1.4. Lyapunov Exponents 15 Then a solution x(·,x0) with x0 = 0 satisfies lim t→+∞ 1 t log x(t, x0) = λj if and only if x0 ∈ Vj \ Vj+1, lim t→−∞ 1 |t| log x(t, x0) = −λj if and only if x0 ∈ Wj \ Wj−1. In particular, limt→±∞ 1 |t| log x(t, x0) = ±λj if and only if x0 ∈ L(λj) = Vj ∩ Wj. Proof. This follows using the arguments in the proof of Theorem 1.4.3: For every j one has Vj+1 ⊂ Vj and for a point x0 ∈ Vj \ Vj+1 the component in L(λj) is nonzero and hence the exponential term eλjt determines the exponential growth rate for t → ∞. By definition, Lj = Vj ∩ Wj,j = 1,..., . Note further that −λ . . . −λ1 are the real parts of the eigenvalues −μ of −A, where μ are the eigenvalues of A. Strictly increasing sequences of subspaces as in (1.4.1), {0} = V +1 ⊂ V ⊂ . . . ⊂ V1 = Rd, {0} = W0 ⊂ W1 ⊂ . . . ⊂ W = Rd, are called flags of subspaces. By definition V = L is the Lyapunov space corresponding to the smallest Lyapunov exponent, and Vj is the sum of the Lyapunov spaces corresponding to the j smallest Lyapunov exponents. Stability Using the concept of Lyapunov exponents and Theorem 1.4.4 we can describe the behavior of solutions of linear differential equations ˙ = Ax as time tends to infinity. By definition, a solution with negative Lyapunov exponent tends to the origin and a solution with positive Lyapunov exponent becomes unbounded (the converse need not be true, as we will see in a moment). It is appropriate to formulate the relevant stability concepts not just for linear differential equations, but for general nonlinear differential equations of the form ˙ = f(x) where f : Rd → Rd. We assume that there are unique solutions ϕ(t, x0),t ≥ 0, of every initial value problem of the form (1.4.2) ˙ = f(x), x(0) = x0 ∈ Rd. Remark 1.4.5. Suppose that f is a locally Lipschitz continuous vector field, i.e., for every x ∈ Rd there are an ε-neighborhood N(x, ε) = {y ∈ Rd | y − x ε} with ε 0 and a Lipschitz constant L 0 such that f(y) − f(x)≤ L y − x for all y ∈ N(x, ε).

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