Mathematical Investigation of Static Pattern Formation with a Locally Active Memristor Model

We present the mathematical investigation of static pattern formation in a Memristor Cellular Nonlinear Network (M-CNN), in consideration of the theory of local activity. The M-CNN has a planar grid form composed of identical memristive cells, which are purely resistively coupled to each other. The single cell contains a DC voltage source, a bias resistor, and a locally active memristor in parallel with a capacitor. The memristor model employed has a simple generic form which helps to reduce the simulation time, and has a functional AC equivalent circuit which facilitates further calculations. We adopt a circuit theoretical approach for the stability analysis of the single cell and a 3-cell ring configuration, as well as the examination of local activity, edge-of-chaos, and sharp-edge-of-chaos domains, which helps us to interpret the results in a better way. The emergence of static patterns is successfully confirmed by simulating the proposed resistively coupled M-CNN utilizing locally active memristors.