A multiple-timing analysis of temporal ratcheting

Abstract

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 $\omega$ and $\alpha \omega$ , where $\alpha$ is a rational number. If $\alpha$ is a ratio of odd and even integers (e.g., $\frac{2}{1},\frac{3}{2},\frac{4}{3}$ ), 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 $\alpha =2$ . Furthermore, we demonstrate, for a reduced model, that the temporal ratcheting effect for $\alpha =\frac{3}{2}$ and $\frac{4}{3}$ 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 $\alpha$ values. On a broader scale, our methodology offers a new mathematical approach to study the complicated nature of temporal ratchets in physical systems.