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

The differential equation $\epsilon \ddot{\phi }+\dot{\phi }-\frac{1}{2}\alpha sin\left(2\mathrm{}\phi \right)=I+\Sigma {n=-\infty }^{\infty }{A}_n\delta (t-t_n)$ describing the periodically driven damped pendulum is analyzed in the strong damping limit $\epsilon \ll 1$, 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 $\omega$. For infinite damping, or zero mass ($\epsilon =0\mathrm{}$), the equation can be transformed to the Schrödinger 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 $\omega$ is a smooth function of the parameters, so there are no phase-locked subharmonic plateaus in the $\omega \mathrm{}\left(I\right)\mathrm{}$ curve, or the $I-V$ characteristics for the CDW or Josephson-junction systems. For small nonzero $\epsilon$ the return map expressing the phase $\phi \left(t_{n+1\mathrm{}}\right)$ as a function of the phase $\phi \left(t_n\right)$ 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.