RESEARCH ON BROWNIAN MOVEMENT BASED ON GENERALIZED MANDELBROT–JULIA SETS FROM A CLASS COMPLEX MAPPING SYSTEM

Abstract

This article analyzes the typical Langevin problem, i.e. the dynamics of a charged particle under the circularly successive influence of several simple impulse functions moving in a double-well potential and a time-dependent magnetic field. Using the stroboscopic sampling, by selecting an appropriate magnetic intensity and time interval, we reduce the Langevin equation to a class complex mapping system. Through an experimental mathematical method, the authors study the structures of generalized M–J (Mandelbrot–Julia) sets generated by the complex mapping system, and expatiate the theory of Brownian movement. The authors find that:.(1) This paper extends Shirriff constructed M sets by combining two simple complex mappings;.(2) The fractal structure of the generalized M–J sets may visually depict the rule of Brownian movement, and the infinite overlapping embedment self-similar structure reflects the complexity of Brownian movement;.(3) Whether the selected time interval is significant or not determines the continuity of the fractal structure for the generalized M–J sets;.(4) The changing rule of particle velocity depends on the different choices of the principal range of phase angle;.(5) If we change the choices of the magnetic intensity and time interval, for example, we choose a randomly fluctuating magnetic field, the generalized J sets may emerge the interior-filling structure feature, i.e. "explosion" phenomena appear in the closure of the unstable periodic orbits of the particle in the velocity space.