Unfolding Nonlinear Dynamics in Analogue Systems With Mem-Elements

The paper considers a relevant class of networks containing memristors and (possibly) nonlinear capacitors and inductors. The goal is to unfold the nonlinear dynamics of these networks by highlighting some main features that are potentially useful for real-time signal processing and in-memory computing. In particular, an analytic treatment is provided for dynamic phenomena as the presence of invariant manifolds, the coexistence of different regimes, complex dynamics and attractors and the phenomenon of bifurcations without parameters, i.e., bifurcations due to changing the initial conditions of the state variables for a fixed set of circuit parameters. The paper also addresses the issue of how to design pulse independent voltage or current sources to steer the network dynamics through different manifolds and attractors. Two relevant examples are worked out in details, namely, a variant of Chua's circuit with a memristor and a nonlinear capacitor and a relaxation oscillator with a memristor and a nonlinear inductor. In the latter example, the paper also studies the effect on manifolds and coexisting dynamics when real memristive devices are accounted for using a class of extended memristor models. The analysis is conducted by means of a recently developed technique named flux-charge analysis method (FCAM). Numerical simulations are presented to confirm the theoretic findings.