1.4. Lyapunov Exponents 13 is defined as λ(x0,A) = lim sup t→∞ 1 t log x(t, x0) , where log denotes the natural logarithm and · is any norm in Rd. Let Ek = E(μk),k = 1,...,r, be the real generalized eigenspaces, denote the distinct real parts of the eigenvalues μk by λj, and order them as λ1 . . . λ , 1 r. Define the Lyapunov space of λj as Lj = L(λj) := Ek, where the direct sum is taken over all generalized real eigenspaces associated to eigenvalues with real part equal to λj. Note that Rd = L(λ1) . . . L(λ ). When the considered matrix A is clear from the context, we write the Lya- punov exponent as λ(x0). We will clarify in a moment the relation between Lyapunov exponents, eigenvalues, and Lyapunov spaces. It is helpful to look at the scalar case first: For ˙ = λx, λ R, the solutions are x(t, x0) = eλtx0. Hence the Lyapunov exponent is a limit (not just a limit superior) and lim t→±∞ 1 t log eλtx0 = lim t→±∞ 1 t log eλt + lim t→±∞ 1 t log |x0| = λ. Thus −λ is the Lyapunov exponent of the time-reversed equation ˙ = −λx. First we state that the Lyapunov exponents do not depend on the norm and that they remain constant under similarity transformations of the matrix. Lemma 1.4.2. (i) The Lyapunov exponent does not depend on the norm in Rd used in its definition. (ii) Let A, B gl(d, R) with B = S−1AS for some S Gl(d, R). Then the Lyapunov exponents λ(x0,A) and λ(y0,B) of the solutions x(t, x0) of ˙ = Ax and y(t, y0) of ˙ = By, respectively, are related by λ(y0,B) = λ(Sy0,A). Proof. (i) This is left as Exercise 1.6.10. (ii) Using Proposition 1.3.1 we find λ(y0,B) = lim sup t→∞ 1 t log y(t, y0) = lim sup t→∞ 1 t log S−1x(t, Sy0) lim sup t→∞ 1 t log S−1 + lim sup t→∞ 1 t log x(t, Sy0) = λ(Sy0,A). Writing x0 = S ( S−1x0 ) , one obtains also the converse inequality. The following result clarifies the relationship between the Lyapunov ex- ponents of ˙ = Ax and the real parts of the eigenvalues of A. It is the main result of this chapter concerning systems in continuous time and explains the relation between Lyapunov exponents for ˙ = Ax and the matrix A, hence establishes a first relation between dynamical systems and linear algebra.
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