  We aim to increase the ability of a of coupled phase oscillators to maintain
the synchronization when the system is affected by stochastic disturbances. We
model the disturbances by Gaussian noise and use the mean first hitting time
when the state hits the boundary of a secure domain, that is a subset of the
basin of the attraction, to measure the synchronization stability. Based on the
invariant probability distribution of a system of phase oscillators subject to
Gaussian disturbances, we propose an optimization method to increase the mean
first hitting time, and thus increase the synchronization stability. In this
method, a new metric for the synchronization stability is defined as the
probability of the state being absent from the secure domain, which reflects
the impact of all the system parameters and the strength of the disturbances.
Furthermore, by this new metric, one may identify those edges which may lead to
desynchronization with a high risk. A case study shows that the mean first
hitting time is dramatically increased after solving the corresponding
optimization problems and the vulnerable edges are effectively identified. It
is also found that optimizing the synchronization by maximizing the order
parameter or the phase cohesiveness may dramatically increase the value of the
metric and decrease the mean first hitting time, thus decrease the
synchronization stability.
