12 1. Autonomous Linear Differential and Difference Equations Example 1.3.6. Let J be a real Jordan block of dimension 2m associated with the complex eigenvalue pair μ = λ ± ıν of a matrix A gl(d, R). With D := λ −ν ν λ , R := R(t) := cos νt sin νt sin νt cos νt , and I := 1 0 0 1 , one obtains for J =⎢ D I · · · · · · · I D that eJt =eλt R tR t2 2! R · · tm−1 (m−1)! R · · · · · · · · · · t2 2! R · tR R . In other words, for x0 = [x1,y1,...,xm,ym] R2m the solution of ˙ = Jx is given with j = 1,...,m, by xj(t, x0) = eλt m k=j tk−j (k j)! (xk cos νt yk sin νt), (1.3.3a) yj(t, y0) = eλt m k=j tk−j (k j)! (xk sin νt + yk cos νt). (1.3.3b) Remark 1.3.7. Consider a Jordan block for a real eigenvalue μ as in Exam- ple 1.3.4. Then the k-dimensional subspace generated by the first k canonical basis vectors (1, 0,..., 0) , . . . , (0,... 0, 1, 0,..., 0) , 1 k m, is invariant under eJt. For a complex-conjugate eigenvalue pair μ, ¯ as in Example 1.3.6 the subspace generated by the first 2k canonical basis vectors is invariant. Remark 1.3.8. Consider a solution x(t),t R, of ˙ = Ax. Then the chain rule shows that the function y(t) := x(−t),t R, satisfies d dt y(t) = d dt x(−t) = −Ax(−t) = −Ay(t),t R. Hence we call ˙ = −Ax the time-reversed equation. 1.4. Lyapunov Exponents The asymptotic behavior of the solutions x(t, x0) = eAtx0 of the linear dif- ferential equation ˙ = Ax plays a key role in understanding the connections between linear algebra and dynamical systems. For this purpose, we in- troduce Lyapunov exponents, a concept that is fundamental for this book, since it also applies to time-varying systems. Definition 1.4.1. Let x(·,x0) be a solution of the linear differential equation ˙ = Ax. Its Lyapunov exponent or exponential growth rate for x0 = 0
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