Square Root of the Monodromy Map Associated with the Equation of RSJ Model of Josephson Junction
- PDF / 649,799 Bytes
- 36 Pages / 439.37 x 666.142 pts Page_size
- 44 Downloads / 159 Views
Results in Mathematics
Square Root of the Monodromy Map Associated with the Equation of RSJ Model of Josephson Junction Sergey I. Tertychniy Abstract. An explicit representation of the monodromy transformation of the space of solutions to the nonlinear ODE, utilized for the modeling of Josephson junctions, is given. In case of positive integer order, the transformation, interpreted as the square root of the noted monodromy transformation, is derived by making use of a symmetry possessed by the associated double confluent Heun equation. Mathematics Subject Classification. Primary: 33E30; Secondary: 34A25, 34C14, 34C20, 34M03, 34M15, 34M35, 44A20. Keywords. Riccati equation, monodromy map, special double confluent Heun equation, symmetry of solution space.
1. Preliminaries In the present work, we discuss some noteworthy properties of the nonlinear differential equation ϕ˙ + sin ϕ = B + A cos ωt
(1)
in which ϕ = ϕ(t) is the unknown function, the symbols A, B, ω stand for some real constants, t is a free real variable, and the overdot denotes derivation with respect to t. Eq. (1) and its generalizations are of interest especially in view of their applications in several models in physics, mechanics, geometry, and the theory of dynamical systems [2,4–6,8]. In particular, Eq. (1) is widely known for its utilization in the so-called ‘Resistively Shunted Junction’ (RSJ-) model Supported in part by RFBR Grant N 17-01-00192. 0123456789().: V,-vol
139
Page 2 of 36
S. I. Tertychniy
Results Math
of the Josephson junction [1,9–11,14] which applies if the effect of the junction electric capacitance is negligible. An efficient way of investigating Eq. (1) assumes its appropriate complexification. Specifically, it is easy to see that Eq. (1) is equivalent to the Riccati equation Φ = (2 i ω z)−1 (1 − Φ2 ) + ( z −1 + μ(1 + z −2 ))Φ
(2)
in which Φ = Φ(z) is an unknown holomorphic function of the free variable z, the symbol denotes derivative with respect to the latter, and the symbols , μ stand for constant parameters. Indeed, formal substitutions z Ø eiωt , Φ(z) Ø eiϕ(t)
(3)
convert Eq. (2) into Eq. (1) with the parameters , μ obtained from the equations A = 2ωμ, B = ω. The right-hand side of the nonlinear equation (2) is a function of z holomorphic everywhere, except at zero, but its solutions may diverge for some other values of argument.1 However, it is easy to show the following [15]: Proposition 1. Let the constants , μ, and ω > 0 be real and let Φ(z) be a solution to Eq. (2) holomorphic at z = 1 and obeying the constraint |Φ(1)| = 1. Then, Φ(z) is also holomorphic in some vicinity of the curve2 |z| = 1; moreover, if |z| = 1 then |Φ(z)| = 1. Indeed, any solution to Eq. (1) can be extended to the whole real axis R, the result being a real analytic function. Carrying out the analytic continuation of ϕ(t) from R to some its vicinity in C, one may define the function Φ = exp(iϕ(t)), which is holomorphic in t and which can also be considered as a holomorphic function of the variable z = exp(iωt) varying in some open set
Data Loading...
