On the application of geometric singular perturbation theory to some classical two point boundary value problems

In this tutorial, we illustrate how geometric singular perturbation theory 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 criterion of matching corresponds to the criterion of transverse intersection of manifolds of solutions. This theme is studied in three classes of problems, linear: epsilon y" + alpha y' + beta y = 0, semilinear: epsilon 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 conditions y(0) A, y(l) = B hold. Chosen for their relatively simple structure, these problems provide a useful introduction to the methods of geometric singular perturbation theory that are now widely used in dynamical systems, from reaction-diffusion equations with traveling waves to perturbed N-degree-of-freedom Hamiltonian systems, and in applications to a variety of fields.