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Particle dynamics in a symmetrically driven underdamped inhomogeneous periodic potential system

We numerically solve the underdamped Langevin equation to obtain the trajectories of a particle in a sinusoidal potential driven by a temporally sinusoidal force in a medium with coefficient of friction periodic in space as the potential but with a phase difference. With the appropriate choice of system parameters, like the mean friction coefficient and the period of the applied field, only two kinds of periodic trajectories are obtained for all possible initial conditions at low noise strengths: one with a large amplitude and a large phase lag with respect to the applied field and the other with a small amplitude and a small phase lag. Thus, the periodic potential system is effectively mapped dynamically into a bistable system. Though the directional asymmetry, brought about only by the frictional inhomogeneity, is weak we find both the phenomena of stochastic resonance, with ready explanation in terms of the two dynamical states of trajectories, and ratchet effect simultaneously in the same parameter space. We analyse the results in detail attempting to find plausible explanations for each.