scholarly journals Dynamics and topological entropy of 1D Greenberg–Hastings cellular automata

2020 ◽  
pp. 1-34
Author(s):  
M. KESSEBÖHMER ◽  
J. D. M. RADEMACHER ◽  
D. ULBRICH
Keyword(s):  
Dynamical System ◽  
Topological Entropy ◽  
Travelling Waves ◽  
Excitable Media ◽  
Skew Product ◽  
Invariant Subset ◽  
One Dimensional ◽  
Markov System ◽  
Positive Topological Entropy ◽  
Shift Dynamics

In this paper we analyse the non-wandering set of one-dimensional Greenberg–Hastings cellular automaton models for excitable media with $e\geqslant 1$ excited and $r\geqslant 1$ refractory states and determine its (strictly positive) topological entropy. We show that it results from a Devaney chaotic closed invariant subset of the non-wandering set that consists of colliding and annihilating travelling waves, which is conjugate to a skew-product dynamical system of coupled shift dynamics. Moreover, we determine the remaining part of the non-wandering set explicitly as a Markov system with strictly less topological entropy that also scales differently for large $e,r$ .

A variation on the variational principle and applications to entropy pairs

1997 ◽  
Vol 17 (1) ◽  
pp. 29-43 ◽  
Author(s):  
F. BLANCHARD ◽  
E. GLASNER ◽  
B. HOST
Keyword(s):  
Dynamical System ◽  
Variational Principle ◽  
Invariant Measure ◽  
Topological Entropy ◽  
Invariant Measures ◽  
Open Cover ◽  
Topological Dynamical System ◽  
Positive Topological Entropy ◽  
Finite Open Cover

The variational principle states that the topological entropy of a topological dynamical system is equal to the sup of the entropies of invariant measures. It is proved that for any finite open cover there is an invariant measure such that the topological entropy of this cover is less than or equal to the entropies of all finer partitions. One consequence of this result is that for any dynamical system with positive topological entropy there exists an invariant measure whose set of entropy pairs is equal to the set of topological entropy pairs.

One-Dimensional Semiconjugacy Revisited

2003 ◽  
Vol 13 (07) ◽  
pp. 1657-1663 ◽  
Author(s):  
J. F. Alves ◽  
J. Sousa Ramos
Keyword(s):  
Topological Entropy ◽  
Piecewise Linear ◽  
Linear Map ◽  
One Dimensional ◽  
Piecewise Linear Map ◽  
Piecewise Monotone ◽  
Positive Topological Entropy ◽  
Interval Map

Let f be a piecewise monotone interval map with positive topological entropy h(f)= log (s). Milnor and Thurston showed that f is topological semiconjugated to a piecewise linear map having slope s. Here we prove that these semiconjugacies are the eigenvectors of a certain linear endomorphism associated to f. Using this characterization, we prove a conjecture presented by those authors.

Zero-entropy invariant measures for skew product diffeomorphisms

2009 ◽  
Vol 30 (3) ◽  
pp. 923-930 ◽  
Author(s):  
PENG SUN
Keyword(s):  
Invariant Measure ◽  
Topological Entropy ◽  
Invariant Measures ◽  
Affirmative Answer ◽  
Skew Product ◽  
Hyperbolic Structure ◽  
Combinatorial Properties ◽  
Positive Topological Entropy ◽  
Zero Entropy ◽  
Uniquely Ergodic

AbstractIn this paper, we study some skew product diffeomorphisms with non-uniformly hyperbolic structure along fibers and show that there is an invariant measure with zero entropy which has atomic conditional measures along fibers. For such diffeomorphisms, our result gives an affirmative answer to the question posed by Herman as to whether a smooth diffeomorphism of positive topological entropy would fail to be uniquely ergodic. The proof is based on some techniques that are analogous to those developed by Pesin and Katok, together with an investigation of certain combinatorial properties of the projected return map on the base.

ON THE MONOTONICITY OF ENTROPY FOR MULTILAYER CELLULAR NEURAL NETWORKS

2009 ◽  
Vol 19 (11) ◽  
pp. 3657-3670 ◽  
Author(s):  
JUNG-CHAO BAN ◽  
CHIH-HUNG CHANG
Keyword(s):  
Dynamical System ◽  
Neural Networks ◽  
Topological Entropy ◽  
Cellular Neural Networks ◽  
Strict Monotonicity ◽  
One Dimensional ◽  
Number Of Layers ◽  
Nested Sequence

This work investigates the monotonicity of topological entropy for one-dimensional multilayer cellular neural networks. The interacting radius and number of layers are treated as parameters. Fix either one of them; the set of topological entropies grows as a strictly nested sequence with respect to one another. Apart from the comparison of the set of topological entropies, maximal and minimal templates are indicators of a dynamical system. Our results demonstrate that maximal and minimal templates of larger interacting radius (respectively number of layers) dominate those of smaller one. To be precise, the strict monotonicity of topological entropy is demonstrated through the comparison of the maximal and minimal templates as the parameters are varied.

