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

TL;DR

This paper analyzes the stability and bifurcation behavior of a periodically driven overdamped pendulum, deriving approximate conditions for bifurcations and applying findings to semiconductor superlattices and Josephson junctions.

Contribution

It introduces an approximate bifurcation condition for the overdamped pendulum with symmetries, validated by numerical data, and applies this to microstructure models.

Findings

Bifurcation is a degenerate pitchfork exchanging stability between symmetric modes.

Approximate bifurcation condition matches numerical results well, especially at high drive amplitudes.

Application to superlattices and Josephson junctions explains rectification and amplification effects.

Abstract

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.