  We study the noise-induced escape from a stable limit cycle of a non-gradient
dynamical system driven by a small additive noise. The fact that the optimal
transition path in this case is infinitely long imposes a severe numerical
challenge to resolve it in the minimum action method. We first consider the
landscape of the quasi-potential near the limit cycle, which characterizes the
minimal cost of the noise to drive the system far away form the limit cycle. We
derive and compute the quadratic approximation of this quasi-potential near the
limit cy- cle in the form of a positive definite solution to a matrix-valued
periodic Riccati differential equation on the limit cycle. We then combine this
local approxima- tion in the neighbourhood of the limit cycle with the minimum
action method applied outside of the neighbourhood. The neighbourhood size is
selected to be compatible with the path discretization error. By several
numerical examples, we show that this strategy effectively improve the minimum
action method to compute the spiral optimal escape path from limit cycles in
various systems.
