Limit cycles for extended bistable stochastic resonance system

Bistable system is a typical model in stochastic resonance (SR) researching. And it has been known that the deterministic dynamics of a noised system plays an important role in making SR phenomena occur. By embedding a non-autonomous system onto a cylinder, or extending the one-dimensional non-autonomous system as a three-dimensional autonomous system restricted on a cylinder, this paper theoretically analyzes the dynamics of the well-known periodically driven bistable system. After three limit cycles (equivalent to the periodic states of the un-extended system) are proved existent for subthreshold case, they are shown to be preserved even in the suprathreshold case with large driving frequency. Moreover, it is also proved in suprathreshold case that the three ones keep touching each other and combine into one larger globally stable limit cycle as the driving frequency decreases to some certain degree.