Subharmonic Solutions in Singular Systems

We consider the problem of bifurcation from homoclinic towards periodic orbits for a periodic singular system of differential equations. We use a functional analytic approach and the Lyapunov–Schmidt method to obtain a bifurcation function for this problem which tends, as →0+, to the well-known Melnikov function for the bifurcation towards transverse homoclinic orbits. We also study the behavior of the bifurcation functions as the period of the periodic orbit tends to infinity.