Melnikov vector and heteroclinic manifolds (Q1340021)

Using the exponential dichotomies, the transversality theory, and the generalized Melnikov vectors, the author finds conditions sufficient for the persistence and the transversality of a singular orbit, with high degeneracy, disposed on a heteroclinic or a homoclinic manifold under a perturbation. The obtained results extend, include, and improve in part corresponding results of \textit{S. Wiggins} [Global bifurcations and chaos -- analytical methods, Springer-Verlag, New York (1988; Zbl 0661.58001)] for the case of completely integrable Hamiltonian system and results of \textit{K. J. Palmer} [J. Differ. Equations 55, 225-256 (1984; Zbl 0539.58028)] and of \textit{F. Battelli} and \textit{C. Lazzarri} [J. Differ. Equations 86, 342-366 (1990; Zbl 0713.34055)] for the case of nonautonomous perturbations when a singular orbit has less degeneracy than that considered in the paper.