Heteroclinic Bifurcation and Singularly Perturbed Boundary Value Problems

We study a singularly perturbed boundary value problrm in R”““: i =.f(s. J, E). cj= glx, .r, s), B,(s(w,), .tj(uo), E)=O, B,(.~(m,+to), ~(w,+oJ), E) =O. Given a candidate for the 0th order approximation which exhibits both boundary layers and interior layers, we present a complete procedure to compute higher order expansions and a procedure to compute the real solution near a truncated asymptotic expansion assuming the hyperbolicity of the regular layers and some generic assumptions. Similar results concerning the existence of periodic solutions (relaxation oscillations) are also presented. Several ideas from dynamical systems theory are employed, e.g., exponential dichotomies, Fredholm alternatives, and heteroclinic bifurcations. ( 1990 Acadermc Press. Inc.