Classification of homoclinic tangencies for periodically perturbed systems

Classification of homoclinic tangencies for periodically perturbed systems is discussed. A relationship between the order of Melnikov function's zeros and the harmonic components of a dynamical system is derived. By applying the singularity theory to the Melnikov function, possible types of homoclinic tangencies are studied for realization of the classification. In addition, certain multi-harmonically perturbed systems are investigated, showing the corresponding homoclinic bifurcation with their bifurcation diagrams.