Linearly independent homoclinic bifurcations parameterized by a small function

In this paper we discuss a small nonautonomous perturbation of an autonomous system on which has a homoclinic solution. Regarding the small perturbation as a parameter in an appropriate space of functions we discuss various situations of co-existence of homoclinic orbits. Those conditions of various co-existence actually define bifurcation manifolds in the space of functions for linearly independent homoclinic bifurcations.