Mechanism of producing a saddle-node bifurcation with the coalescence of two unstable periodic orbits

A saddle-node bifurcation with the coalescence of a stable periodic orbit and an unstable periodic orbit is a common phenomenon in nonlinear systems. This study investigates the mechanism of producing another saddle-node bifurcation with the coalescence of two unstable periodic orbits. The saddle-node bifurcation results from a codimension-two bifurcation that a period doubling bifurcation line tangentially intersects a saddle-node bifurcation line in a parameter plane. Based on the bifurcation theory, the saddle-node bifurcation with the coalescence of two unstable periodic orbits is studied using the codimension-two bifurcation.