Analytical results on the periodically driven damped pendulum. Application to sliding charge-density waves and Josephson junctions

Abstract

The differential equation $\ensuremath{\epsilon}\stackrel{\ifmmode\ddot\else\textasciidieresis\fi{}}{\ensuremath{\varphi}}+\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}\ensuremath{-}\frac{1}{2}\ensuremath{\alpha}sin(2\ensuremath{\varphi})=I+\ensuremath{\Sigma}{n=\ensuremath{-}\ensuremath{\infty}}^{\ensuremath{\infty}}{A}_{n}\ensuremath{\delta}(t\ensuremath{-}{t}_{n})$ describing the periodically driven damped pendulum is analyzed in the strong damping limit $\ensuremath{\epsilon}\ensuremath{\ll}1$, using first-order perturbation theory. The equation may represent the motion of a sliding charge-density wave (CDW) in ac plus dc electric fields, and the resistively shunted Josephson junction driven by dc and microwave currents. When the torque $I$ exceeds a critical value the pendulum rotates with a frequency $\ensuremath{\omega}$. For infinite damping, or zero mass ($\ensuremath{\epsilon}=0$), the equation can be transformed to the Schr\"odinger equation of the Kronig-Penney model. When ${A}_{n}$ is random the pendulum exhibits chaotic motion. In the regular case ${A}_{n}=A$ the frequency $\ensuremath{\omega}$ is a smooth function of the parameters, so there are no phase-locked subharmonic plateaus in the $\ensuremath{\omega}(I)$ curve, or the $I\ensuremath{-}V$ characteristics for the CDW or Josephson-junction systems. For small nonzero $\ensuremath{\epsilon}$ the return map expressing the phase $\ensuremath{\varphi}({t}_{n+1})$ as a function of the phase $\ensuremath{\varphi}({t}_{n})$ is a one-dimensional circle map. Applying known analytical results for the circle map one finds narrow subharmonic plateaus at all rational frequencies, in agreement with experiments on CDW systems.