On bifurcations of systems with homoclinic loops to a saddle-focus with saddle index 1/2

The instability regions in a neighborhood of a bifurcated system with homoclinic loops to a saddle-focus with saddle index half and zero divergence are determined. It is shown that if the leading divergence in the saddle-focus is less than zero or the saddle index is larger than half, then the system under consideration or the system close to it has stable periodic orbits. An unperturbed system having a saddle-focus equilibrium with certain eigenvalues has two-dimensional stable manifold and one-dimensional unstable manifold. For a sufficiently small neighborhood of the closure of the homoclinic trajectory, there exists no completely unstable periodic orbits of the system. The results also show the presence of countable set of bounded domains whose boundaries contain disjoint intervals.