We propose a topological framework for the detection of Hopf bifurcations directly from time series, based on persistent homology applied to phase space reconstructions via Takens embedding within the framework of Topological Data Analysis. The central idea is that changes in the dynamical regime are reflected in the emergence or disappearance of a dominant one-dimensional homological features in the reconstructed attractor. To quantify this behavior, we introduce a simple and interpretable scalar topological functional defined as the maximum persistence of homology classes in dimension one. This functional is used to construct a computable criterion for identifying critical parameters in families of dynamical systems without requiring knowledge of the underlying equations. The proposed approach is validated on representative systems of increasing complexity, showing consistent detection of the bifurcation point. The results support the interpretation of dynamical transitions as topological phase transitions and demonstrate the potential of topological data analysis as a model-free tool for the quantitative analysis of nonlinear time series.

翻译：我们提出了一种直接从时间序列检测Hopf分岔的拓扑框架，该框架基于拓扑数据分析范畴内通过Takens嵌入进行相空间重构所应用的持续同调方法。其核心思想在于：动力学 regime 的变化会反映在重构吸引子中主导性一维同调特征的出现或消失上。为量化这一行为，我们引入一个简洁且可解释的标量拓扑泛函，定义为一维同调类的最大持续度。该泛函用于构建一个可计算的判据，以识别动力系统族中的临界参数，而无需了解其底层方程。所提方法在复杂度递增的代表性系统上进行了验证，结果表明能一致地检测出分岔点。这些结果支持将动力学转变解释为拓扑相变，并展示了拓扑数据分析作为无模型工具用于非线性时间序列定量分析的潜力。