Exponential dichotomies and transversal homoclinic orbits in degenerate cases

In this paper, we first give a sufficient condition which assures that a linear differential equation depending on a small parameter admits an exponential dichotomy onR, then we use the result obtained here on exponential dichotomies to investigate the existence of transversal homoclinic orbits of perturbed differential systems in two degenerate cases and obtain a Melnikov-type vector. The results on exponential dichotomies of this paper provide us a tool of proving the transversality of homoclinic orbits in studying degenerate bifurcations.This work is supported by NSF of China.