18 1. Autonomous Linear Differential and Difference Equations This result shows, in particular, that Lyapunov exponents alone do not allow us to characterize stability for linear systems. They are related to exponential or, equivalently, to asymptotic stability, and they do not detect polynomial instabilities. The following example is a damped linear oscillator. Example 1.4.11. The second order differential equation ¨ + 2b ˙ + x = 0 is equivalent to the system ˙ 1 ˙ 2 = 0 1 −1 −2b x1 x2 . The eigenvalues are μ1,2 = −b ± √ b2 − 1. For b 0 the real parts of the eigenvalues are negative and hence the stable subspace coincides with R2. Hence b is called a damping parameter. Note also that for every solution x(·) the function y(t) := ebtx(t),t ∈ R, is a solution of the equation ¨+(1−b2)y = 0. 1.5. The Discrete-Time Case: Linear Difference Equations This section discusses solution formulas and stability properties, in partic- ular, Lyapunov exponents for autonomous linear difference equations. In this discrete-time case, the time domain R is replaced by N0 or Z, and one obtains equations of the form (1.5.1) xn+1 = Axn, where A ∈ gl(d, R). By induction, one sees that for positive time the solu- tions ϕ(n, x0) are given by (1.5.2) ϕ(n, x0) = xn = Anx0,n ∈ N. If A is invertible, i.e., if 0 is not an eigenvalue of A, this formula holds for all n ∈ Z, and An,n ∈ Z, forms a fundamental solution. It is an important feature of the discrete time case that solutions may not exist for negative times. In fact, if A is not of full rank, only for points in the range of A there is x−1 with x0 = Ax−1 and the point x−1 is not unique. For simplicity, we will restrict the discussion in the present section and in the rest of this book to the invertible case A ∈ Gl(d, R) where solutions are defined on Z. Existence of solutions and continuous dependence on initial values is clear from (1.5.2). Similarly, the result on the solution space, Theorem 1.1.1, is immediate: If A ∈ Gl(d, R), the set of solutions (ϕ(n, x0))n∈Z forms a d-dimensional linear space (with pointwise addition and multiplication by scalars). When it comes to the question of solution formulas, the real

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