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 S1×R3 because the system has a periodicity for the invariant transformation. In this paper, we study the properties of periodic solutions winding around S1 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 S1 are calculated by using a suitable Poincar 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.