Complex Systems

Periodic Patterns in the Binary Difference Field Download PDF

André Barbé
Division of Electronics, Systems, Automation and Technology,
Katholieke Universiteit Leuven, Kardinaal Mercierlaan 94,
3030 Heverlee, Belgium


The difference sequence of a binary sequence is the binary sequence representing the presence of a difference in value at two neighboring sites in the original sequence. The difference field is the ordered ensemble of all difference sequences aligned one under the other. It is equivalent to the space-time pattern of a one-dimensional cellular automaton under a simple asymmetric rule. Periodic boundary conditions imposed at the boundaries of the propagation net of changes, which is induced by a finite change of values in the initial state, give rise to periodic bands of tilings along these boundary lines. Width and period of these bands evolve in a well-defined way, exhibiting period and bandwidth doubling. A special kind of self-similarity is apparent, and the pattern has a fractal skeleton. Periodic boundary conditions may result from a conservation law imposed on the states in the propagation net.