Minimal cover of high-dimensional chaotic attractors by embedded recurrent patterns
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2016
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| _version_ | 1866913508516429824 |
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| author | Crane, Daniel L. Davidchack, Ruslan L. Gorban, Alexander N. |
| author_facet | Crane, Daniel L. Davidchack, Ruslan L. Gorban, Alexander N. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1607_02180 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Minimal cover of high-dimensional chaotic attractors by embedded recurrent patterns Crane, Daniel L. Davidchack, Ruslan L. Gorban, Alexander N. Chaotic Dynamics 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. |
| title | Minimal cover of high-dimensional chaotic attractors by embedded recurrent patterns |
| topic | Chaotic Dynamics |
| url | https://arxiv.org/abs/1607.02180 |