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.
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2014 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.