Smooth Solution Space of Slow Subsystem in Nonstandard Singular Perturbation Problem

In this paper, the algebraic structure of smooth solution space of the slow subsystem in a nonstandard singular perturbation problem is investigated. It is shown that all the smooth solutions of the slow subsystem constitutes a linear space whose dimension exactly equals the power of the characteristic polynomial of this system. Based on this result, the well-posed statement of an initial value problem, the existence and uniqueness, and analytical expression of smooth solutions of the slow subsystem are given respectively. The result of this paper generalizes the initial value problem of a normal linear system, which is from a standard singular perturbation problem, to the one of the slow subsystem in the case of a nonstandard singular perturbation problem, which paves a theoretical foundation of applicability of two-time-scale analysis in nonstandard singular perturbation problem.