  We study the fluctuation-activated transition process in a system of two
coupled bistable oscillators, in which each oscillator is driven by one
constant force and an independent Gaussian white noise. The transition pathway
has been identified and the transition rate has been computed as the coupling
strength $\mu$ and the mismatch $\sigma$ in the force constants are varied. For
identical oscillators ($\sigma=0$), the transition undergoes a change from a
two-step process with two candidate pathways to a one-step process with also
two candidate pathways to a one-step process with a single pathway as $\mu$ is
increased. For nonidentical oscillators ($\sigma\neq0$), a novel transition
emerges that is a mixture of a two-step pathway and a one-step pathway.
Interestingly, we find that the total transition rate depends nonmonotonically
on $\mu$: a maximal rate appears in an intermediate magnitude of $\mu$.
Moreover, in the presence of weak coupling the rate also exhibits an unexpected
maximum as a function of $\sigma$. The results are in an excellent agreement
with our numerical simulations by forward flux sampling.
