A coupled system composed of two nonlinear circuit systems is investigated. In this paper, the existence condition and the analytical expressions of equilibrium in higher-dimensional system are derived, and the co-dimension 1 and co-dimension 2 bifurcations of equilibrium are also studied. Furthermore, the complicated bifurcations are obtained through the continuation of limit cycles. It may lead to various dynamical behaviors such as periodic motion, chaos, etc., for the interaction of two subsystems with periodic motions under different coupling parameters. Using the qualitative analysis of equilibrium before and after coupling, the relation between the discontinuity of bifurcation diagram and occurrence of neutral saddle in the case of weak coupling is presented.