Chutani Malayaja, Rao Nithyanand, Nirmal Thyagu N, Gupte Neelima
Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India.
Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India.
Chaos. 2020 Jan;30(1):013109. doi: 10.1063/1.5100362.
We analyze the time series obtained from different dynamical regimes of evolving maps and flows by constructing their equivalent time series networks, using the visibility algorithm. The regimes analyzed include periodic, chaotic, and hyperchaotic regimes, as well as intermittent regimes and regimes at the edge of chaos. We use the methods of algebraic topology, in particular, simplicial complexes, to define simplicial characterizers, which can analyze the simplicial structure of the networks at both the global and local levels. The simplicial characterizers bring out the hierarchical levels of complexity at various topological levels. These hierarchical levels of complexity find the skeleton of the local dynamics embedded in the network, which influence the global dynamical properties of the system and also permit the identification of dominant motifs. We also analyze the same networks using conventional network characterizers such as average path lengths and clustering coefficients. We see that the simplicial characterizers are capable of distinguishing between different dynamical regimes and can pick up subtle differences in dynamical behavior, whereas the usual characterizers provide a coarser characterization. However, the two taken in conjunction can provide information about the dynamical behavior of the time series, as well as the correlations in the evolving system. Our methods can, therefore, provide powerful tools for the analysis of dynamical systems.
我们通过使用可见性算法构建等效时间序列网络，来分析从演化映射和流的不同动力学状态获得的时间序列。所分析的状态包括周期、混沌和超混沌状态，以及间歇状态和混沌边缘状态。我们使用代数拓扑方法，特别是单纯复形，来定义单纯特征描述符，它可以在全局和局部层面分析网络的单纯结构。单纯特征描述符揭示了不同拓扑层面的复杂层次。这些复杂层次找到了嵌入在网络中的局部动力学骨架，其影响系统的全局动力学特性，还能识别主导模式。我们还使用诸如平均路径长度和聚类系数等传统网络特征描述符来分析相同的网络。我们发现，单纯特征描述符能够区分不同的动力学状态，并能捕捉到动力学行为中的细微差异，而通常的特征描述符提供的是更粗略的描述。然而，两者结合可以提供有关时间序列动力学行为以及演化系统中相关性的信息。因此，我们的方法可以为动力系统分析提供强大的工具。
