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

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.