MELNIKOV VECTOR AND HOMOCLINIC BIFURCATIONS IN A DEGENERATE CASE

Abstract The main purpose of this paper is to study homoclinic bifurcation in a degenerate case of differential equations of form x ˙ = f ( x ⋅ ɛ ) and x ˙ = g ( x ) + h ( t , x ⋅ ɛ ) . 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.