Complexity of Shapiro steps

Shapiro steps and chaos in the Frenkel-Kontorova model with substrate lateral vibration

Numerical simulations are used to examine the dynamics of the dc-driven Frenkel-Kontorova model with an oscillation substrate subjected to lateral periodic excitations in overdamped and underdamped cases, respectively. The results reveal that the system exhibits frequency locking and chaotic behaviors due to the fact that the lateral vibration of the substrate potential introduces an additional frequency and degree of freedom into the system. In the overdamped case, we show that the appearance of subharmonic Shapiro steps can be attributed to the deformation of the substrate potential or inertia. The characteristics of the steps are significantly affected by the amplitude and frequency of the lateral vibration. When the vibration frequency is relatively high, the change of the width of the first harmonic Shapiro step with increasing amplitude exhibits a Bessel-type oscillation, but the oscillation deviates from the Bessel curve at lower frequencies. In the amplitude dependence, although the oscillatory behavior of the critical depinning force at the high frequency is anomalous, local maxima (minima) of the first step width correspond to local minima (maxima) of the critical depinning force, and the largest Lyapunov exponent obtained in the pinned state represents a mirror relationship of the critical depinning force. In contrast to the overdamped system, the underdamped one exhibits both subharmonic Shapiro steps and chaotic behaviors. We show the increased inertia of the latter system plays an important role in suppressing the emergence of subharmonic steps, which is opposite to the result of the former. When the dc force changes, chaos appears not only between adjacent subharmonic Shapiro steps but also on some specific steps where chaos should be avoided. The variation regularity of the first step width and the critical depinning force is thus annihilated in vibration amplitude and frequency dependence. However, superlubricity can be achieved by careful adjustment of vibration amplitude and frequency. The findings can serve as a theoretical guideline for technological applications such as device building and voltage standards.