Exponential dichotomy and admissibility of linearized skew-product semiflows defined on a compact positively invariant subset of semiflows

2009 ◽  
Vol 10 (4) ◽  
pp. 2062-2071 ◽  
Author(s):  
Bin-Guo Wang ◽  
Zhi-Cheng Wang
Keyword(s):  
Exponential Dichotomy ◽  
Skew Product ◽  
Invariant Subset

scholarly journals On Exponential Dichotomy of Variational Difference Equations

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
Bogdan Sasu
Keyword(s):  
Difference Equations ◽  
Exponential Dichotomy ◽  
New Method ◽  
Skew Product ◽  
Skew Product Flows

We give very general characterizations for uniform exponential dichotomy of variational difference equations. We propose a new method in the study of exponential dichotomy based on the convergence of some associated series of nonlinear trajectories. The obtained results are applied to difference equations and also to linear skew-product flows.

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 $e,r$ .

scholarly journals On Dichotomous Behavior of Variational Difference Equations and Applications

2009 ◽  
Vol 2009 ◽  
pp. 1-16 ◽  
Author(s):  
Bogdan Sasu
Keyword(s):  
Control System ◽  
Difference Equations ◽  
Exponential Dichotomy ◽  
Sequence Spaces ◽  
Skew Product ◽  
Skew Product Flows

We give new and very general characterizations for uniform exponential dichotomy of variational difference equations in terms of the admissibility of pairs of sequence spaces overℕwith respect to an associated control system. We establish in the variational case the connections between the admissibility of certain pairs of sequence spaces overℕand the admissibility of the corresponding pairs of sequence spaces overℤ. We apply our results to the study of the existence of exponential dichotomy of linear skew-product flows.

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