scholarly journals A Complete Description of the Dynamics of Legal Outer-totalistic Affine Continuous Cellular Automata

2021 ◽  
Author(s):  
Barbara Wolnik ◽  
Marcin Dembowski ◽  
Antoni Augustynowicz ◽  
Bernard De Baets
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Computer Simulations ◽  
Analytical Methods ◽  
Unique Combination ◽  
Binary Case ◽  
Interesting Class

Abstract We present an investigation into the evolution and dynamics of the simplest generalization of binary cellular automata: Affine Continuous Cellular Automata (ACCAs), with [0,1] as state set and local rules that are affine in each variable. We focus on legal outer-totalistic ACCAs, an interesting class of dynamical systems that show some properties that do not occur in the binary case. A unique combination of computer simulations (sometimes quite advanced) and a panoply of analytical methods allow to lay bare the dynamics of each and every one of these cellular automata.

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.

HIGHER ORDER CELLULAR AUTOMATA

2005 ◽  
Vol 08 (02n03) ◽  
pp. 169-192 ◽  
Author(s):  
NILS A. BAAS ◽  
TORBJØRN HELVIK
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Level Structure ◽  
Higher Order ◽  
New Concepts ◽  
Multi Level

We introduce a class of dynamical systems called Higher Order Cellular Automata (HOCA). These are based on ordinary CA, but have a hierarchical, or multi-level, structure and/or dynamics. We present a detailed formalism for HOCA and illustrate the concepts through four examples. Throughout the article we emphasize the principles and ideas behind the construction of HOCA, such that these easily can be applied to other types of dynamical systems. The article also presents new concepts and ideas for describing and studying hierarchial dynamics in general.

scholarly journals Almost all $S$-integer dynamical systems have many periodic points

1998 ◽  
Vol 18 (2) ◽  
pp. 471-486 ◽  
Author(s):  
T. B. WARD
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Zeta Function ◽  
Periodic Points ◽  
Radius Of Convergence ◽  
Dynamical Zeta Function ◽  
Hyperbolic Dynamical Systems ◽  
Almost All

We show that for almost every ergodic $S$-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than $\exp(-\frac{1}{2}h_{\rm top})<1$. In the arithmetic case almost every zeta function is irrational.We conjecture that for almost every ergodic $S$-integer dynamical system the radius of convergence of the zeta function is exactly $\exp(-h_{\rm top})<1$ and the zeta function is irrational.In an important geometric case (the $S$-integer systems corresponding to isometric extensions of the full $p$-shift or, more generally, linear algebraic cellular automata on the full $p$-shift) we show that the conjecture holds with the possible exception of at most two primes $p$.Finally, we explicitly describe the structure of $S$-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems.

Chaotic Dynamics of Partial Difference Equations with Polynomial Maps

2021 ◽  
Vol 31 (09) ◽  
pp. 2150133
Author(s):  
Haihong Guo ◽  
Wei Liang
Keyword(s):  
Dynamical Systems ◽  
Computer Simulations ◽  
Difference Equations ◽  
Chaotic Dynamics ◽  
Discrete Dynamical Systems ◽  
Partial Difference Equations ◽  
Partial Difference ◽  
Polynomial Maps ◽  
Expansion Theory ◽  
Theoretical Results

In this paper, chaotic dynamics of a class of partial difference equations are investigated. With the help of the coupled-expansion theory of general discrete dynamical systems, two chaotification schemes for partial difference equations with polynomial maps are established. These controlled equations are proved to be chaotic either in the sense of Li–Yorke or in the sense of both Li–Yorke and Devaney. One example is provided to illustrate the theoretical results with computer simulations for demonstration.

scholarly journals Estimation of Reverberation Time in Classrooms Using the Residual Minimization Method

2017 ◽  
Vol 42 (4) ◽  
pp. 609-617 ◽  
Author(s):  
Artur Nowoświat ◽  
Marcelina Olechowska
Keyword(s):  
Computer Simulations ◽  
Acoustic Field ◽  
Analytical Methods ◽  
Room Acoustics ◽  
Reverberation Time ◽  
Minimization Method ◽  
Theoretical Methods ◽  
Residual Minimization

Abstract The objective of the residual minimization method is to determine a coefficient correcting the Sabine’s model. The Sabine’s equation is the most commonly applied formula in the designing process of room acoustics with the use of analytical methods. The correction of this model is indispensable for its application in rooms having non-diffusive acoustic field. The authors of the present paper will be using the residual minimization method to work out a suitable correction to be applied for classrooms. For this purpose, five different poorly dampened classrooms were selected, in which the measurements of reverberation time were carried out, and for which reverberation time was calculated with the use of theoretical methods. Three of the selected classrooms had the cubic volume of 258.5 m3 and the remaining two had the cubic volume of 190.8 m3. It was sufficient to estimate the correction for the Sabine’s equation. To verify the results, three other classrooms were selected, in which also the measurements of reverberation time were carried out. The results were verified by means of real measurements of reverberation time and by means of computer simulations in the program ODEON.

Composition and invariance methods for solving some stochastic control problems

1975 ◽  
Vol 7 (02) ◽  
pp. 299-329 ◽  
Author(s):  
V. E. Beneš
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Dynamical Systems ◽  
White Noise ◽  
Stochastic Control ◽  
Analytical Methods ◽  
Control Problems ◽  
Numerical Treatment ◽  
Partial Differential ◽  
Stochastic Control Problems

This paper considers certain stochastic control problems in which control affects the criterion through the process trajectory. Special analytical methods are developed to solve such problems for certain dynamical systems forced by white noise. It is found that some control problems hitherto approachable only through laborious numerical treatment of the non-linear Bellman-Hamilton-Jacobi partial differential equation can now be solved.

MODELING LINEAR DYNAMICAL SYSTEMS BY CONTINUOUS-VALUED CELLULAR AUTOMATA

2007 ◽  
Vol 18 (05) ◽  
pp. 833-848 ◽  
Author(s):  
JUAN CARLOS SECK TUOH MORA ◽  
MANUEL GONZALEZ HERNANDEZ ◽  
NORBERTO HERNANDEZ ROMERO ◽  
AARON RODRIGUEZ TREJO ◽  
SERGIO V. CHAPA VERGARA
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Linear Systems ◽  
Configuration Space ◽  
Integration Method ◽  
Numerical Calculations ◽  
Linear Dynamical Systems ◽  
Original Part

This paper exposes a procedure for modeling and solving linear systems using continuous-valued cellular automata. The original part of this work consists on showing how the cells in the automaton may contain both real values and operators for carrying out numerical calculations and solve a desired problem. In this sense the automaton acts as a program, where data and operators are mixed in the evolution space for obtaining the correct calculations. As an example, Euler's integration method is implemented in the configuration space in order to achieve an approximated solution for a dynamical system. Three examples showing linear behaviors are presented.

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