ON THE APPLICATION OF GEOMETRIC SINGULAR PERTURBATION-THEORY TO SOME CLASSICAL 2 POINT BOUNDARY-VALUE-PROBLEMS

In this tutorial, we illustrate how geometric singular perturbation th eory provides a complementary dynamical systems-based approach to the method of matched asymptotic expansions for some classical singularly- perturbed boundary value problems. The central theme is that the crite rion of matching corresponds to the criterion of transverse intersecti on of manifolds of solutions. This theme is studied in three classes o f problems, linear: epsilon y'' + alpha y' + beta y = 0, semilinear: e psilon y'' + alpha y' + f(y) = 0, and quasilinear: epsilon y'' + g(y)y ' + f(y) = 0, on the interval [0, 1], where t is an element of [0, 1], ' = d/dt, 0 < epsilon much less than 1, and general boundary conditio ns y(0) A, y(l) = B hold. Chosen for their relatively simple structure , these problems provide a useful introduction to the methods of geome tric singular perturbation theory that are now widely used in dynamica l systems, from reaction-diffusion equations with traveling waves to p erturbed N-degree-of-freedom Hamiltonian systems, and in applications to a variety of fields.