Embedding the Dynamics of Forced Nonlinear Systems in Multistable Memristor Circuits

A well-known feature of memristors is that they makes the circuit dynamics much richer than that generated by classical $RLC$ circuits containing nonlinear resistors. In the case of circuits with ideal memristors, such a multistability property, i.e., the coexistence of many different attractors for a fixed set of parameters, is connected to the fact that the state space is composed of a continuum of invariant manifolds where either convergent or oscillatory and more complex behaviors can be displayed. In this paper we investigate the possibility of designing memristor circuits where known attractors are embedded into the invariant manifolds. We consider a class of forced nonlinear systems containing several systems which are known to display complex dynamics, and we investigate under which conditions the dynamics of any given system of the class can be reproduced by a circuit composed of a two-terminal (one port) element connected to a flux-controlled memristor. It is shown that an input-less circuit is capable to replicate the system attractors generated by varying the constant forcing input, once the parameters of the two-terminal element and the memristor nonlinear characteristic are suitably selected. Indeed, there is a one-one correspondence between the dynamics generated by the nonlinear system for a constant value of the input and that displayed on one of the invariant manifolds of the input-less memristor circuit. Some extensions concerning the case of non-constant forcing terms and the use of charge-controlled memristors are also provided. The results are illustrated via FitzHugh-Nagumo model and Duffing oscillator.