to construct the generalized M-J sets. Based on the particle's dynamics characteristic, we discussed the physical meaning of the generalized M-J sets. The authors found that: (1) The fractal structure of the generalized M-J sets may visually reflect the change rule of particle's velocity. (2) Whether the selected time intervals is significative determines whether the fractal structure of the generalized M-J sets has the continuity. (3) The evolution of the generalized M-J sets, i.e., the change rule of particle's velocity, depends on the different choices of the principal range of the phase angle. (4) If we change the choices of the magnetic intensity and time intervals, for example, choose a random fluctuant magnetic field, the generalized J sets may present the interior-filling structure feature, i.e., “explosion" phenomena appear in the closure of the particle's instable periodic orbits in the velocity space.