A Minimum Action Method for Dynamical Systems with Constant Time Delays

Abstract

In this work, we construct a minimum action method for dynamical systems with constant time delays. The minimum action method (MAM) plays an important role in seeking the most probable transition pathway induced by small noise. There exist two formulations of the minimum action method: one is the geometric formulation based on the Maupertuis principle, and the other one is the temporal formulation. The geometric formulation relies on the conservation of Hamiltonian corresponding to the Freidlin--Wentzell action functional. For systems with time delays, the Hamiltonian does not conserve due to the explicit dependence on the time delay, which implies that the geometric MAM is not applicable. We work with the temporal formulation of MAM for problems with time delays. By defining an auxiliary path, we remove the optimization with respect to time through the optimal linear time scaling. The pointwise correspondence between the auxiliary path and the delayed transition path is dealt with by a penalty term included into the action functional. The action functional is then discretized by the finite element method, and strategies for $h$-adaptive mesh refinement have been developed. Numerical examples have been presented to demonstrate the effectiveness of our algorithm.