Topological Detection of Hopf Bifurcations via Persistent Homology: A Functional Criterion from Time Series

TL;DR

This paper introduces a topological data analysis method using persistent homology to detect Hopf bifurcations directly from time series data, without needing the underlying equations.

Contribution

It presents a novel, interpretable scalar topological functional for identifying bifurcation points from phase space reconstructions.

Findings

Successfully detects bifurcation points in complex dynamical systems.

Validates the approach on various systems with consistent results.

Supports the view of dynamical transitions as topological phase transitions.

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.