🤖 AI Summary
This work proposes a model-free method for detecting Hopf bifurcation points directly from time series without prior knowledge of the underlying dynamical equations. By leveraging Takens’ embedding to reconstruct the phase space and applying one-dimensional persistent homology, the approach introduces maximal persistence—a scalar topological functional with clear interpretability—as a bifurcation criterion, framing dynamical transitions as topological phase changes. Experimental validation on several canonical nonlinear systems demonstrates that the method reliably and accurately identifies Hopf bifurcations, confirming its effectiveness and broad applicability. This study thus offers a novel topological perspective for data-driven bifurcation analysis.
📝 Abstract
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.