Delay induced bifurcation of dominant transition pathways

TL;DR

This paper studies how time delay influences the dominant transition pathways in stochastic systems, revealing bifurcations and their effects on transition rates through stability analysis and numerical simulations.

Contribution

It introduces a bifurcation analysis of dominant transition pathways in delayed stochastic systems, extending the minimum action method to include delay effects.

Findings

Delay induces bifurcation of DTPs when delay is large enough

Transition rate constants depend nontrivially on delay time

Bifurcation diagram mapped on the delay and non-conservation parameters

Abstract

We investigate delay effects on dominant transition pathways (DTP) between metastable states of stochastic systems. A modified version of the Maier-Stein model with linear delayed feedback is considered as an example. By a stability analysis of the {"on-axis"} DTP in trajectory space, we find that a bifurcation of DTPs will be induced when time delay $\tau$ is large enough. This finding is soon verified by numerically derived DTPs which are calculated by employing a recently developed minimum action method extended to delayed stochastic systems. Further simulation shows that, the delay-induced bifurcation of DTPs also results in a nontrivial dependence of the transition rate constant on the delay time. Finally, the bifurcation diagram is given on the $\tau-\beta$ plane, where $\beta$ measures the non-conservation of the original Maier-Stein model.