Exact results for discrete dynamical systems on a pair of contours

Abstract

The study of discrete model of movement on a single contour, equivalent to CA 184 in terms of Wolfram cellular automata classification, was conducted in the late 1990s and early 2000s. In similar formulations of problems for contour networks, conflicts of movement take place in nodes. These problems have been poorly studied so far. However, they are very interesting for scientific and research applications relating to the flow theory (traffic, development of new materials, energy metabolism in the body). In this paper, results are obtained for the simplest contour network. This network is a pair of contours with common node. The supporter of the system consists of 2 contours with a common point (node). Particles move on their contour in accordance with rule 184 or 240 of Wolfram cellular automata. We consider the system with a common cell (simple node) or common point between 2 cells on each contour (alternating node). We have developed approaches to the study of two‐contour system such that these approaches can be used to the study of contour networks with more complex architecture. We have obtained the following criterion of that the system comes to the state of free movement (self‐organization). The criterion of self‐organization is inequality ρ1+ρ2≤1/2 in the case of rule CA 184 and simple node and inequality ρ1+ρ2≤1 in the case of rule CA 240 and alternating node. In the general case, presence of self‐organization depends on the initial state of the system. Velocities of particles have been found for different rules of movement. Prospects of future research and possible application are outlined.