A STUDY OF BIFURCATIONS IN A CIRCULAR REAL CELLULAR AUTOMATON

1993 ◽  
Vol 03 (02) ◽  
pp. 293-321 ◽  
Author(s):  
JÜRGEN WEITKÄMPER
Keyword(s):  
Hopf Bifurcation ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Parallel Computers ◽  
Discrete Dynamical Systems ◽  
Two Dimensional ◽  
Bifurcation Structure ◽  
Logistic Mapping ◽  
Henon Mapping

Real cellular automata (RCA) are time-discrete dynamical systems on ℝN. Like cellular automata they can be obtained from discretizing partial differential equations. Due to their structure RCA are ideally suited to implementation on parallel computers with a large number of processors. In a way similar to the Hénon mapping, the system we consider here embeds the logistic mapping in a system on ℝN, N>1. But in contrast to the Hénon system an RCA in general is not invertible. We present some results about the bifurcation structure of such systems, mostly restricting ourselves, due to the complexity of the problem, to the two-dimensional case. Among others we observe cascades of cusp bifurcations forming generalized crossroad areas and crossroad areas with the flip curves replaced by Hopf bifurcation curves.

scholarly journals Investigating Transformational Complexity: Counting Functions a Region Induces on Another in Elementary Cellular Automata

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Martin Biehl ◽  
Olaf Witkowski
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Complex Systems ◽  
Cellular Automaton ◽  
Artificial Life ◽  
Discrete Dynamical Systems ◽  
Living Systems ◽  
Finite Region ◽  
Counting Functions ◽  
Elementary Cellular Automaton

Over the years, the field of artificial life has attempted to capture significant properties of life in artificial systems. By measuring quantities within such complex systems, the hope is to capture the reasons for the explosion of complexity in living systems. A major effort has been in discrete dynamical systems such as cellular automata, where very few rules lead to high levels of complexity. In this paper, for every elementary cellular automaton, we count the number of ways a finite region can transform an enclosed finite region. We discuss the relation of this count to existing notions of controllability, physical universality, and constructor theory. Numerically, we find that particular sizes of surrounding regions have preferred sizes of enclosed regions on which they can induce more transformations. We also find three particularly powerful rules (90, 105, 150) from this perspective.

scholarly journals Embedding Bratteli–Vershik systems in cellular automata

2009 ◽  
Vol 30 (5) ◽  
pp. 1561-1572 ◽  
Author(s):  
MARCUS PIVATO ◽  
REEM YASSAWI
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Finite Type ◽  
Two Dimensional ◽  
Time Diagram ◽  
Constant Length ◽  
Subshift Of Finite Type ◽  
One Dimensional ◽  
Toeplitz Systems

AbstractMany dynamical systems can be naturally represented as Bratteli–Vershik (or adic) systems, which provide an appealing combinatorial description of their dynamics. If an adic system X is linearly recurrent, then we show how to represent X using a two-dimensional subshift of finite type Y; each ‘row’ in a Y-admissible configuration corresponds to an infinite path in the Bratteli diagram of X, and the vertical shift on Y corresponds to the ‘successor’ map of X. Any Y-admissible configuration can then be recoded as the space-time diagram of a one-dimensional cellular automaton Φ; in this way X is embedded in Φ (i.e. X is conjugate to a subsystem of Φ). With this technique, we can embed many odometers, Toeplitz systems, and constant-length substitution systems in one-dimensional cellular automata.

scholarly journals On the Network Analysis of the State Space of Discrete Dynamical Systems

2017 ◽  
Vol 27 (04) ◽  
pp. 1750062 ◽  
Author(s):  
Cheng Xu ◽  
Chengqing Li ◽  
Jinhu Lü ◽  
Shi Shu
Keyword(s):  
Dynamical Systems ◽  
Network Analysis ◽  
Cellular Automata ◽  
State Space ◽  
Discrete Dynamical Systems ◽  
The State ◽  
Dynamical Complexity ◽  
Two Parameters ◽  
Theoretical Analyses

This paper discusses the letter entitled “Network analysis of the state space of discrete dynamical systems” by A. Shreim et al. [Phys. Rev. Lett. 98, 198701 (2007)]. We found that some theoretical analyses are wrong and the proposed indicators based on two parameters of the state-mapping network cannot discriminate the dynamical complexity of the discrete dynamical systems composed of a 1D cellular automata.

scholarly journals The stability theorems for discrete dynamical systems on two-dimensional manifolds

1983 ◽  
Vol 90 ◽  
pp. 1-55 ◽  
Author(s):  
Atsuro Sannami
Keyword(s):  
Dynamical Systems ◽  
Smooth Manifold ◽  
Riemannian Metric ◽  
Discrete Dynamical Systems ◽  
Two Dimensional ◽  
Stability Theorems ◽  
The Stability

One of the basic problems in the theory of dynamical systems is the characterization of stable systems.Let M be a closed (i.e. compact without boundary) connected smooth manifold with a smooth Riemannian metric and Diffr (M) (r ≥ 1) denote the space of Cr diffeomorphisms on M with the uniform Cr topology.

