Bifurcation formula for transition paths in stochastic dynamical systems by spectral flow

Jinqiao Duan; Zhihao Zhao

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arXiv · 2508.01954

Bifurcation formula for transition paths in stochastic dynamical systems by spectral flow

Jinqiao Duan·Zhihao Zhao

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Abstract

This paper investigates bifurcation phenomena and stability of most probable transition paths (MPTPs) in stochastic dynamical systems through a combined variational and spectral flow approach. Within the Onsager-Machlup framework, MPTPs are characterized as minimizers of an energy-dependent Lagrangian functional incorporating noise intensity. Existence criteria for such minimizers are established through critical value analysis and variational techniques. The main theoretical advancement is a spectral flow formula that detects bifurcation points and quantifies stability changes under noise perturbations. Specifically, the analysis reveals: (i) noise-sensitive MPTPs where variations in noise intensity destroy the minimizer property, and (ii) noise-robust MPTPs where stability is maintained despite finite noise fluctuations. These results establish a correspondence between Lagrangian bifurcations and stochastic phase transitions, providing a mathematical foundation for predicting noise-driven transition mechanisms in stochastic systems.

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Jinqiao Duan, Zhihao Zhao. 2025-08-03. Bifurcation formula for transition paths in stochastic dynamical systems by spectral flow. https://arxiv.org/abs/2508.01954

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