Chua Corsage Memristor: Phase Portraits, Basin of Attraction, and Coexisting Pinched Hysteresis Loops

The Chua Corsage Memristor is the simplest example of a passive but locally active memristor endowed with two asymptotically stable equilibrium points Q(0) and Q(1) when powered by an E-volt battery, where - 10V < E < 10 V. The basin of attraction is defined by x(0) < 30-E, E < 10V for Q(0), and x(0) > 30-E, E > - 10V for Q(1). By adding an inductor of appropriate value L > 0H in series with the battery, the resulting circuit undergoes a supercritical Hopf bifurcation and becomes an oscillator for - 10V < E < -3.334 V. Applying a sinusoidal voltage source v(t) = Asin(2 pi ft) across the Chua corsage memristor, one finds two distinct coexisting stable periodic responses, depicted by their associated pinched hysteresis loops, of the same frequency f whose basin of attraction is defined by x(0) <= x*(0), and x(0) > x*(0), respectively, where x*(0) depends on both amplitude A and frequency f. An in-depth and comprehensive analysis of the above global nonlinear phenomena is presented using tools from nonlinear circuit theory, such as Chua's dynamic route method, and from nonlinear dynamics, such as phase portrait analysis and bifurcation theory.