Dimension and Lyapunov Exponents

Abstract

An important concept for (chaotic) dynamical systems is the notion of Lyapunov exponents, introduced by Oseledec (1968). Lyapunov exponents are numbers which describe the average behavior of the derivative of a map along a trajectory. Let F be a differentiable map from the n-dimensional phase space to itself. For each point x in the phase space, the trajectory (or orbit) of x is the sequence x, F(x), F(F(x)), F(F(F(x))), … . For each point x in the phase space, we consider the Jacobian matrix

$${\rm DF^k(x)}$$

of partial derivatives of the kth iterate of the map F at x, where k is any positive integer. For the discrete time system x(k+l) = F(x(k)) starting from x(0), the Jacobian matrix

$${\rm J_{(k)}}\ =\ {\rm DF^k(x(0))}$$

is the product DF(x(k−l))DF(x(k−2))…DF(x(0)). The matrices J(k) can be used to estimate the exponential rate at which nearby orbits are separated. For the discrete time system x(k+l) = F(x(k)), the separation of two initial points x(0) and y(0) after time k is x(k)−y(k). If these two initial conditions are close to each other, then the separation of these two points under forward iteration of the map F is approximately the matrix J(k) times the difference vector x(0)−y(0), Lyapunov exponents depend on a trajectory x(0), x(l), x(2), …, x(k), …, where x(k+l) = F(x(k)). Write

$${\rm B}_0$$

for the unit ball in n-dimensional phase space (that is, x is an n-dimensional vector), and denote the successive iterates by

$${\rm B_{k+1}}\ =\ {\rm DF(x(k))\ B_k}$$

Notice that each B_K is an ellipsoid. Let 1 ≤ j ≤ n. Let β_j,k = the length of the jth largest axis of B_k . We define the jth Lyapunov number L(j) of F at x(0) to be

$${\rm L_{(j)}}\ =\ {\rm lim_{k\rightarrow\infty}\ ({\beta}_{j,k})^{1/k}}$$

where we assume here that the limit exists. The trajectory would be extremely unusual if the limit did not exist. Notice that the quantity L(j) is the average factor by which the jth largest axis grows per unit time. Here k is time. The Lyapunov exponents of the trajectory are the natural logarithms

$${\rm\lambda_j}\ =\ {\rm log\ L(j)}$$