Tetsushi Ueta, Hiroshi Kawakami, Bifurcation of heteroclinic orbits in a circuit containing a Josephson junction element, John Wiiley & Sons, Inc, Electronics and Communications in Japan, Part 3 : Fundamental Electronic Science, Vol.77, No.2, pp.85-93, Feb. 1994.
Abstract: The circuit equation of the circuit containing a Josephson junction and the equation of motion of a pendulum in which an elastic restoring force is applied in the direction of rotation are described in terms of a second-order ordinary differential equation containing the sum of a trigonometric term on the state and a linear term. In such a dynamic system, the number of saddles and sinks changes according to the parameter values and no periodicity exists with respect to the location of the saddles and sinks, thus analysis is difficult. In this paper, a circuit containing a Josephson junction is chosen as an example of such a dynamic system. Classification by the tangent bifurcation of the saddles and sinks and the results of the analysis based on the heteroclinic orbits on the phase plane are presented. The heteroclinic orbit presents a global bifurcation on the phase plane due to its structural instability. The parameters generated by this orbit are used as the values for the bifurcation, and the bifurcation set was derived on the parameter plane. As a result, the unique linking number is defined for a particular parameter region surrounded by the bifurcation curve. This value can be used as a guideline for rotation of the orbit on the phase plane. Further, by combining it with the bifurcation ensemble, a classification of the stable sinks in the basin is made possible.