The Lyapunov Exponent: Multidimensional Case

Abstract

How can we measure the randomness of a differentiable mapping T on a multidimensional space? First, find its linear approximation at a point x by the Jacobian matrix DT (x) and observe that a sphere of a small radius centered at x is approximately mapped to an ellipsoid. By applying DT (T k x), k ≥ 1, repeatedly along the orbit of x, we observe that the image of the given sphere is approximated by an ellipsoid and that the lengths of its semi-axes increase exponentially. The average of the exponents of growth are called the Lyapunov exponents, and they are obtained using singular values of D(T n) as n → ∞. The largest Lyapunov exponent is equal to the divergence speed of two nearby points. For a comprehensive survey consult [Y1]. 10.1 Singular Values of a Matrix In this section we give a brief introduction to singular values. In this chapter we need facts only for square matrices but we consider general matrices because the argument for rectangular matrices is the same. Let A be an n × m real matrix. Its transpose is denoted by A T. The m×m symmetric matrix A T A has real eigenvalues, and the corresponding eigenvectors are also real and pairwise orthogonal. If λ ∈ R satisfies A T Av = λv for some v ∈ R m , ||v|| = 1, then ||Av|| 2 = (Av, Av) = (A T Av, v) = (λv, v) = λ , and so λ ≥ 0. Let λ 1 ≤ · · · ≤ λ m be the eigenvalues of A T A with corresponding eigenvectors v i , ||v i || = 1. Put σ i = λ i , 1 ≤ i ≤ m. The σ i are called the singular values of A. Note that σ i = ||Av i || and σ 1 ≤ · · · ≤ σ m .