Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 133, pp. 1-20.

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

Author: Sergey I. Tertychniy (VNIIFTRI, Mendeleevo, Russia)

Abstract: 
 The first order nonlinear ordinary differential equation
 $\dot \varphi(t) + \sin\varphi(t)=B+A\cos\omega t$,
 which is commonly used as a simple model  of an overdamped
 Josephson junction in superconductors is investigated.
 Its general solution is obtained in the case known as
 phase-lock where all but one solution converge to a
 common `essentially periodic' attractor.
 The general solution is represented in explicit form
 in terms of the Floquet solution of a double confluent Heun equation.
 In turn, the latter 
 solution is represented through the Laurent series
 which defines an analytic function on the Riemann sphere with
 punctured poles.
 The series coefficients are given in terms of infinite  products
 of $2\times2$ matrices with a single zero element.
 The closed form of the phase-lock condition is obtained and
 represented as the condition for existence of a real root of
 a transcendental function.
 The efficient phase-lock criterion is conjectured, and
 its plausibility is confirmed in numerical tests.

Submitted May 25, 2007. Published October 12, 2007.
Math Subject Classifications: 33E30, 34A05, 34A25, 34B30, 34B60, 34M05, 34M35, 70K40
Key Words: Overdamped Josephson junction; nonlinear first order ODE; 
           linear second order  ODE; double confluent Heun equation; 
	   phase lock; general solution.