Global dynamics of Chua Corsage Memristor circuit family: fixed-point loci, Hopf bifurcation, and coexisting dynamic attractors

Abstract

This paper presents an in-depth and rigorous mathematical analysis of a family of nonlinear dynamical circuits whose only nonlinear component is a Chua Corsage Memristor (CCM) characterized by an explicit seven-segment piecewise-linear equation. When connected across an external circuit powered by a DC battery, or a sinusoidal voltage source, the resulting circuits are shown to exhibit four asymptotically stable equilibrium points, a unique stable limit cycle spawn from a supercritical Hopf bifurcation along with three static attractors, four coexisting dynamic attractors of an associated non-autonomous nonlinear differential equation, and four corresponding coexistingpinched hysteresis loops. Thebasin of attractions of the above static and dynamic attractors is derived numerically via global nonlinear analysis. When driven by a battery, the resulting CCM circuit exhibits a contiguous fixed-point loci, along with its DC V–I curve described analytically by two explicit parametric equations. We also proved the fundamental feature of theedge of chaos property; namely, it is possible to destabilize a stable circuit (i.e., without oscillation) and make it oscillate, by merely adding a passive circuit element, namely $$L >0$$. The CCM circuit family is one of the few known example of a strongly nonlinear dynamical system that is endowed with numerous coexisting static and dynamic attractors that can be studied both experimentally, and mathematically, via exact formulas.