SOME NONROBUST BERNOULLI-SHIFT RULES

2009 ◽  
Vol 19 (10) ◽  
pp. 3407-3415 ◽  
Author(s):  
LIN CHEN ◽  
FANGYUE CHEN ◽  
WEIFENG JIN ◽  
FANGFANG CHEN ◽  
GUANRONG CHEN
Keyword(s):  
Cellular Automata ◽  
Topological Entropy ◽  
Sequence Space ◽  
Bernoulli Shift ◽  
Binary Sequence ◽  
Class I ◽  
Positive Topological Entropy ◽  
Elementary Cellular Automata ◽  
Topologically Mixing ◽  
Shift Dynamics

In this paper, it is shown that elementary cellular automata rule 172, as a member of the Chua's robust period-1 rules and the Wolfram class I, is also a nonrobust Bernoulli-shift rule. This rule actually exhibits complex Bernoulli-shift dynamics in the bi-infinite binary sequence space. More precisely, in this paper, it is rigorously proved that rule 172 is topologically mixing and has positive topological entropy on a subsystem. Hence, rule 172 is chaotic in the sense of both Li–Yorke and Devaney. The method developed in this paper is also applicable to checking the subshifts contained in other robust period-1 rules, for example, rules 168 and 40, which also represent nonrobust Bernoulli-shift dynamics.

Topological Entropy and Mixing Invariant Extremal Distributional Chaos

2017 ◽  
Vol 27 (09) ◽  
pp. 1750139 ◽  
Author(s):  
Lidong Wang ◽  
Nan Li ◽  
Fengchun Lei ◽  
Zhenyan Chu
Keyword(s):  
Dynamical System ◽  
Topological Entropy ◽  
Distributional Chaos ◽  
Complex Dynamical System ◽  
Positive Topological Entropy

We show that there exists a mixing dynamical system with an invariant, extremal and transitive distributionally scrambled set. Meanwhile, we prove that such a complex dynamical system has zero topological entropy. Finally, we extend the method of constructing the “strong” distributionally scrambled set and show that the new dynamical system has a positive topological entropy.

Entrainment and Annihilation of Reentrant Excitation in a Periodically Stimulated Ring of Excitable Media

1997 ◽  
Vol 36 (04/05) ◽  
pp. 290-293
Author(s):  
L. Glass ◽  
T. Nomura
Keyword(s):  
Cardiac Arrhythmias ◽  
Initial Phase ◽  
Human Heart ◽  
Periodic Wave ◽  
Excitable Media ◽  
One Dimensional ◽  
Clinical Observations ◽  
Periodic Stimulation ◽  
Reentrant Excitation ◽  
Stimulation Of

Abstract:Excitable media, such as nerve, heart and the Belousov-Zhabo- tinsky reaction, exhibit a large excursion from equilibrium in response to a small but finite perturbation. Assuming a one-dimensional ring geometry of sufficient length, excitable media support a periodic wave of circulation. As in the periodic stimulation of oscillations in ordinary differential equations, the effects of periodic stimuli of the periodically circulating wave can be described by a one-dimensional Poincaré map. Depending on the period and intensity of the stimulus as well as its initial phase, either entrainment or termination of the original circulating wave is observed. These phenomena are directly related to clinical observations concerning periodic stimulation of a class of cardiac arrhythmias caused by reentrant wave propagation in the human heart.

SIMPLE AND COMPLEX DYNAMICS FOR ONE-DIMENSIONAL MANIFOLD MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1351-1355
Author(s):  
VLADIMIR FEDORENKO
Keyword(s):  
Topological Entropy ◽  
Complex Dynamics ◽  
Periodic Points ◽  
Dimensional Manifold ◽  
One Dimensional ◽  
Circle Maps ◽  
Interval Maps ◽  
Chain Recurrent Set ◽  
Recurrent Set

We give a characterization of complex and simple interval maps and circle maps (in the sense of positive or zero topological entropy respectively), formulated in terms of the description of the dynamics of the map on its chain recurrent set. We also describe the behavior of complex maps on their periodic points.

Sign in / Sign up

Export Citation Format

Share Document