Study of the driven damped pendulum: Application to Josephson junctions and charge-density-wave systems

Abstract

The equation of motion of the driven damped pendulum is related at low dissipation to a current-fed Josephson junction and at high dissipation to transport in charge-density-wave (CDW) systems. We report on an extensive numerical investigation of these equations. At low dissipation we find broad bands of chaotic solutions as a function of the frequency and amplitude of the driving force. It is pointed out that periodic solutions may possess a symmetry corresponding to the invariance of the equations of motion under a simultaneous spatial (phase) inversion and a shift in the phase of the driving force by an odd multiple of $\pi$. At low dissipation chaos is usually approached via a sequence of period-doubling bifurcations if this symmetry has been broken and directly from period 1 with associated intermittency behavior if the symmetry is not broken. At high dissipation no chaotic behavior is found but broad bands of symmetry-broken solutions, which may be related to recently reported hysteresis phenomena in CDW systems, occur. Discussions of properties of the Poincaré maps and of the fractal dimension of the strange attractors associated with the chaotic solutions have been included.