14 1. Autonomous Linear Differential and Difference Equations Theorem 1.4.3. For ˙ = Ax, A gl(d, R), there are exactly Lyapunov exponents λ(x0), the distinct real parts λj of the eigenvalues of A. For a solution x(·,x0) (with x0 = 0) one has λ(x0) = limt→±∞ 1 t log x(t, x0) = λj if and only if x0 L(λj). Proof. Using Lemma 1.4.2 we may assume that A is given in real Jordan form. Then the assertions of the theorem can be derived from the solution formulas in the generalized eigenspaces. For a Jordan block for a real eigen- value μ = λj, formula (1.3.2) yields for every component xi(t, x0) of the solution x(t, x0) and |t| 1, log |xi(t, x0)| = μt + log m k=i tk−i (k i)! xk μt + m log |t| + log max k |xk| , log |xi(t, x0)| μt m log |t| log max k |xk| . Since 1 t log |t| 0 for t ±∞, it follows that limt→±∞ 1 t log |xi(t, x0)| = μ = λj. With a bit more writing effort, one sees that this is also valid for every component of x0 in a subspace for a Jordan block corresponding to a complex-conjugate pair of eigenvalues with real part equal to λj: By (1.3.3) one obtains the product of eλjt with a polynomial in t and sin and cos functions. The logarithm of the second factor, divided by t, converges to 0 for t ±∞. The Lyapunov space L(λj) is obtained as the direct sum of such subspaces and every component of a corresponding solution has exponential growth rate λj. Since we may take the maximum-norm on Rd, this shows that every solution starting in L(λj) has exponential growth rate λj for t ±∞. The only if part will follow from Theorem 1.4.4. We emphasize that a characterization of the Lyapunov spaces L(λj) via the exponential growth rates of solutions needs the limits for t +∞ and for t −∞. In other words, if one only considers the exponential growth rate for positive time, i.e., for time tending to +∞, one cannot characterize the Lyapunov spaces. Nevertheless, we can extend the result of Theorem 1.4.3 by describing the exponential growth rates for positive and negative times and arbitrary initial points. Theorem 1.4.4. Consider the linear autonomous differential equation ˙ = Ax with A gl(d, R) and corresponding Lyapunov spaces Lj := L(λj),λ1 . . . λ . Let V +1 = W0 := {0} and for j = 1,..., define (1.4.1) Vj := L . . . Lj and Wj := Lj . . . L1.
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