24 1. Autonomous Linear Differential and Difference Equations exponentially stable if there exist α 1,η 0 and β (0, 1) such that for all x0 N(x∗,η) the solution satisfies ϕ(n, x0) x∗≤ αβn x0 x∗ for all n N unstable if it is not stable. The origin 0 Rd is a fixed point of any linear difference equation. The following definition referring to the Lyapunov spaces will be useful. Definition 1.5.10. The stable, center, and unstable subspaces associated with the matrix A Gl(d, R) are defined as L− = λ j 0 L(λj),L0 = L(0), and L+ = λ j 0 L(λj). One obtains a characterization of asymptotic and exponential stability of the origin for xn+1 = Axn in terms of the eigenvalues of A. Theorem 1.5.11. For a linear difference equation xn+1 = Axn in Rd the following statements are equivalent: (i) The origin 0 Rd is asymptotically stable. (ii) The origin 0 Rd is exponentially stable. (iii) All Lyapunov exponents are negative (i.e., all moduli of the eigen- values are less than 1.) (iv) The stable subspace L− satisfies L− = Rd. Proof. The proof is completely analogous to the proof for differential equa- tions see Theorem 1.4.8. It remains to characterize stability of the origin. Theorem 1.5.12. The origin 0 Rd is stable for the linear difference equation xn+1 = Axn if and only if all Lyapunov exponents are nonpositive and the eigenvalues with modulus equal to 1 are semisimple. Proof. The proof is completely analogous to the proof for differential equa- tions see Theorem 1.4.10. Again we see that Lyapunov exponents alone do not allow us to charac- terize stability. They are related to exponential stability or, equivalently, to asymptotic stability and they do not detect polynomial instabilities. 1.6. Exercises Exercise 1.6.1. One can draw the solutions x(t, x0) R2 of ˙ = Ax with A gl(2, R) either componentwise as functions of t R or as sin- gle parametrized curves in R2. The latter representation of all solutions
Previous Page Next Page