Abstract: The main purpose of this paper is to study homocliaic bifurcation in a degenerate case of differential equations of form x=f(x, ε) and x=g(x) + h(t, x. ε). By using theory of exponential dichotomies and Liapunov-Schmidt method, the authors obtain the Melnikov-like vectors by which the existence of homoclinic orbits can be detected.

Key words: Exponential dichotomies, Liapunov-Schmidt method, Homoclinic orbits