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

The differential equation ${ε}{{}}{{φ}}+{ ̇ \.{}{}}{{φ}}{-}1/2{α}sin(2{φ})=I+{Σ}{n={-}{∞}}^{{∞}}{A}ₙ{δ}(t{-}{t}ₙ)$ describing the periodically driven damped pendulum is analyzed in the strong damping limit ${ε}{}1$, 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 ω. For infinite damping, or zero mass (ε=0), the equation can be transformed to the Schr\"odinger equation of the Kronig-Penney model. When Aₙ is random the pendulum exhibits chaotic motion. In the regular case Aₙ=A the frequency ω is a smooth function of the parameters, so there are no phase-locked subharmonic plateaus in the ω(I) curve, or the I-V characteristics for the CDW or Josephson-junction systems. For small nonzero ε the return map expressing the phase φ(tₙ₊₁) as a function of the phase φ(tₙ) 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.