Memristor Circuits: Bifurcations without Parameters

The present manuscript relies on the companion paper entitled Memristor Circuits: Flux-Charge Analysis Method,which has introduced a comprehensive analysis method to study the nonlinear dynamics of memristor circuits in the flux-charge (phi, q)-domain. The Flux-Charge Analysis Method is based on Kirchhoff Flux and Charge Laws and constitutive relations of circuit elements in terms of incremental fluxes and incremental charges. The straightforward application of the method has previously provided a full portrait of the nonlinear dynamics and bifurcations of the simplest memristor circuit composed by a capacitor and a flux-controlled memristor. This paper aims to show that the method is effective to analyze nonlinear dynamics and bifurcations in memristor circuits with more complex dynamics including Hopf bifurcations (originating persistent oscillations) and period-doubling cascades (leading to chaotic behavior). One key feature of the method is that it makes clear how initial conditions give rise to bifurcations for an otherwise fixed set of circuit parameters. To the best of the authors' knowledge, these represent the first results that relate such bifurcations, which are referred to in the paper as Bifurcations without Parameters, with physical circuit variables as the initial conditions of dynamic circuit elements.