General solution of overdamped Josephson junction equation in the case of phase-lock

Long-term behavior of solutions of the equation \dotϕ+ \sinϕ=f with periodic f and the modeling of dynamics of overdamped Josephson junctions: Unlectured notes

The method of efficient description of long-term behavior of solutions of the non-linear first order ODE \dotϕ+\sinϕ=f for arbitrary periodic $f$ is discussed. The criterion enabling one to separate and identify the qualitatively different solutions is established. The applications of the method to the modeling of dynamics of overdamped Josephson junctions in superconductors are outlined.

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 …

On families of constrictions in model of overdamped Josephson junction and Painlevé 3 equation

The tunneling effect predicted by B.Josephson (Nobel Prize, 1973) concerns the Josephson junction: two superconductors separated by a narrow dielectric. It states existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by a family of differential equations on 2-torus depending on 3 parameters: $B$ (abscissa), $A$ (ordinate), $ω$ (frequency). We study its rotation number $ρ(B,A;ω)$ as a function of $(B,A)$ with fixed $ω$. The phase-lock areas are the level sets $L_r:=\{ρ=r\}$ with non-empty interiors; they exist for $r\in\mathbb Z$ (Buchst…

The interrelation of the special double confluent Heun equation and the equation of RSJ model of Josephson junction revisited

The explicit formulas for the maps interconnecting the sets of solutions of the special double confluent Heun equation and the equation of the RSJ model of overdamped Josephson junction in case of shifted sinusoidal bias are given. The approach these are based upon leans on the extensive application of eigenfunctions of certain linear operator acting on functions holomorphic on the universal cover of the punctured complex plane. The functional equation the eigenfunctions noted obey is derived, the matrix form of the monodromy transformation they infer is given.

Josephson effect and slow-fast systems

In order to model the processes taking place in systems with Josephson contacts, a differential equation on a torus with three parameters is used. One of the parameters of the system can be considered small and the methods of the fast--slow systems theory can be applied. The properties of the phase--lock areas -- the subsets in the parameter space, in which the changing of a current doesn't affect the voltage -- are important in practical applications. The phase-lock areas coincide with the Arnold tongues of a Poincare map along the period. A description of the limit properties of Arnold tongu…

RSJ模型中包含两个约瑟夫森调和函数的所有分数肖普步骤

All fractional Shapiro steps in the RSJ model with two Josephson harmonics

本文研究了RSJ模型中通过包含两个约瑟夫森谐波产生所有可能分数肖普步骤的现象，并揭示了正负分数步骤的不对称性。