Güzel İsmail, Munch Elizabeth, Khasawneh Firas A
Department of Mathematics Engineering, İstanbul Technical University, Maslak, İstanbul 34469, Turkey.
Department of Computational Mathematics, Science and Engineering and Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA.
Chaos. 2022 Sep;32(9):093111. doi: 10.1063/5.0102421.
Existing tools for bifurcation detection from signals of dynamical systems typically are either limited to a special class of systems or they require carefully chosen input parameters and a significant expertise to interpret the results. Therefore, we describe an alternative method based on persistent homology-a tool from topological data analysis-that utilizes Betti numbers and CROCKER plots. Betti numbers are topological invariants of topological spaces, while the CROCKER plot is a coarsened but easy to visualize data representation of a one-parameter varying family of persistence barcodes. The specific bifurcations we investigate are transitions from periodic to chaotic behavior or vice versa in a one-parameter collection of differential equations. We validate our methods using numerical experiments on ten dynamical systems and contrast the results with existing tools that use the maximum Lyapunov exponent. We further prove the relationship between the Wasserstein distance to the empty diagram and the norm of the Betti vector, which shows that an even more simplified version of the information has the potential to provide insight into the bifurcation parameter. The results show that our approach reveals more information about the shape of the periodic attractor than standard tools, and it has more favorable computational time in comparison with the Rösenstein algorithm for computing the maximum Lyapunov exponent.
现有的用于从动力系统信号中检测分岔的工具通常要么局限于某一特定类别的系统，要么需要精心选择输入参数且需具备相当专业知识才能解读结果。因此，我们描述了一种基于持久同调的替代方法——拓扑数据分析中的一种工具——该方法利用贝蒂数和克罗克图。贝蒂数是拓扑空间的拓扑不变量，而克罗克图是一个单参数变化的持久条形码族的粗化但易于可视化的数据表示。我们所研究的特定分岔是在一个单参数微分方程集合中从周期行为到混沌行为的转变，反之亦然。我们通过对十个动力系统进行数值实验来验证我们的方法，并将结果与使用最大李雅普诺夫指数的现有工具进行对比。我们进一步证明了到空图的瓦瑟斯坦距离与贝蒂向量范数之间的关系，这表明信息的一个更简化版本有可能为分岔参数提供深入见解。结果表明，我们的方法比标准工具揭示了更多关于周期吸引子形状的信息，并且与用于计算最大李雅普诺夫指数的罗森斯坦算法相比，具有更有利的计算时间。
