The topology of a chaotic attractor in the Kuramoto-Sivashinsky equation

被引:1
作者
Abadie, Marie [1 ]
Beck, Pierre [2 ]
Parker, Jeremy P. [3 ]
Schneider, Tobias M. [2 ]
机构
[1] Univ Luxembourg, Dept Math, 6 Ave Fonte, L-4364 Esch Sur Alzette, Luxembourg
[2] Ecole Polytech Fed Lausanne, Emergent Complex Phys Syst Lab ECPS, CH-1015 Lausanne, Switzerland
[3] Univ Dundee, Div Math, Dundee DD1 4HN, Scotland
基金
欧洲研究理事会;
关键词
UNSTABLE PERIODIC-ORBITS; TEMPLATE ANALYSIS; SYSTEMS;
D O I
10.1063/5.0237476
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
The Birman-Williams theorem gives a connection between the collection of unstable periodic orbits (UPOs) contained within a chaotic attractor and the topology of that attractor, for three-dimensional systems. In certain cases, the fractal dimension of a chaotic attractor in a partial differential equation (PDE) is less than three, even though that attractor is embedded within an infinite-dimensional space. Here, we study the Kuramoto-Sivashinsky PDE at the onset of chaos. We use two different dimensionality-reduction techniques-proper orthogonal decomposition and an autoencoder neural network-to find two different mappings of the chaotic attractor into three dimensions. By finding the image of the attractor's UPOs in these reduced spaces and examining their linking numbers, we construct templates for the branched manifold, which encodes the topological properties of the attractor. The templates obtained using two different dimensionality reduction methods are equivalent. The organization of the periodic orbits is identical and consistent symbolic sequences for low-period UPOs are derived. While this is not a formal mathematical proof, this agreement is strong evidence that the dimensional reduction is robust, in this case, and that an accurate topological characterization of the chaotic attractor of the chaotic PDE has been achieved. (c) 2025 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/)
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页数:14
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