Mandelbrot Set in Coupled Iterative Maps and in the Continuous Systemes

Olga B. Isaeva

Department of non-linear processes, Saratov State University, Russian Federation

Abstract: One of reach and fascinating sub-disciplines in nonlinear dynamics is a theory of the iterative complex analytic mappings. The well-known example is the complex quadratic map, from which the Mandelbrot set as most popular example of fractal arise. We reduce the complex quadratic map to the set of two coupled real logistic maps and use this representation to construct an electronic schema with switched capacitors, which gives a possibility to observe dynamical phenomena intrinsic to complex analytic iterative maps in a real physical device. Experimental results demonstrate the Mandelbrot set on the parameter plane of the system. Perhaps, this is the first observation of the Mandelbrot set in a real physical experiment. The aim of the work is to suggest the examples of physical systems, which manifest phenomena of complex analytic dynamics and is practically realizable. Hence, it may be expected that properly arranged coupling between two period-doubling elements of any nature might ensure the whole system to demonstrate the phenomena of complex analytic dynamics. For example, we consider Ikeda map, Henon map, Rossler system, non-linear oscilators.