Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems

TL;DR

This paper introduces BuZZ, a novel method using zigzag persistent homology to efficiently detect bifurcations in dynamical systems through a single persistence diagram, reducing computational costs.

Contribution

The paper presents BuZZ, a new one-step zigzag persistence-based approach for detecting bifurcations, improving efficiency over traditional methods that analyze multiple diagrams.

Findings

Successfully detects bifurcations in synthetic examples.

Effective in identifying behavior changes in a real dynamical system.

Reduces computational complexity compared to existing methods.

Abstract

Bifurcations in dynamical systems characterize qualitative changes in the system behavior. Therefore, their detection is important because they can signal the transition from normal system operation to imminent failure. While standard persistent homology has been used in this setting, it usually requires analyzing a collection of persistence diagrams, which in turn drives up the computational cost considerably. Using zigzag persistence, we can capture topological changes in the state space of the dynamical system in only one persistence diagram. Here we present Bifurcations using ZigZag (BuZZ), a one-step method to study and detect bifurcations using zigzag persistence. The BuZZ method is successfully able to detect this type of behavior in two synthetic examples as well as an example dynamical system.