Stability properties of periodically driven overdamped pendula and their implications to physics of semiconductor superlattices and Josephson junctions

We consider the first order differential equation with a sinusoidal nonlinearity and periodic time dependence, that is, the periodically driven overdamped pendulum. The problem is studied in the case that the explicit time-dependence has symmetries common to pure ac-driven systems. The only bifurcation that exists in the system is a degenerate pitchfork bifurcation, which describes an exchange of stability between two symmetric nonlinear modes. Using a type of Prufer transform to a pair of linear differential equations, we derive an approximate condition of the bifurcation. This approximation is in very good agreement with our numerical data. In particular, it works well in the limit of large drive amplitudes and low external frequencies. We demonstrate the usefulness of the theory applying it to the models of pure ac-driven semiconductor superlattices and Josephson junctions. We show how the knowledge of bifurcations in the overdamped pendulum model can be utilized to describe effects of rectification and amplification of electric fields in these microstructures.Comment: 15 pages, 7 figures, Revtex 4.1. Revised and expanded following referee's report. Submitted to journal Chaos

Symmetry breaking in driven and strongly damped pendulum

We examine the conditions for appearance of symmetry breaking bifurcation in damped and periodically driven pendulum in the case of strong damping. We show that symmetry breaking, unlike other nonlinear phenomena, can exist at high dissipation. We prove that symmetry breaking phases exist between phases of symmetric normal and symmetric inverted oscillations. We find that symmetry broken solutions occupy a sufficiently smaller region of pendulum's parameter space in comparison to the statements made in earlier considerations [McDonald and Plischke, Phys. Rev. B 27 (1983) 201]. Our research on symmetry breaking in a strongly damped pendulum is relevant to an understanding of phenomena of dynamic symmetry breaking and rectification in a pure ac driven semiconductor superlattices.Comment: 11 pages, 4 color figures, RevTeX