Journal of East China Normal University. Natural Science Edition. Huadong Shifan Daxue Xuebao. Ziran Kexue Ban (J. East China Norm. Univ. Natur. Sci. Ed.) (20130101), no.~3, 140-148. ISSN: 1000-5641 (print).
Subject
34 Ordinary differential equations -- 34B Boundary value problems 34B08 Parameter dependent boundary value problems
The authors consider general singularly perturbed systems of ordinary differential equations, in slow-fast form. The main assumption is that the limit (fast) system, obtained by setting the singular parameter equal to zero, has a normally hyperbolic manifold of equilibria. As is well known, standard geometric singular perturbation theory implies that this manifold perturbs smoothly for small values of the singular parameter. The main result of the authors is that, under appropriate assumptions, the two-point boundary value problem associated to the system has a locally unique solution which traces that of the limit problem which is assumed to exist and to satisfy certain nondegeneracy conditions. This is accomplished as an application of the well-known exchange lemma of Jones and Kopell. Moreover, as an interesting application of their abstract result, they provide an elegant geometric singular perturbation proof of an old result of C. Schmeiser and R. Weiss [SIAM J. Math. Anal. {\bf 17} (1986), no.~3, 560--579; MR0838241 (87i:34061)].