Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 3.15

Nonautonomous Linear Systems

Mathematics — Calculus · perturbation-stability-2x2

Problem

Let AkA_k be 2×22 \times 2 matrices over R\mathbb{R}, and let fk(x)=Akxf_k(x) = A_k x, k=1,2k = 1, 2. (a) Assume that the origin is asymptotically stable for f1f_1. Prove that there exists a δ>0\delta > 0 (depending on A1A_1) such that if ∥A1−A2∥<δ\|A_1 - A_2\| < \delta, then the origin is asymptotically stable for f2f_2. (b) Is the preceding statement true if ``asymptotically stable'' is replaced by ``stable''? Explain.

Browse the problem library