Bifurcation of limit cycles and separatrix loops in singular Lienard systems

We give sufficient conditions for the existence of one or two limit cycles of singular Lienard systems through the construction of a Poincare-Bendixson domain. With the help of the theory of rotated vector fields,we develop a method to compute bifurcation value at Saddle-node bifurcation of limit cycles and homoclinic or symmetric heteroclinic bifurcations. We also present application examples and prove the existence of duck cycles