Unfolding Nonlinear Dynamics in Analogue Systems With Mem-Elements

Mauro Di Marco,Mauro Forti,Fernando Corinto,Leon Chua

doi:10.1109/tcsi.2020.3024248

Abstract

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.

Highlights

C ONVENTIONAL computing architectures are facing fundamental challenges including the heat and memory wall, the end of Moore’s law and the von Neumann bottleneck, i.e., the high costs associated with constant data movements between the memory and the processor [1], [2]
The goal is to extend flux-charge analysis method (FCAM) in order to unfold the nonlinear dynamics of networks in N by highlighting some main peculiar features that are potentially useful for real-time signal processing and in-memory computing
The state equation (SE) representation in the (φ, q)-domain and (v, i )-domain, in combination with an extension of the FCAM theory in [14]–[16], have been proved effective to unfold the dynamics of a class of memristor circuits with nonlinear storage elements

Summary

INTRODUCTION

C ONVENTIONAL computing architectures are facing fundamental challenges including the heat and memory wall, the end of Moore’s law and the von Neumann bottleneck, i.e., the high (energy and speed) costs associated with constant data movements between the memory and the processor [1], [2]. For modeling purposes at nanoscale, mem-elements are sometimes used in combination with nonlinear inductors or capacitors. A class N of mem-circuits containing any number of memristors and (possibly) nonlinear capacitors and inductors, is considered. The goal is to extend FCAM in order to unfold the nonlinear dynamics of networks in N by highlighting some main peculiar features that are potentially useful for real-time signal processing and in-memory computing. DI MARCO et al.: UNFOLDING NONLINEAR DYNAMICS IN ANALOGUE SYSTEMS WITH MEM-ELEMENTS dynamics and attractors, by applying suitable pulse independent voltage or current sources. The first one concerns a variant of Chua’s circuit with a memristor and a nonlinear capacitor, for which it is shown that there is coexistence of different regimes and attractors as equilibrium points, cycles and complex attractors, for a fixed set of circuit parameters. The second one concerns a relaxation oscillator with a class of extended memristors and a nonlinear inductor, for which there is coexistence of equilibrium points and limit cycles

A CLASS OF CIRCUITS WITH MEM-ELEMENTS

INVARIANT MANIFOLDS AND COEXISTING DYNAMICS

CHUA’S CIRCUIT WITH MEMRISTOR AND NONLINEAR CAPACITOR

RELAXATION OSCILLATOR WITH MEMRISTOR AND NONLINEAR INDUCTOR

FCAM for Circuits Including an Extended Memristor

CONCLUSION