Abstract
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 $e,r$.
Highlights
Following Greenberg, Hastings and Hassard [6], we consider a basic cellular automaton model of an excitable medium based on the alphabetA := {0, 1, 2, . . . , e, e + 1, e + 2, . . . , e + r }, of cardinality #A := card(A) = a + 1 for some positive integers e, r and a := e + r
In this paper we analyse the non-wandering set of one-dimensional Greenberg– Hastings cellular automaton models for excitable media with e 1 excited and r 1 refractory states and determine its 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 skewproduct dynamical system of coupled shift dynamics
Ulbrich scientific contexts such as theoretical cardiology, neuroscience, chemistry, transition to turbulence, and surface catalysis, and it is a paradigm of nonlinear dynamics, selforganization and pattern formation [9, 11]
Summary
Following Greenberg, Hastings and Hassard [6], we consider a basic cellular automaton model of an excitable medium based on the alphabet. As will be discussed more in the sequel, Z consists of configurations which are either purely left- or right-moving under the dynamics of T , or sequences of local pulses wL (leftwards) and wR (rightwards) glued at one position, which annihilate each other in time. The proof is an adaption and more complete exposition of the technique in [5] for the case e = r = 1, which is special as Y = Z , which has been shown by the third author in [19], that is, eventual image and pulse-collision subsystem coincide in this case only We remark that it is purely a combinatorial counting argument of space– time windows and in this sense independent of the topology. In this proof the largest growth rate of different space–time windows, and the topological entropy, stems from counting enduring annihilations, that is, elements in the set Z∞. 1}, on which the standard right shift σR,− and left shift σL,+ are defined with pseudo-inverses for m < 0 given by σRm,−((. . . , x−1, x0)) := (. . . , x−1, x0, 0m ), σLm,+((x0, x1, . . .)) := (0m , x0, x1, . . .)
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