Turing patterns via pinning control in the simplest memristive cellular nonlinear networks

Abstract

Complex patterns are commonly retrieved in spatially-extended systems formed by coupled nonlinear dynamical units. In particular, Turing patterns have been extensively studied investigating mathematical models pertaining to different fields, such as chemistry, physics, biology, mechanics, and electronics. In this paper, we focus on the emergence of Turing patterns in memristive cellular nonlinear networks by means of spatial pinning control. The circuit architecture is made by coupled units formed by only two elements, namely, a capacitor and a memristor. The analytical conditions for which Turing patterns can be derived in the proposed architecture are discussed in order to suitably design the circuit parameters. In particular, we derive the conditions on the density of the controlled nodes for which a Turing pattern is globally generated. Finally, it is worth to note that the proposed architecture can be considered as the simplest ideal electronic circuit able to undergo Turing instability and give rise to pattern formation.