CHARACTERIZING THE ABILITY OF PARALLEL ARRAY GENERATORS ON REVERSIBLE PARTITIONED CELLULAR AUTOMATA

1999 ◽  
Vol 13 (04) ◽  
pp. 523-538 ◽  
Author(s):  
KENICHI MORITA ◽  
SATOSHI UENO ◽  
KATSUNOBU IMAI
Keyword(s):  
Pattern Recognition ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Inverse System ◽  
Two Dimensional ◽  
Turing Machines ◽  
Recognition Mechanism ◽  
Parallel Array ◽  
Parallel Pattern ◽  
Monotonic Constraint

A PCAAG introduced by Morita and Ueno is a parallel array generator on a partitioned cellular automaton (PCA) that generates an array language (i.e. a set of symbol arrays). A "reversible" PCAAG (RPCAAG) is a backward deterministic PCAAG, and thus parsing of two-dimensional patterns can be performed without backtracking by an "inverse" system of the RPCAAG. Hence, a parallel pattern recognition mechanism on a deterministic cellular automaton can be directly obtained from a RPCAAG that generates the pattern set. In this paper, we investigate the generating ability of RPCAAGs and their subclass. It is shown that the ability of RPCAAGs is characterized by two-dimensional deterministic Turing machines, i.e. they are universal in their generating ability. We then investigate a monotonic RPCAAG (MRPCAAG), which is a special type of an RPCAAG that satisfies monotonic constraint. We show that the generating ability of MRPCAAGs is exactly characterized by two-dimensional deterministic linear-bounded automata.

Definition and Application of a Five-Parameter Characterization of One-Dimensional Cellular Automata Rule Space

2001 ◽  
Vol 7 (3) ◽  
pp. 277-301 ◽  
Author(s):  
Gina M. B. Oliveira ◽  
Pedro P. B. de Oliveira ◽  
Nizam Omar
Keyword(s):  
Phase Transition ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Dynamic Behavior ◽  
Discrete Dynamical Systems ◽  
Transition Diagram ◽  
One Dimensional ◽  
Rule Space ◽  
Spatially Extended

Cellular automata (CA) are important as prototypical, spatially extended, discrete dynamical systems. Because the problem of forecasting dynamic behavior of CA is undecidable, various parameter-based approximations have been developed to address the problem. Out of the analysis of the most important parameters available to this end we proposed some guidelines that should be followed when defining a parameter of that kind. Based upon the guidelines, new parameters were proposed and a set of five parameters was selected; two of them were drawn from the literature and three are new ones, defined here. This article presents all of them and makes their qualities evident. Then, two results are described, related to the use of the parameter set in the Elementary Rule Space: a phase transition diagram, and some general heuristics for forecasting the dynamics of one-dimensional CA. Finally, as an example of the application of the selected parameters in high cardinality spaces, results are presented from experiments involving the evolution of radius-3 CA in the Density Classification Task, and radius-2 CA in the Synchronization Task.

scholarly journals Larger than Life: Digital Creatures in a Family of Two-Dimensional Cellular Automata

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Kellie M. Evans
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Nearest Neighbor ◽  
Two Dimensional ◽  
Game Of Life ◽  
Large Neighborhood ◽  
International Audience ◽  
Parameter Values

International audience We introduce the Larger than Life family of two-dimensional two-state cellular automata that generalize certain nearest neighbor outer totalistic cellular automaton rules to large neighborhoods. We describe linear and quadratic rescalings of John Conway's celebrated Game of Life to these large neighborhood cellular automaton rules and present corresponding generalizations of Life's famous gliders and spaceships. We show that, as is becoming well known for nearest neighbor cellular automaton rules, these ``digital creatures'' are ubiquitous for certain parameter values.

Cellular Automata

2017 ◽  
Vol 29 (1) ◽  
pp. 42-50 ◽  
Author(s):  
Rupali Bhardwaj ◽  
Anil Upadhyay
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Empirical Study ◽  
Time Evolution ◽  
Discrete Dynamical Systems ◽  
The State ◽  
Dimensional Lattice ◽  
Elementary Cellular Automata

Cellular automata (CA) are discrete dynamical systems consist of a regular finite grid of cell; each cell encapsulating an equal portion of the state, and arranged spatially in a regular fashion to form an n-dimensional lattice. A cellular automata is like computers, data represented by initial configurations which is processed by time evolution to produce output. This paper is an empirical study of elementary cellular automata which includes concepts of rule equivalence, evolution of cellular automata and classification of cellular automata. In addition, explanation of behaviour of cellular automata is revealed through example.

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