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

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 Prüfer 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 the effects of rectification and amplification of electric fields in these microstructures.

Exactly solvable model for drift of suspended ferromagnetic particles induced by the Magnus force

“…(4b) governing the evolution of the periodically driven overdamped pendulum. Some properties of its solutions have already been studied, e.g., in the context of Josephson junctions [19,20]. However, to the best of our knowledge, there are no known steady-state solutions of this 80 equation.…”

“…Moreover, under certain conditions (they can be realized, e.g., for a spinning sphere moving in a 15 rarefied gas [8,9]) there may exist the inverse, rather than classical, Magnus effect, in which the Magnus force direction is opposite to that predicted by Bernoulli's principle. However, in the case of smooth spherical particles, whose translational and rotational motions are characterized by small Reynolds numbers, the Magnus effect is classical and the Magnus force is 20 determined analytically [10]. Although in general these conditions are rather restrictive, they can be easily realized for small particles suspended in a viscous fluid.…”

Rectification of terahertz radiation in semiconductor superlattices in the absence of domains

“…The remaining constants are a, , and e, which are the superlattice period, the reduced Planck constant, and the electron charge, respectively. This phenomena bears close resemblance to the inverse ac-Josephson effect [25,26], and is in fact just one example of similarity between Josephson junctions and ssls [27,18,28]. Unquantized spontaneous dc has also been predicted for lateral semiconductor superlattices [29,28] that only differ by their geometry from the bulk ssls studied here.…”

“…If the governing equations remain invariant in transformation S, we say that they are symmetric or have symmetry S. In our previous works we have focused on the appearance of a spontaneous dc voltage via dynamical symmetry breaking [21,18,29,40,28]. Let u 0 be the generated dc voltage:…”

Finite-temperature dynamics of matter-wave dark solitons in linear and periodic potentials: An example of an antidamped Josephson junction

“…Nevertheless, the dynamical system (14) exhibits a rich behavior for γ < 0, a case corresponding to a damped Josephson junction (see, e.g., Refs. [60][61][62]). As an additional comment in this vein, suitable parallels between the two cases can be drawn on the basis of the invariance of Eq.…”