Exponential trichotomy and homoclinic bifurcation with saddle-center equilibrium

In this paper the bifurcation of a homoclinic orbit is studied for an ordinary differential equation with periodic perturbation. Exponential trichotomy theory with the method of Lyapunov–Schmidt is used to obtain some sufficient conditions to guarantee the existence of homoclinic solutions and periodic solutions for this problem. Some known results are extended.