Exponential dichotomies and heteroclinic bifurcations in degenerate cases (Q1895461)

The small non-autonomous perturbations of a smooth autonomous \(n\)- dimensional system having a heteroclinic manifold \(S\) connecting two hyperbolic saddle points \(p_1\) and \(p_2\), are considered. Let \(I = n_1 + n_2 - n\), where \(n_1\) and \(n_2\) are dimensions of the unstable manifold at \(p_1\) and of the stable one at \(p_2\). The authors concern with the case \(I \geq 0\). It is proved that provided the Melnikov vector is nondegenerate, there exists near \(S\) an invariant manifold consisting of bounded solutions of the perturbed system. Earlier \textit{M. Yamashita} [Nonlinear Anal., Theory Methods Appl. 18, No. 7, 657- 670 (1992; Zbl 0769.34035)] proved the same result in the case \(p_1 = p_2\) (then \(I = 0)\).