Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme Multistability, and Bifurcations Without Parameters

Chapter 5 has developed the flux-charge analysis method (FCAM) for the analysis of a class \(\mathcal {L}\mathcal {M}\) of memristor circuits containing memristors, linear resistors, inductors, capacitors, and independent voltage and current sources. The formulation of circuit equations (DAEs and SEs) has been provided in the (φ, q)-domain and in the (v, i)-domain and relationships between the analysis in the two domains have been discussed. A fundamental property is that the SE description in the (φ, q)-domain of a circuit in \(\mathcal {L}\mathcal {M}\) has a lower order with respect to the corresponding description in the (v, i)-domain, so that it is expected that the dynamic analysis in the former domain is simpler. The goal of this chapter is to highlight some main advantages of FCAM by studying the dynamics of a number of low-order autonomous memristor circuits in the (φ, q)-domain, namely, circuits described by a first-order SE in the (φ, q)-domain, oscillatory circuits described by a second-order SE in the (φ, q)-domain, and chaotic circuits described by a third-order SE in the (φ, q)-domain. Such a study permits to show that memristor circuits display a number of fundamental peculiar dynamic features. First of all, the state space in the (v, i)-domain can be foliated in a continuum of invariant manifolds, thus implying the coexistence of infinitely many different reduced-order dynamics and attractors. Moreover, such circuits display bifurcations due to changing the initial conditions for a fixed set of circuit parameters that are named bifurcations without parameters.