The theory of LΩEGF and the scaling theory on multidimensional Fokker-Planck equations with gradient potential are applied to derive the probability transition rate for Brownian motion in a symmetric two-dimensional bistable potential. Based on the rate method in adiabatic elimination, the phenomenon of stochastic resonance is studied in this system. Compared with the known results, the analytic expression for signal-to-noise ratio derived in this study is independent of the intensity of friction,i.e. not only applicable to an overdamped system, but also applicable to a lightly damped system. Numerical simulations fruther verify the significance of the analytically approximate result.
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