Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 3.13

Nonautonomous Linear Systems

Mathematics — Calculus · stability-independence-of-initial-time

Problem

Let O⊂Rn\mathcal{O} \subset \mathbb{R}^n be an open set, and set Ω=R×O\Omega = \mathbb{R} \times \mathcal{O}. Suppose that f:Ω→Rnf : \Omega \to \mathbb{R}^n satisfies the hypotheses of Theorems 3.2 and 3.3. Suppose also that xˉ∈O\bar{x} \in \mathcal{O} is a stable equilibrium for ff, according to Definition 3.7. Prove that for any ε>0\varepsilon > 0 and t0∈Rt_0 \in \mathbb{R}, there exists a δ>0\delta > 0 such that if ∥x0−xˉ∥<δ\|x_0 - \bar{x}\| < \delta, then the solution of the initial value problem x(t,t0,x0)x(t, t_0, x_0) exists for all t≥t0t \geq t_0 and ∥x(t,t0,x0)−xˉ∥<ε,for t≥t0.\|x(t, t_0, x_0) - \bar{x}\| < \varepsilon, \quad \text{for } t \geq t_0. (Thus, the notion of stability does not depend upon the choice of the initial time.)

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