Border-collision bifurcations from stable fixed points to any number of coexisting chaotic attractors

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-10-09 DOI:10.1080/10236198.2023.2265495

D. J. W. Simpson

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引用次数: 3

Abstract

AbstractIn diverse physical systems stable oscillatory solutions devolve into more complicated solutions through border-collision bifurcations. Mathematically these occur when a stable fixed point of a piecewise-smooth map collides with a switching manifold as parameters are varied. The purpose of this paper is to highlight the extreme complexity possible in the subsequent dynamics. By perturbing instances of the n-dimensional border-collision normal form for which the nth iterate is a direct product of chaotic skew tent maps, it is shown that many chaotic attractors can arise. Burnside's lemma is used to count the attractors; chaoticity is proved by demonstrating that some iterate of the map is piecewise-expanding. The resulting transition from a stable fixed point to many coexisting chaotic attractors occurs throughout open subsets of parameter space and is not destroyed by higher order terms, hence can be expected to occur generically in mathematical models.Keywords: Piecewise-linearpiecewise-smoothborder-collision bifurcationrobust chaosBurnside's lemmaMathematics Subject Classifications: 37G3539A28 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work was supported by Marsden Fund contract MAU1809, managed by Royal Society Te Apārangi. The author thanks Paul Glendinning and Chris Tuffley for discussions that helped improve the results.

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从稳定不动点到任意数量共存混沌吸引子的边界碰撞分岔

在多种物理系统中，稳定振荡解通过边界碰撞分岔演化为更复杂的解。在数学上，当一个分段平滑映射的稳定不动点与一个随参数变化的切换流形发生碰撞时，就会发生这种情况。本文的目的是强调后续动力学中可能存在的极端复杂性。通过扰动n维边界碰撞范式的实例，其中n次迭代是混沌斜帐篷映射的直接乘积，证明了可以产生许多混沌吸引子。Burnside引理用于计算吸引子;混沌性是通过证明映射的某些迭代是分段展开的来证明的。由此产生的从一个稳定不动点到许多共存混沌吸引子的转变发生在参数空间的开放子集中，并且不被高阶项破坏，因此可以预期在数学模型中一般会发生。关键词:分段-线性分段-平滑边界-碰撞分岔鲁棒混沌burnside定理数学学科分类:37G3539A28披露声明作者未报告潜在利益冲突。这项工作得到了马斯登基金合同MAU1809的支持，由皇家学会网站Apārangi管理。作者感谢Paul Glendinning和Chris Tuffley的讨论，帮助改进了结果。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.