Dynamical Behavior of Cellular Automata

Cellular Automata are simple computational models which can be leveraged to model a wide variety of dynamical systems. Composed of a lattice of discrete cells that take finite number of states based on previous iterations these models differ greatly from dynamic systems that vary continuously in space or time. However given their ability to model many of continuous systems, it could be postulated that certain cellular automata exhibit chaotic dynamics. This paper examines the the chaotic analysis of a specific class of Cellular Automata, Elementary Cellular Automata (ECA), using the Lyapunov Exponent as our primary measure of chaos. The Lyapunov Exponent for these cellular automata does not have a standardized definition, given the unique nature of this model and its relative youth as an area of study. We focused on two methods used to calculate the Lyapunov Exponent, both of which generally define the Lyapunov Exponent to be the rate of error propagation in time, yet differ in specific method. Given these two theoretical frameworks we stove to create an efficient numerical application to calculate the Lyapunov Exponent of ECA of arbitrary rule number.