Exploring Parameter Control Techniques for Discrete Memristive Maps in Network Structures

Abstract

This paper proposes an adaptive control structure of parameters for discrete time based on Lyapunov function for memristive Rulkov neuron maps in coupled network structures with unknown and potentially time-varying system parameters. The standard Rulkov map is modified by adding a discrete flux-controlled memristor, which introduces a state variable on top of it.

There are four control objectives systematically tackled: stabilization of single map trajectories, estimation of parameters, drive-response synchronization, and extension to an N-node ring network. For each objective, both control laws and parameter update laws are obtained, and stability is guaranteed using Lyapunov functions. Theoretical analysis shows asymptotic convergence of synchronization and zero estimation errors for the parameters. The framework enables numerical simulations to be conducted. The results presented have shown that validation is theoretically possible and feasible for the construction of an adaptive controller. On the other hand, in the literature, almost nothing exists about the behavior of discrete memristive neuronal systems, which are mainly concerned with the behavior of continuous memristive neurons or non-memristive discrete maps.