A multiple-timing analysis of temporal ratcheting

We develop a two-timing perturbation analysis to provide quantitative insights on the existence of temporal ratchets in an exemplary system of a particle moving in a tank of fluid in response to an external vibration of the tank. We consider two-mode vibrations with angular frequencies ω ω and α ω α ω , where α α is a rational number. If α α is a ratio of odd and even integers (e.g., 2/1,\,3/2,\,4/3 2 1 , 3 2 , 4 3 ), the system yields a net response: here, a nonzero time-average particle velocity. Our first-order perturbation solution predicts the existence of temporal ratchets for α =2 α = 2 . Furthermore, we demonstrate, for a reduced model, that the temporal ratcheting effect for α =3/2 α = 3 2 and 4/3 4 3 appears at the third-order perturbation solution. More importantly, we find closed-form formulas for the magnitude and direction of the induced net velocities for these α α values. On a broader scale, our methodology offers a new mathematical approach to study the complicated nature of temporal ratchets in physical systems. Graphic abstract