Singularly perturbed systems

Abstract

This paper discusses existence and nonexistence of C1 quasi-steady-states to singularly perturbed problems near a singular point. In contrast to the existence and uniqueness result well known for the same problem near a regular point, the answer depends on generic conditions involving both the differential and the transcendental equations of the system. When existence is guaranteed, multiple C 1 solutions will typically appear but their number cannot exceed an explicit upper bound. By comparison, infinitely many generalized solutions exist in the same situation. The verification of the generic conditions as well as the practical determination of the exact number of solutions reduces to algebraic calculations. The bulk of our approach combines results in bifurcation theory (generalizations of the Morse lemma to vector-valued functions) with standard methods in ODE's, Arguments from algebraic geometry (generalized Bezout's theorem) and homotopy theory (parametrized Sard's theorem) are also involved, at a le...