A criterion for the invariance of characteristic exponents of linear systems with a small parameter multiplying the derivative (Q1778245)

The paper is a continuation of author's work [\textit{N. A. Izobov} and the author, Differ. Equations 34, No. 8, 1052--1059 (1998; Zbl 1126.34341) and the author, Dokl. Nats. Akad. Nauk Belarusi 46, No. 3, 35--37 (2002; Zbl 1074.34052)]. Consider the singular linear system \[ \varepsilon \dot x = A(t)x, \quad x\in \mathbb{R}^n, \quad t\geq 0, \] together with the perturbed system \[ \varepsilon \dot x = (A(t)+Q(t))x. \] Here, \(A(t)\) is a piecewise continuous bounded coefficient matrix, \(Q(t)\) is some piecewise continuous perturbation. The author derives a necessary and sufficient condition for the preservation of the characteristic set of the original system.