Degeneration phenomenon in linear ordinary differential equations

Abstract

Given a linear constant coefficient ODE depending on a parameter, when this parameter approaches zero, the solution set converges to the solution set of the limit differential equation if the leading coefficient does not vanish. The situation is very subtle in the singular case, i.e., in the case when this coefficient becomes zero. The solution set then may even collapse completely. In this note, a formalism is developed in which the solution set of a linear constant coefficient ODE always depends continuously on the equation coefficients.