Analytical Derivation of Sharp-Edge-of-Chaos Domain in a One-Dimensional Memristor Array

In this work, we present analytical derivation of Sharp-Edge-of-Chaos (SEOC) domain for one dimensional (1D) reaction-diffusion arrays where we assume 1-port coupling with periodic boundary conditions. We consider a general form for the complexity function of the uncoupled cell and following an iterative approach, we derive the analytical formula of the destabilization condition for the 1D reaction-diffusion array with n elements. The destabilization condition further gives the critical value of the coupling resistor element for the emergence of pattern formation across the array. In order to demonstrate the functionality of the analytical derivations, we examine the normalized version of the complexity function of a practical memristive cell, and investigate the evolution of the critical value of the coupling resistor of the 1D array with respect to the parameter values of the complexity function and to the array size. In this way, we reveal a time-efficient simulation method for the determination of the destabilization condition in 1D memristive reaction-diffusion arrays which can be adopted for arrays of higher dimensions as well as for n-port couplings in the future.