Symbolic Dynamics of Coupled Map Lattices

2006 ◽  
Vol 96 (3) ◽  
Author(s):  
Shawn D. Pethel ◽  
Ned J. Corron ◽  
Erik Bollt
Keyword(s):  
Symbolic Dynamics ◽  
Coupled Map Lattices

A method of recovering the initial vectors of globally coupled map lattices based on symbolic dynamics

2011 ◽  
Vol 20 (12) ◽  
pp. 120507 ◽  
Author(s):  
Li-Sha Sun ◽  
Xiao-Yun Kang ◽  
Qiong Zhang ◽  
Lan-Xin Lin
Keyword(s):  
Symbolic Dynamics ◽  
Coupled Map Lattices

scholarly journals Extensive bounds on the topological entropy of repellers in piecewise expanding coupled map lattices

2012 ◽  
Vol 33 (3) ◽  
pp. 870-895 ◽  
Author(s):  
RICARDO COUTINHO ◽  
BASTIEN FERNANDEZ
Keyword(s):  
Topological Entropy ◽  
Symbolic Dynamics ◽  
Interaction Strength ◽  
Coupled Map Lattices ◽  
Piecewise Affine ◽  
Small Coupling ◽  
Rigorous Description ◽  
A Chain ◽  
The Individual ◽  
Bounded Below

AbstractBeyond the uncoupled regime, the rigorous description of the dynamics of (piecewise) expanding coupled map lattices remains largely incomplete. To address this issue, we study repellers of periodic chains of linearly coupled Lorenz-type maps which we analyze by means of symbolic dynamics. Whereas all symbolic codes are admissible for sufficiently small coupling intensity, when the interaction strength exceeds a chain length independent threshold, we prove that a large bunch of codes is pruned and an extensive decay follows suit for the topological entropy. This quantity, however, does not immediately drop off to 0. Instead, it is shown to be continuous at the threshold and to remain extensively bounded below by a positive number in a large part of the expanding regime. The analysis is firstly accomplished in a piecewise affine setting where all calculations are explicit and is then extended by continuation to coupled map lattices based on$C^1$-perturbations of the individual map.

scholarly journals Initial condition estimate of coupled map lattices system based on symbolic dynamics

2009 ◽  
Vol 58 (5) ◽  
pp. 2921
Author(s):  
Shen Min-Fen ◽  
Lin Lan-Xin ◽  
Li Xiao-Yan ◽  
Chang Chun-Qi
Keyword(s):  
Symbolic Dynamics ◽  
Coupled Map Lattices ◽  
Initial Condition

ASYMPTOTIC STATES IN COUPLED MAP LATTICES

2008 ◽  
Vol 18 (02) ◽  
pp. 285-311
Author(s):  
ALEXANDRE TAVARES BARAVIERA ◽  
MARCELO MENDES DISCONZI
Keyword(s):  
Symbolic Dynamics ◽  
Numerical Experiments ◽  
Invariant Measures ◽  
Asymptotic State ◽  
Coupled Map Lattices ◽  
Small Contribution ◽  
Common Language ◽  
Rigorous Results ◽  
Asymptotic Behaviors ◽  
And Mathematics

Coupled Map Lattices (CML) are a kind of dynamical systems that appear naturally in some contexts, like the discretization of partial differential equations, and as a simple model of coupling between nonlinear systems. The coupling creates new and rich properties, that has been the object of intense investigation during the last decades. In this work we have two goals: first we give a nontechnical introduction to the theory of invariant measures and equilibrium in dynamics (with analogies with equilibrium in statistical mechanics) because we believe that sometimes a lot of interesting problems on the interface between physics and mathematics are not being developed simply due to the lack of a common language. Our second goal is to make a small contribution to the theory of equilibrium states for CML. More specifically, we show that a certain family of Coupled Map Lattices presents different asymptotic behaviors when some parameters (including coupling) are changed. The goal is to show that we start in a configuration with infinitely many different measures and, with a slight change in coupling, get an asymptotic state with only one measure describing the dynamics of most orbits coupling. Associating a symbolic dynamics with symbols +1, 0 and -1 to the system we describe a transition characterized by an asymptotic state composed only of symbols +1 or only of symbols -1. We rigorously prove our assertions and provide numerical experiments with two goals: first, as illustration of our rigorous results and second, to motivate some conjectures concerning the problem and some of its possible variations.

D-test: A New Test for Analyzing Scale Invariance Using Symbolic Dynamics and Symbolic Entropy

Methodology ◽  
2011 ◽  
Vol 7 (3) ◽  
pp. 88-95 ◽  
Author(s):  
Jose A. Martínez ◽  
Manuel Ruiz Marín
Keyword(s):  
Symbolic Dynamics ◽  
Scale Invariance ◽  
Rating Scales ◽  
Rating Scale ◽  
Marketing Research ◽  
Response Patterns ◽  
Measurement Scales ◽  
Improve Measurement ◽  
Entropy Measures ◽  
The Difference

The aim of this study is to improve measurement in marketing research by constructing a new, simple, nonparametric, consistent, and powerful test to study scale invariance. The test is called D-test. D-test is constructed using symbolic dynamics and symbolic entropy as a measure of the difference between the response patterns which comes from two measurement scales. We also give a standard asymptotic distribution of our statistic. Given that the test is based on entropy measures, it avoids smoothed nonparametric estimation. We applied D-test to a real marketing research to study if scale invariance holds when measuring service quality in a sports service. We considered a free-scale as a reference scale and then we compared it with three widely used rating scales: Likert-type scale from 1 to 5 and from 1 to 7, and semantic-differential scale from −3 to +3. Scale invariance holds for the two latter scales. This test overcomes the shortcomings of other procedures for analyzing scale invariance; and it provides researchers a tool to decide the appropriate rating scale to study specific marketing problems, and how the results of prior studies can be questioned.

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