The numerical solution of a challenging class of turning point problems

A curious class of challenging singularly perturbed turning point problems is considered and properties of the solutions to corresponding initial value problems are studied. The solutions are exponentially small near the turning point and become unstable after passing it. Various state-of-the-art codes available in MATLAB as well as one-step and multistep methods on a uniform mesh are tested. By examining a number of examples, one finds that the usual error control strategies may not work when the solution near the turning point is small, while one-step and multistep methods on a uniform mesh work only for a moderately small perturbation parameter. A scale amplification transformation, however, seems to give the correct solution when the solution is extremely small and/or zero at the turning point. Extensions to problems with more equilibria are also briefly considered.