Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 3.9

Nonautonomous Linear Systems

Mathematics — Calculus · flow-map-ivp

Problem

Suppose that for all t,s∈Rt, s \in \mathbb{R}, Φt,s:Rn→Rn\Phi_{t,s} : \mathbb{R}^n \to \mathbb{R}^n satisfies the properties of a flow, namely: • (t,s,y)↦Φt,s(y)(t, s, y) \mapsto \Phi_{t,s}(y) is continuous from R2+n\mathbb{R}^{2+n} into Rn\mathbb{R}^n. • t↦Φt,s(y)t \mapsto \Phi_{t,s}(y) is C1C^1 from R\mathbb{R} into Rn\mathbb{R}^n, for all (s,y)∈Rn+1(s, y) \in \mathbb{R}^{n+1}. • Φt,s∘Φs,τ=Φt,τ\Phi_{t,s} \circ \Phi_{s,\tau} = \Phi_{t,\tau}, for all t,s,τ∈Rt, s, \tau \in \mathbb{R}. • Φt,t\Phi_{t,t} is the identity map on Rn\mathbb{R}^n. Prove that x(t,s,y)=Φt,s(y)x(t, s, y) = \Phi_{t,s}(y) solves initial value problem x′=f(t,x),x(s)=y,x' = f(t, x), \quad x(s) = y, where the vector field ff is defined by f(s,y)=ddtΦt,s(y)∣t=s,for all (s,y)∈Ω.f(s, y) = \frac{d}{dt} \Phi_{t,s}(y)\bigg|_{t=s}, \quad \text{for all } (s, y) \in \Omega.

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