State equations of memristor circuits with nonlinear lossless elements in the Flux-charge domain

Recent works have introduced an effective technique to analyze nonlinear dynamics of a class LM of circuits containing ideal flux or charge controlled memristors and linear lossless elements (i.e. ideal capacitors and inductors). The technique, named Flux Charge Analysis Method (FCAM), is based on analyzing the circuits in the flux charge domain instead of the traditional voltage current domain. Goal of this paper is to extend the FCAM to a larger class N of circuits containing also nonlinear capacitors and inductors. Nonlinear circuits with memristors and nonlinear lossless elements are widely used to several real nanoscale devices including the well-known Josephson junction. After deriving the constitutive relation in the flux charge domain of each two terminal element in N, the work focuses on a relevant subclass of N for which a state equation description can be obtained. State Equations (SE) formulation provides the fundamental basis for studying the chief features of the nonlinear dynamics: presence of invariant manifolds in autonomous circuits; coexistence of infinitely many different reduced order dynamics on the manifolds; bifurcations due to changing of initial conditions for a fixed set of parameters, a.k.a. bifurcations without parameters.