  We consider stochastic resonance for a diffusion with drift given by a
potential, which has two metastable states and two pathways between them.
Depending on the direction of the forcing, the height of the two barriers, one
for each path, will either oscillate alternating or in synchronisation. We
consider a simplified model given by a continuous time Markov Chains with two
states. This was done for alternating and synchronised wells. The invariant
measures are derived for both cases and shown to be constant for the
synchronised case. A PDF for the escape time from an oscillatory potential is
studied. Methods of detecting stochastic resonance are presented, which are
linear response, signal-noise ratio, energy, out-of-phase measures, relative
entropy and entropy. A new statistical test called the conditional
Kolmogorov-Smirnov test is developed, which can be used to analyse stochastic
resonance. An explicit two dimensional potential is introduced, the critical
point structure derived and the dynamics, the invariant state and escape time
studied numerically. The six measures are unable to detect the stochastic
resonance in the case of synchronised saddles. The distribution of escape times
however not only shows a clear sign of stochastic resonance, but changing the
direction of the forcing from alternating to synchronised saddles an additional
resonance at double the forcing frequency starts to appear. The conditional KS
test reliably detects the stochastic resonance. This paper is mainly based on
the thesis "Stochastic Resonance for a Model with Two Pathways"
