1.5. The Discrete-Time Case: Linear Difference Equations 23 Then a solution ϕ(·,x0) with x0 = 0 satisfies lim n→+∞ 1 n log ϕ(n, x0) = λj if and only if x0 Vj \ Vj+1, lim n→−∞ 1 |n| log ϕ(n, x0) = −λj if and only if x0 Wj \ Wj−1. In particular, limn→±∞ 1 |n| log ϕ(n, x0) = ±λj if and only if x0 L(λj) = Vj Wj. Proof. This follows using the solution formulas and the arguments given in the proof of Theorem 1.5.6. Here the time-reversed equation has the form xn+1 = A−1xn,n Z. The eigenvalues of A−1 are given by the inverses of the eigenvalues μ of A and the Lyapunov exponents are −λ . . . −λ1, since log ( μ−1 ) = log |μ| . The generalized eigenspaces and hence the Lyapunov spaces L(−λj) coincide with the corresponding generalized eigenspaces and Lyapunov spaces L(λj), respectively. Stability Using the concept of Lyapunov exponents and Theorem 1.5.8 we can describe the behavior of solutions of linear difference equations xn+1 = Axn as time tends to infinity. By definition, a solution with negative Lyapunov exponent tends to the origin and a solution with positive Lyapunov exponent becomes unbounded. It is appropriate to formulate the relevant stability concepts not just for linear differential equations, but for general nonlinear difference equations of the form xn+1 = f(xn), where f : Rd Rd. In general, the solutions ϕ(n, x0) are only defined for n 0. Various stability concepts characterize the asymptotic behavior of ϕ(n, x0) for n ∞. Definition 1.5.9. Let x∗ Rd be a fixed point of the difference equation xn+1 = f(xn), i.e., the solution ϕ(n, x∗) with initial value ϕ(0,x∗) = x∗ satisfies ϕ(n, x∗) x∗. Then the point x∗ is called: stable if for all ε 0 there exists a δ 0 such that ϕ(n, x0) N(x∗,ε) for all n N whenever x0 N(x∗,δ) asymptotically stable if it is stable and there exists a γ 0 such that limn→∞ ϕ(n, x0) = x∗ whenever x0 N(x∗,γ)
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