Mandelbrot set in coupled logistic maps and in an electronic experiment

Abstract

We suggest an approach to constructing physical systems with dynamical characteristics of the complex analytic iterative maps. The idea follows from a simple notion that the complex quadratic map by a variable change may be transformed into a set of two identical real one-dimensional quadratic maps with a particular coupling. Hence, dynamical behavior of similar nature may occur in coupled dissipative nonlinear systems, which relate to the Feigenbaum universality class. To substantiate the feasibility of this concept, we consider an electronic system, which exhibits dynamical phenomena intrinsic to complex analytic maps. Experimental results are presented, providing the Mandelbrot set in the parameter plane of this physical system. PACS number(s): 05.45.Df, 05.45.Xt

Dynamics of a discrete system with the operator of evolution given by an implicit function: from the Mandelbrot map to a unitary map

“…Вместе с тем, системы из этого класса интересны не только с математической точки зрения. Они оказы-ваются физически реализуемыми [24][25][26][27] и применимыми в весьма разнообразных отрас-лях: в теории фазовых переходов [28,29], в теории перколяции [30], в теории квантового туннелирования [31], в КАМ-теории [32], в вакуумной микроэлектронике [33] и для опи-сания ограниченной диффузией агрегации [34]. Канонической моделью для комплексной динамики является отображение Мандельброта -комплексифицированное логистическое отображение [3,4,35].…”

“…[16,22]. На базе этих би-фуркаций организуются бесконечные каскады со своими специфическими скейлинговыми характеристиками [15,22,25,33]. На рисунке 2 представлена карта динамических режимов отображения (5) на плоско-сти (ϕ = Arg ρ, r = |ρ|).…”

Effect of noise on the dynamics of a complex map at the period-tripling accumulation point

“…Another approach developed in Ref. [22] is based on a construction of a system of two coupled units, each of which can demonstrate Feigenbaum's period-doubling cascade, and the coupling is of some special kind. Moreover, this idea has been verified in an experiment with coupled electronic circuits.…”

Fourier spectrum of the signal generated at the point of period tripling bifurcation accumulation

“…Based on this idea, in the paper [9] the electronic analog device, modelling the dynamics of the coupled logistic maps is proposed. In this system the first observation of the Mandelbrot set in physical experiment is carried out.…”

“…Furthermore, the universal properties of the spectrum at the GSK point are demonstrated in mentioned physical experiment. With the aid of the described in [9] device, several first periodtripling bifurcations have been successively observed in physical experiment (to get immediately the GSK point is impossible problem because of noise and experimental error).…”