resonance.

Consider the case $\Omega=0$ and where the noise $\epsilon$ is very small. The particle will stay very close to one of the wells of the potential and will occasionally escape to the other well. The time of the actual transition from one well to the other is very short compare to the time it stays in any particular well.

Now consider the case where $\Omega>0$. For particular choices of $\Omega>0$ and $\epsilon>0$, these transitions between the two wells will become synchronised with the driving frequency $\Omega$. This is called stochastic resonance. Thus the term *noise induced synchronisation* was used for systems where the amplitude of the forcing $F$ was not large [@tk_14; @tk_15] (see also the discussions in [@tk_16]). New insights into the exact manner of these synchronised transitions will be studied in this paper, which may be more appropriate in light of the results obtained in this paper.

For small noise $\epsilon$, one would expect that stochastic resonance depends only on the essential properties of the system, such as the height difference between the wells and the pathways for escape. We investigate what effects these multiple pathways have on the appearance of stochastic resonance. Varying $F$, $\Omega$ and 