Abadie Marie, Beck Pierre, Parker Jeremy P, Schneider Tobias M
Department of Mathematics, University of Luxembourg, 6, Av. de la Fonte, 4364 Esch-sur-Alzette, Luxembourg.
Emergent Complexity in Physical Systems Laboratory (ECPS), École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland.
Chaos. 2025 Jan 1;35(1). doi: 10.1063/5.0237476.
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.
对于三维系统，伯曼 - 威廉姆斯定理给出了混沌吸引子中包含的不稳定周期轨道（UPOs）集合与该吸引子拓扑结构之间的联系。在某些情况下，偏微分方程（PDE）中混沌吸引子的分形维数小于3，尽管该吸引子嵌入在无限维空间中。在此，我们研究混沌起始时的Kuramoto - Sivashinsky偏微分方程。我们使用两种不同的降维技术——本征正交分解和自编码器神经网络——来找到混沌吸引子到三维空间的两种不同映射。通过在这些降维空间中找到吸引子的UPOs的图像并检查它们的环绕数，我们构建了分支流形的模板，该模板编码了吸引子的拓扑性质。使用两种不同降维方法获得的模板是等效的。周期轨道的组织是相同的，并且推导了低周期UPOs的一致符号序列。虽然这不是一个正式的数学证明，但这种一致性是有力的证据，表明在这种情况下降维是可靠的，并且已经实现了对混沌偏微分方程混沌吸引子的准确拓扑表征。
