VISUALIZATION OF CHAOTIC DYNAMICAL SYSTEMS BASED ON MANDELBROT SET METHODOLOGY

Abstract

Although there are lots of methods to analyze chaotic systems, they tend to be of local nature and reveal only limited information about a certain group of control parameters. This paper explores a visualization technique on the basis of Mandelbrot set (M-set) methodology to give an overall view of a chaotic system's dynamical performance in the parameter space. Firstly, the Lyapunov spectra with regard to different points in the parameter space are calculated, according to which the color of these points is defined. Then points in the given parameter space are mapped to the computer screen and a colorful image named Lyapunov distribution map (LDM) is generated, which conveys a wealth of information about a system's dynamic behavior across variations in its control parameters. The results of visualizing two typical chaotic systems proved the feasibility and validity of this technique. The study suggests that desired dynamical performance of a system can be conveniently achieved by selecting a point from some color region in its LDM and adjusting the control parameters according to the point value. Furthermore, there is no need for any prior knowledge of the system under evaluation and the complicated mathematical analysis before the formulation of appropriate rules for a new system can also be avoided since uniform rules are employed in the classification of parameter points for any system and the LDM is generated via the same algorithm.