Complex Systems

The Relationship between One-dimensional Continuous Cellular Automata and One-dimensional Nonlinear Dynamical Systems Download PDF

Randall E. Rausch
Department of Mathematical Sciences,
University of Texas at Dallas,
Richardson, Texas


In this paper the relationship between systems of real-valued, or continuous, one-dimensional cellular automata (CA) and a corresponding one-dimensional nonlinear dynamical system is investigated. The CA will be defined in a periodic domain and will satisfy requirements on the function used to assign new values; the corresponding one-dimensional dynamical system will be derived from the function used to assign new values. It is shown that the qualitative behavior of the CA--the existence of a stable orbit with the same value at all cells--is determined by the corresponding one-dimensional system and the geometry of the CA.