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 \(\omega \) and \(\alpha \omega \), where \(\alpha \) is a rational number. If \(\alpha \) is a ratio of odd and even integers (e.g., \(\tfrac{2}{1},\,\tfrac{3}{2},\,\tfrac{4}{3}\)), 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 \(\alpha =\tfrac{3}{2}\) and \(\tfrac{4}{3}\) 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.