20 1. Autonomous Linear Differential and Difference Equations Example 1.5.3. Let J be a real Jordan block of dimension 2m associated with the pair of complex eigenvalues μ = α ± ıβ of a matrix A gl(d, R). With D = α −β β α and I2 = [ 1 0 0 1 ], one obtains that J has the form D I2 · · · · · · · I2 D = D 0 · · · · · · · 0 D + 0 I2 · · · · · · · I2 0 . Thus J is the sum of a block-diagonal matrix ˜ with blocks D and a nilpo- tent matrix N with N m−1 = 0. Then, observing that ˜ and N commute, i.e., ˜ DN = N ˜, one computes for n m 1, Jn = ( ˜ + N)n = m−2 i=0 n i ˜ n−i N i = n 0 ˜ n I + n 1 ˜ n−1 N + . . . + n m 2 ˜ n−(m−2) N m−1 . Note that |μ| = α2 + β2, hence with ϕ [0, 2π) determined by cos ϕ = α α2+β2 (thus μ = |μ| eıϕ) one can write the matrix D as D = α −β β α = |μ| R with R := cos ϕ sin ϕ sin ϕ cos ϕ . Thus D describes a rotation by the angle ϕ followed by multiplication by |μ|. Write ˜ for the block diagonal matrix with blocks R. One obtains for n m 2 the solution formula ϕ(n, y0) = Jny0 = m−2 i=0 n i |μ|n−i ˜n−iN i y0 = |μ|n+2−m m−2 i=0 n i |μ|m−2−i ˜n−iN i y0. (1.5.4) As in the continuous-time case, the asymptotic behavior of the solutions ϕ(n, x0) = Anx0 of the linear difference equation xn+1 = Axn plays a key role in understanding the connections between linear algebra and dynamical systems. For this purpose, we introduce Lyapunov exponents. Definition 1.5.4. Let ϕ(n, x0) = Anx0,n Z, be a solution of the lin- ear difference equation xn+1 = Axn. Its Lyapunov exponent or exponential growth rate for x0 = 0 is defined as λ(x0) = lim sup n→∞ 1 n log ϕ(n, x0) , where log denotes the natural logarithm and · is any norm in Rd.
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