Complex Systems
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Volume 25, Number 1 (2016)

Coexistence of Dynamics for Two-Dimensional Cellular Automata
Ricardo Severino, Maria Joana Soares, and Maria Emilia Athayde

This paper is concerned with the study of six rules from the family of square Boolean cellular automata (CAs) having a neighborhood consisting of four peripheral neighbors and with periodic boundary conditions. Based on intensive computations, we are able to conclude, with statistical support, that these rules have a common feature: they all show coexistence of dynamics, in the sense that as the size n of the side of the square increases with fixed parity, the relative size of the basin of attraction of the homogeneous final state tends to a constant value that is neither zero nor one. It is also statistically shown that the values of the constant levels—one for n odd and the other for n even—can be considered as equal for five of the rules, while for the sixth rule these values are one-half of the others. Some results obtained for the one-dimensional CAs with four peripheral neighbors are also reported, to support our claim that with periodic boundary conditions, the coexistence of dynamics can only appear for automata with dimension higher than one.

Commutative Cellular Automata
Victor Duy Phan

Commutative cellular automata are a class of cellular automata that portray certain characteristics of commutative behavior. We develop the notion of neighborhood partitions and neighborhood equivalence classes to analyze and enumerate these automata.

How External Environment and Internal Structure Change the Behavior of Discrete Systems
Jim Hay and David Flynn

This paper continues our computer studies using virtual systems to examine the behavior of subsystems in a system as a model of the behavior of social systems made up of individuals. The subsystems in our virtual system are global cellular automata (GCAs) as suggested by Wolfram [1], placed at the vertices of a GCA network (GCAN) as developed by Chandler [2]. The behavioral results are based on the four classes of cellular automata output patterns as identified by Wolfram [3] and are measured by the fraction of ordered GCAs in a GCAN.

Our objective has been to show how our theory of social dynamics explains this behavior. That theory states that the behavior of a social system and of our virtual systems model depends upon the external environment of the system defined as centrality and the internal structure of the system defined as the four parameters of differentiation, namely, diversity, connectedness, interdependence, and adaptability of the subsystems in the system. In previous papers we have shown the effect of diversity and connectedness. In this paper we show that behavior becomes more ordered and focused as interdependence and adaptability increase and as centrality decreases.

A Symbolic Dynamics Perspective of the Game of Three-Dimensional Life
Bo Chen, Fangyue Chen, Genaro J. Martínez, and Danli Tong

The games of three-dimensional life are the extension models of Conway's Game of Life. Under the framework of symbolic dynamics, we undertake an analysis of the complexity of gliders in games of three-dimensional life rules by the directed graph representation and transition matrix. More specifically, the gliders here are topologically mixing and possess positive topological entropy on their concrete subsystems. Finally, the method presented in this paper is also applicable to other gliders in different D-dimensions.

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