@tran2014]. Further comparative studies of the stochastic resonance for the diffusion case $X^\epsilon_t$ versus the Markov Chain $Y^\epsilon_t$ case were done by Hermann, Imkeller, Pavlyukevich and Peithmann in [@ld_3_comp_4_imkeller2004stochastic; @comp_2_Herrmann2005; @ld_2_comp_3_herrmann2005large; @herrmann2005]. A collection of papers on comparative studies between stochastic resonance in diffusion and Markov Chains can be found in the monograph [@tran2014]. One of the main conclusions in [@ld_2_comp_3_herrmann2005large; @herrmann2005; @comp_2_Herrmann2005; @tran2014] is rigorously showing that using linear response and signal-to-noise ratio to analyse stochastic resonance in the diffusion case $X^\epsilon_t$ gives a different result to analysing the Markov Chain case $Y^\epsilon_t=\pm1$ with the same techniques even asymptotically in the small noise limit. Other common methods used to study stochastic resonance include invariant measures and Fourier transforms. We consider six measures of stochastic resonance frequently used and considered by Pavlyukevich in his thesis [@pav_thesis; @tran2014] which are linear response, signal-to-noise ratio, energy, out-of-phase measure, relative entropy and entropy.

In this paper we will study stochastic resonance on a two dimensional toy model, in both the diffusion and Markov Chain cases, and where there are two independent pathways between the wells going through two different saddles. The escape times and the six measures of stochastic resonance introduced above are 