Homoclinic bifurcations in a planar dynamical system

The homoclinic bifurcation properties of a planar dynamical system are analyzed and the corresponding bifurcation diagram is presented. The occurrence of two Bogdanov-Takens bifurcation points provides two local existing curves of homoclinic orbits to a saddle excluding the separatrices not belonging to the homoclinic orbits. Using numerical techniques, these curves are continued in the parameter space. Two further curves of homoclinic orbits to a saddle including the separatrices not belonging to the homoclinic orbits are calculated by numerical methods. All these curves of homoclinic orbits have a unique intersection point, at which there exists a double homoclinic orbit. The local homoclinic bifurcation diagram of both the double homoclinic orbit point and the points of homoclinic orbits to a saddle-node are also gained by numerical computation and simulation.