Directional Metric Entropy and Lyapunov Exponents for Dynamical Systems Generated by Cellular Automata

2015 ◽  
pp. 11-27
Author(s):  
Maurice Courbage ◽  
Brunon Kamiński
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Lyapunov Exponents ◽  
Metric Entropy

scholarly journals Upper bounds on measure-theoretic tail entropy for dominated splittings

2019 ◽  
Vol 40 (9) ◽  
pp. 2305-2316
Author(s):  
YONGLUO CAO ◽  
GANG LIAO ◽  
ZHIYUAN YOU
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Upper Bounds ◽  
Metric Entropy

For differentiable dynamical systems with dominated splittings, we give upper estimates on the measure-theoretic tail entropy in terms of Lyapunov exponents. As our primary application, we verify the upper semi-continuity of metric entropy in various settings with domination.

The metric entropy of random dynamical systems in a Banach space: Ruelle inequality

2012 ◽  
Vol 34 (2) ◽  
pp. 594-615 ◽  
Author(s):  
ZHIMING LI ◽  
LIN SHU
Keyword(s):  
Banach Space ◽  
Dynamical Systems ◽  
Probability Measure ◽  
Lyapunov Exponents ◽  
Compact Set ◽  
Metric Entropy ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Integrability Conditions ◽  
Random Compact Set

AbstractConsider a random cocycle Φ on a separable infinite-dimensional Banach space preserving a probability measure, which is supported on a random compact set. We show that if Φ satisfies some mild integrability conditions on the differentials, then Ruelle’s inequality relating entropy and positive Lyapunov exponents holds.

RELATIVE KOLMOGOROV ENTROPY OF A CHAOTIC SYSTEM IN THE PRESENCE OF NOISE

2008 ◽  
Vol 18 (09) ◽  
pp. 2851-2855 ◽  
Author(s):  
VADIM S. ANISHCHENKO ◽  
SERGEY ASTAKHOV
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Chaotic System ◽  
Stochastic System ◽  
Chaotic Systems ◽  
Metric Entropy ◽  
Kolmogorov Entropy ◽  
Entropy Estimation ◽  
Relative Metric ◽  
Mixing Property

The mixing property is characterized by the metric entropy that is introduced by Kolmogorov for dynamical systems. The Kolmogorov entropy is infinite for a stochastic system. In this work, a relative metric entropy is considered. The relative metric entropy allows to estimate the level of mixing in noisy dynamical systems. An algorithm for calculating the relative metric entropy is described and examples of the metric entropy estimation are provided for certain chaotic systems with various noise intensities. The results are compared to the entropy estimation given by the positive Lyapunov exponents.

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.

scholarly journals A computational framework for finding parameter sets associated with chaotic dynamics

2021 ◽  
pp. 1-11
Author(s):  
S. Koshy-Chenthittayil ◽  
E. Dimitrova ◽  
E.W. Jenkins ◽  
B.C. Dean
Keyword(s):  
Experimental Data ◽  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Parameter Space ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Computational Framework ◽  
Independent Variables ◽  
Systems Model ◽  
Small Support

Many biological ecosystems exhibit chaotic behavior, demonstrated either analytically using parameter choices in an associated dynamical systems model or empirically through analysis of experimental data. In this paper, we use existing software tools (COPASI, R) to explore dynamical systems and uncover regions with positive Lyapunov exponents where thus chaos exists. We evaluate the ability of the software’s optimization algorithms to find these positive values with several dynamical systems used to model biological populations. The algorithms have been able to identify parameter sets which lead to positive Lyapunov exponents, even when those exponents lie in regions with small support. For one of the examined systems, we observed that positive Lyapunov exponents were not uncovered when executing a search over the parameter space with small spacings between values of the independent variables.

Lyapunov Exponents Vs Expansivity and Sensitivity in Cellular Automata

ACRI ’96 ◽  
1997 ◽  
pp. 57-71 ◽  
Author(s):  
Michele Finelli ◽  
Giovanni Manzini ◽  
Luciano Margara
Keyword(s):  
Cellular Automata ◽  
Lyapunov Exponents
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