16 1. Autonomous Linear Differential and Difference Equations Then the initial value problems above have unique solutions which, in gen- eral, are only defined on open intervals containing t = 0 (we will see an example for this in the beginning of Chapter 4). If the solutions remain bounded on bounded intervals, one can show that they are defined on R. Various stability concepts characterize the asymptotic behavior of the solutions ϕ(t, x0) for t ±∞. Definition 1.4.6. Let x∗ Rd be a fixed point of the differential equation ˙ = f(x), i.e., the solution ϕ(t, x∗) with initial value ϕ(0,x∗) = x∗ satisfies ϕ(t, x∗) x∗. Then the point x∗ is called: stable if for all ε 0 there exists a δ 0 such that ϕ(t, x0) N(x∗,ε) for all t 0 whenever x0 N(x∗,δ) asymptotically stable if it is stable and there exists a γ 0 such that limt→∞ ϕ(t, x0) = x∗ whenever x0 N(x∗,γ) exponentially stable if there exist α 1 and β, η 0 such that for all x0 N(x∗,η) the solution satisfies ϕ(t, x0) x∗≤ α x0 x∗ e−βt for all t 0 unstable if it is not stable. It is immediate to see that a point x∗ is a fixed point of ˙ = Ax if and only if x∗ ker A, the kernel of A. The origin 0 Rd is a fixed point of any linear differential equation. Definition 1.4.7. The stable, center, and unstable subspaces associated with the matrix A gl(d, R) are defined as L− = λj 0 L(λj),L0 = L(0), and L+ = λj 0 L(λj). The following theorem characterizes asymptotic and exponential stabil- ity of the origin for ˙ = Ax in terms of the eigenvalues of A. Theorem 1.4.8. For a linear differential equation ˙ = Ax in Rd the fol- lowing statements are equivalent: (i) The origin 0 Rd is asymptotically stable. (ii) The origin 0 Rd is exponentially stable. (iii) All Lyapunov exponents (i.e., all real parts of the eigenvalues) are negative. (iv) The stable subspace L− satisfies L− = Rd. Proof. First observe that by linearity, asymptotic and exponential stability of the fixed point x∗ = 0 Rd in a neighborhood N(x∗,γ) implies asymp- totic and exponential stability, respectively, for all points x0 Rd. In fact,
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