On cellular automata, traffic and dynamical systems in graphs

Abstract

Qualitative studies of discrete dynamical systems behavior on networks are relevant in many fields such as system biology, transportation, information traffic, material sciences and so on. We consider cellular automata on one-dimensional and two-dimensional toroidal supporters. At every discrete time moment, each cell of a cellular automaton is in one of two states 0 and 1. We introduce concept of the cellular automaton mass at fixed time. The cellular automaton mass is the quantity of cells such that these cells are in the state 1. The mass conservation law takes place if the mass of cellular automaton is the same at every time. Concepts of explosion and annihilation have been introduced. Explosion takes place if the mass of cellular automaton increases at each iteration until all cells are in the state 1. Annihilation takes place if the mass of cellular automaton decreases at each iteration until all cells are in the state 0. We consider classes of cellular automata such that the state of cell at the next time depends on the state at current time and states of neighboring cells belonging to fixed set. We have found sets of cellular automata such that the mass conservation law, explosion or annihilation takes place for these automata.