A geometrical interpretation of the chaotic state of inhomogeneous deterministic cellular automata (original) (raw)

Chaos in networks of two-dimensional homogeneous cellular automata

Physica A: Statistical Mechanics and its Applications, 1991

We carefully selected 16 rules from all 65 536 possible rules on a two-dimensional network to create the corresponding 16 homogeneous cellular automata. VaD'ing the initial state in the computer simulation we investigate systematically the properties of the final states of these cellular automata.

On the Dynamical Behavior of Chaotic Cellular Automata

Theoretical Computer Science, 1999

The shift (bi-infinite) cellular automaton is a chaotic dynamical system according to all the definitions of deterministic chaos given for discrete time dynamical systems (e.g., those given by Devaney [6] and by Knudsen [10]). The main motivation to this fact is that the temporal evolution of the shift cellular automaton under finite description of the initial state is unpredictable. Even

Topological chaos for elementary cellular automata

Lecture Notes in Computer Science, 1997

We apply the definition of chaos given by Devaney for discrete time dynamical systems to the case of elementary cellular automata, i.e., 1-dimensional binary cellular automata with radius 1. A discrete time dynamical system is chaotic according to the Devaney's definition of chaos if it is topologically transitive, is sensitive to initial conditions, and has dense periodic orbits. We enucleate an easy-to-check property of the local rule on which a cellular automaton is based which is a necessary condition for chaotic behavior. We prove that this property is also sufficient for a large class of elementary cellular automata. The main contribution of this paper is the formal proof of chaoticity for many non additive elementary cellular automata. Finally, we prove that the above mentioned property does not remain a necessary condition for chaoticity in the case of non elementary cellular automata.

A Study of Chaos in Non-uniform Cellular Automata

Communications in Nonlinear Science and Numerical Simulation, 2019

This paper studies chaotic behavior of 1-dimensional non-uniform cellular automata (CAs). The chaotic system is unpredictable in the long run. To understand the source of unpredictability, we add an information to a cell and check how it flows to its neighbors. That is, once the state of a cell is changed, the probability to which the change affects its neighbors, is calculated. The probabilistic value decides whether one cell communicates with another cell. Depending on the communication among cells, a binary relation over the set of cells is defined in this paper which is known as communication relation. It is shown that the relation should be an equivalence relation to make a cellular automaton (CA) chaotic. The equivalence relation among the cells form an equivalence class which is named as communication class. Another important property of a chaotic CA is transitivity of the CA. Transitivity of a CA does not allow disjoint subsets of a configuration space. Based on the communication class and transitivity property of a CA, we newly define the chaos in CA. Sometimes one cell may not be able to communicate with another cell due to blocking word. The presence of blocking word in a CA makes the CA nontransitive. This paper shows a method to find out such blocking words. Finally, a parametrization technique is developed in this work based on the transitivity, communication class and blocking words. Depending on the parameter value of a non-uniform CA, one can predict whether the behavior of the CA is chaotic.

Some results about the chaotic behavior of cellular automata

Theoretical Computer Science, 2005

We study the behavior of cellular automata (CA for short) in the Cantor, Besicovitch and Weyl topologies. We solve an open problem about the existence of transitive CA in the Besicovitch topology. The proof of this result has some interest of its own since it is obtained by using Kolmogorov complexity. To our knowledge it is the first result about discrete dynamical systems obtained using Kolmogorov complexity. We also prove that in the Besicovitch topology every CA has either a unique periodic point (thus a fixed point) or an uncountable set of periodic points. This result underlines the fact that CA have a great degree of stability; it may be considered a further step towards the understanding of CA periodic behavior.

Investigating topological chaos by elementary cellular automata dynamics

Theoretical Computer Science, 2000

We apply the two di erent deÿnitions of chaos given by Devaney and by Knudsen for general discrete time dynamical systems (DTDS) to the case of elementary cellular automata, i.e., 1-dimensional binary cellular automata with radius 1. A DTDS is chaotic according to the Devaney's deÿnition of chaos i it is topologically transitive, has dense periodic orbits, and it is sensitive to initial conditions. A DTDS is chaotic according to the Knudsen's deÿnition of chaos i it has a dense orbit and it is sensitive to initial conditions. We enucleate an easy-to-check property (left or rightmost permutivity) of the local rule associated with a cellular automaton which is a su cient condition for D-chaotic behavior. It turns out that this property is also necessary for the class of elementary cellular automata. Finally, we prove that the above mentioned property does not remain a necessary condition for chaoticity in the case of non elementary cellular automata.

Cellular automata and dynamical systems

1989

In this thesis we investigate the theoretical nature of the mathematical structures termed cellular automata. Chapter 1: Reviews the origin and history of cellular automata in order to place the current work into context. Chapter 2: Develops a cellular automata framework which contains the main aspects of cellular automata structure which have appeared in the literature. We present a scheme for specifying the cellular automata rules for this general model and present six examples of cellular automata within the model. Chapter 3: Here we develop a statistical mechanical model of cellular automata behaviour. We consider the relationship between variations within the model and their relationship to dynamical systems. We obtain results on the variance of the state changes, scaling of the cellular automata lattice, the equivalence of noise, spatial mixing of the lattice states and entropy, synchronous and asynchronous cellular automata and the equivalence of the rule probability and the ...

Complex chaotic behavior of a class of subshift cellular automata

1993

A class of parameterized boolean, one-dimensional, biinfinite cellular autom at a has been st udied and t heir behavior observed when some param eters of t he local function are cha nged. T hese aut omata are equivalent to a par ticular class of boolean neur al networks and t he change in t he paramet ers corres ponds to a change in t he symmetricity of t he connect ion matrix. T he purpo se is to ana lyze t he different dynamics, beginning wit h a symmet ric connect ion matrix and moving toward an ant isymmet ric one. We have observed t hat simp le dynamics corres ponds to t he symmetric sit uation, whereas t he antisymmet rical case yields more complex behavior. On t he bas is of t hese observatio ns, we have identified a new class of cellular auto mata t ha t is characterized by shiftlike dynamics (simple and complex subshift rules); t hese cellular automata corres pond to t he asymmet ric sit uations and t hey are chaotic dyn amical syst ems.

The effects of randomness in asynchronous 1D cellular automata

proceedings of ALIFE IV, 1994

The first major effect is that certain 1D-ACA, which generate non-chaotic patterns when not randomized, generate “edge-of-chaos” patterns when randomized. Some of these patterns are similar to those generated using Wolfram's class IV automata or coupled map lattices. ...

Complex dynamics of elementary cellular automata emerging in chaotic rules

2010

We show techniques of analyzing complex dynamics of cellular automata (CA) with chaotic behaviour. CA are well known computational substrates for studying emergent collective behaviour, complexity, randomness and interaction between order and chaotic systems. A number of attempts have been made to classify CA functions on their space-time dynamics and to predict behaviour of any given function. Examples include mechanical computation, λ and Zparameters, mean field theory, differential equations and number conserving features. We aim to classify CA based on their behaviour when they act in a historical mode, i.e. as CA with memory. We demonstrate that cell-state transition rules enriched with memory quickly transform a chaotic system converging to a complex global behaviour from almost any initial condition. Thus just in few steps we can select chaotic rules without exhaustive computational experiments or recurring to additional parameters. We provide analysis of well-known chaotic functions in onedimensional CA, and decompose dynamics of the automata using majority memory exploring glider dynamics and reactions.