Tetsushi Ueta, Hiroshi Kawakami, Bifurcation of an inductively coupled Josephson junction circuit, Institute of Electronics, Information and Communication Engineers, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, Vol.E77-A, No.11, 1758-1763, Nov. 1994.
Abstract: Some qualitative properties of an inductively coupled circuit containing two Josephson junction elements with a dc source are investigated. The system is described by a four-dimensional autonomous differential equation. However, the phase space can be regarded as S^1×R^3 because the system has a periodicity for the invariant transformation. In this paper, we study the properties of periodic solutions winding around S^1 as a bifurcation problem. Firstly, we analyze equilibria in this system. The bifurcation diagram of equilibria and its topological classification are given. Secondly, the bifurcation diagram of the periodic solutions winding around S^1 are calculated by using a suitable Poincare mapping, and some properties of periodic solutions are discussed. From these analyses, we clarify that a periodic solution so-called "caterpillar solution"  is observed when the two Josephson junction circuits are weakly coupled.