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

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.

Figures (6)

• Figure 1: Persistence diagrams associated with the two-dimensional Takens embeddings of the Hopf normal form for representative values of the bifurcation parameter $\mu$: (a) $\mu=-1.0$, (b) $\mu=-0.5$, (c) $\mu=-0.1$, (d) $\mu=-0.05$, (e) $\mu=0.0$, (f) $\mu=0.05$, (g) $\mu=0.1$, (h) $\mu=0.5$, and (i) $\mu=1.0$. Blue points correspond to $H_0$ classes and red points to $H_1$ classes. For negative values of $\mu$, the diagrams remain concentrated near the diagonal, indicating the absence of a robust one-dimensional topological structure. As $\mu$ approaches the critical value and becomes positive, a persistent $H_1$ class emerges and becomes increasingly separated from the diagonal, consistently reflecting the onset of the limit cycle generated by the Hopf bifurcation.
• Figure 2: Topological functionals associated with the proposed criterion for detecting the Hopf bifurcation. (A) The maximum persistence in $H_1$, $\max(d-b)$, serves as the primary observable, capturing the emergence of a dominant homology class near the critical value $\mu_c=0$ and its subsequent strengthening for $\mu>0$. (B) The $L^1$ norm of the Betti vector, $\|Bv_1\|_1$, provides a complementary validation, exhibiting a consistent transition from near-zero to positive values as the limit cycle develops. Together, these functionals confirm the presence and localization of the topological transition associated with the Hopf bifurcation.
• Figure 3: Sensitivity of the topological functional $\mathcal{H}(\mu)$ to the embedding parameters in the Hopf normal form. (A) Variation with respect to the embedding dimension $m$ for fixed delay $\tau = 26$. (B) Variation with respect to the delay $\tau$ for fixed embedding dimension $m = 2$. In both cases, while the amplitude of the functional changes slightly, the qualitative behavior is preserved and the transition is consistently detected near the critical parameter value.
• Figure 4: Topological observables for the Lorenz system.
• Figure 5: Topological analysis of the BZ reaction.

Theorems & Definitions (6)

• Remark 1: Parameter selection
• Definition 1: Dominant topological functional
• Theorem 1: Hopf topological criterion
• Remark 2
• proof : Proof sketch
• Definition 2: Topological estimator