Dynamics and topological entropy of 1D Greenberg–Hastings cellular automata
Dynamics and topological entropy of 1D Greenberg–Hastings cellular automata
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DOI：
10.1017/etds.2020.18
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发表时间：
2019-03
影响因子：
0.9
通讯作者：
Marc Kessebohmer;J. Rademacher;Dennis Ulbrich
中科院分区：
文献类型：
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作者：
Marc Kessebohmer;J. Rademacher;Dennis Ulbrich
在本文中，我们分析了具有 $e\geqslant 1$ 激发态和 $r\geqslant 1$ 不应态的可激发介质的一维 Greenberg-Hastings 元胞自动机模型的非游走集，并确定其（严格正的）拓扑熵。我们证明它是由非游走集的德瓦尼混沌闭合不变子集产生的，该子集由碰撞和湮灭行波组成，与耦合位移动力学的斜积动力学系统共轭。此外，我们将非游走集的剩余部分明确确定为马尔可夫系统，其拓扑熵严格较小，对于大的 $e,r$ 也有不同的缩放比例。
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 $e,r$ .