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arXiv · 1607.02180

Minimal cover of high-dimensional chaotic attractors by embedded recurrent patterns

Daniel L. Crane·Ruslan L. Davidchack·Alexander N. Gorban
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Abstract

We propose a general method for constructing a minimal cover of high-dimensional chaotic attractors by embedded unstable recurrent patterns. By minimal cover we mean a subset of available patterns such that the approximation of chaotic dynamics by a minimal cover with a predefined proximity threshold is as good as the approximation by the full available set. The proximity measure, based on the concept of a directed Hausdorff distance, can be chosen with considerable freedom and adapted to the properties of a given chaotic system. In the context of a spatiotemporally chaotic attractor of the Kuramoto--Sivashinsky system on a periodic domain, we demonstrate that the minimal cover can be faithfully constructed even when the proximity measure is defined within a subspace of dimension much smaller than the dimension of space containing the attractor. We discuss how the minimal cover can be used to provide a reduced description of the attractor structure and the dynamics on it.

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Daniel L. Crane, Ruslan L. Davidchack, Alexander N. Gorban. 2016-07-07. Minimal cover of high-dimensional chaotic attractors by embedded recurrent patterns. https://doi.org/10.1016/j.cnsns.2024.108345

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