Extensive bounds on the topological entropy of repellers in piecewise expanding coupled map lattices
Extensive bounds on the topological entropy of repellers in piecewise expanding coupled map lattices
复制标题
分段扩展耦合映射晶格中排斥极拓扑熵的广泛界限
DOI：
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发表时间：
2012
影响因子：
0.9
通讯作者：
B. Fernandez
中科院分区：
文献类型：
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作者：
R. Coutinho;B. Fernandez
摘要 除了非耦合状态之外，对（分段）扩展耦合映射晶格动力学的严格描述在很大程度上仍然不完整。为了解决这个问题，我们研究了线性耦合洛伦兹型映射的周期链的排斥子，并通过符号动力学进行分析。尽管所有符号代码在足够小的耦合强度下都是可接受的，但当相互作用强度超过与链长度无关的阈值时，我们证明一大堆代码被修剪，并且拓扑熵会发生广泛的衰减。然而，这个数量并不会立即下降到 0。相反，它在阈值处是连续的，并且在扩张区域的大部分区域中保持在正数下方。该分析首先在分段仿射设置中完成，其中所有计算都是显式的，然后通过延续到基于单个映射的 $C^1$ 扰动的耦合映射晶格来扩展。
Abstract Beyond 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.