A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science Part IX: Quasi-Ergodicity (original) (raw)
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Complex Dynamic Behaviors in Cellular Automata Rule 14
Discrete Dynamics in Nature and Society, 2012
Wolfram divided the 256 elementary cellular automata rules informally into four classes using dynamical concepts like periodicity, stability, and chaos. Rule 14, which is Bernoulli σ τ -shift rule and is a member of Wolfram's class II, is said to be simple as periodic before. Therefore, it is worthwhile studying dynamical behaviors of rule 14, whether it possesses chaotic attractors or not. In this paper, the complex dynamical behaviors of rule 14 of one-dimensional cellular automata are investigated from the viewpoint of symbolic dynamics. We find that rule 14 is chaotic in the sense of both Li-Yorke and Devaney on its attractor. Then, we prove that there exist fixed points in rule 14. Finally, we use diagrams to explain the attractor of rule 14, where characteristic function is used to describe that all points fall into Bernoulli-shift map after two iterations under rule 14.
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.
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.
Complex Dynamics of Elementary Cellular Automata Emerging from Chaotic Rules
International Journal of Bifurcation and Chaos, 2012
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.
Generalized Sub-Shifts in Elementary Cellular Automata: The "Strange Case" of Chaotic Rule 180
Theoretical Computer Science, 1998
We study the dynamical behavior of elementary cellular automaton 180. This rule gives rise to a global dynamics on the phase space of all one-dimensional bi-infinite configurations which is Devaney topologically chaotic. The dense sub-dynamical system of configurations in background of 0s is a generalized sub-shift, i.e., multiple sub-shift whose multiplicity constant depends on the initial configuration. This sub-dynamical system
2012 International Conference on High Performance Computing & Simulation (HPCS), 2012
De Bruijn diagrams have been used as a useful tool for the systematic analysis of one-dimensional cellular automata (CA). They can be used to calculate particular kind of configurations, ancestors, complex patterns, cycles, Garden of Eden configurations and formal languages. However, there is few progress in two dimensions because its complexity increases exponentially. In this paper, we will offer a way to explore systematically such patterns by de Bruijn diagrams from initial configurations. Such analysis is concentrated mainly in two evolution rules: the famous Game of Life (complex CA) and the Diffusion Rule (chaotic CA). We will display some preliminary results and benefits to use de Bruijn diagrams in these CA.
On the quantitative behavior of the linear cellular automata
2013
In this paper, we study the quantitative behavior of one-dimensional linear cellular automata T f [−r,r] , defined by local rule f (x −r , . . . , xr) = r i=−r λ i x i (mod m), acting on the space of all doubly infinite sequences with values in a finite ring Zm, m ≥ 2. Once generalize the formulas given by Ban et al. [J. Cellular Automata 6 (2011) 385-397] for measure-theoretic entropy and topological pressure of one-dimensional cellular automata, we calculate the measure entropy and the topological pressure of the linear cellular automata with respect to the Bernoulli measure on the set Z Z m . Also, it is shown that the uniform Bernoulli measure is the unique equilibrium measure for linear cellular automata. We compare values of topological entropy and topological directional entropy by using the formula obtained by Akın [J. Computation and Appl. Math. 225 (2) (2009) 459-466]. The topological directional entropy is interpreted by means of figures. As an application, we demonstrate that the Hausdorff of the limit set of a linear cellular automaton is the unique root of Bowen's equation. Some open problems remain to be of interest.
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.
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 ...