next up previous
Next: Dimensions and entropies Up: Lyapunov exponents Previous: The maximal exponent

The Lyapunov spectrum

The computation of the full Lyapunov spectrum requires considerably more effort than just the maximal exponent. An essential ingredient is some estimate of the local Jacobians, i.e. of the linearized dynamics, which rules the growth of infinitesimal perturbations. One either finds it from direct fits of local linear models of the type tex2html_wrap_inline7515, such that the first row of the Jacobian is the vector tex2html_wrap_inline6979, and tex2html_wrap_inline7519 for tex2html_wrap_inline7521, where m is the embedding dimension. The tex2html_wrap_inline6979 is given by the least squares minimization tex2html_wrap_inline7527 where tex2html_wrap_inline7529 is the set of neighbors of tex2html_wrap_inline6691 [45, 71]. Or one constructs a global nonlinear model and computes its local Jacobians by taking derivatives. In both cases, one multiplies the Jacobians one by one, following the trajectory, to as many different vectors tex2html_wrap_inline7533 in tangent space as one wants to compute Lyapunov exponents. Every few steps, one applies a Gram-Schmidt orthonormalization procedure to the set of tex2html_wrap_inline7533, and accumulates the logarithms of their rescaling factors. Their average, in the order of the Gram-Schmidt procedure, give the Lyapunov exponents in descending order. The routine lyap_spec uses this method, which goes back to [71] and [45], employing local linear fits. Apart from the problem of spurious exponents, this method contains some other pitfalls: It assumes that there exist well defined Jacobians, and does not test for their relevance. In particular, when attractors are thin in the embedding space, some (or all) of the local Jacobians might be estimated very badly. Then the whole product can suffer from these bad estimates and the exponents are correspondingly wrong. Thus the global nonlinear approach can be superior, if a modeling has been successful, see Sec. gif.

In Table gif we show the exponents of the stroboscopic NMR laser data in a three dimensional embedding as a function of the neighborhood size. Using global nonlinear models, we find the numbers given in the last two rows. More material is discussed in [2]. The spread of values in the table for this rather clean data set reflects the difficulty of estimating Lyapunov spectra from time series, which has to be done with great care. In particular, when the algorithm is blindly applied to data from a random process, it cannot internally check for the consistency of the assumption of an underlying dynamical system. Therefore a Lyapunov spectrum is computed which now is completely meaningless.

method tex2html_wrap_inline7463 tex2html_wrap_inline7539 tex2html_wrap_inline7541
local linear k=20 0.32 -0.40 -1.13
`` k=40 0.30 -0.51 -1.21
`` k=160 0.28 -0.68 -1.31
radial basis functions 0.27 -0.64 -1.31
polynomial 0.27 -0.64 -1.15

Table:   Lyapunov exponents of the NMR laser data, determined with a three-dimensional embedding. The algorithms described in Sec. gif give tex2html_wrap_inline7549 for the largest exponent.

The computation of the first part of the Lyapunov spectrum allows for some interesting cross-checks. It was conjectured [72], and is found to be correct in most physical situations, that the Lyapunov spectrum and the fractal dimension of an attractor are closely related. If the expanding and least contracting directions in space are continuously filled and only one partial dimension is fractal, then one can ask for the dimensionality of a (fractal) volume such that it is invariant, i.e. such that the sum of the corresponding Lyapunov exponents vanishes, where the last one is weighted with the non-integer part of the dimension:
where k is the maximum integer such that the sum of the k largest exponents is still non-negative. tex2html_wrap_inline7555 is conjectured to coincide with the information dimension.

The Pesin identity is valid under the same assumptions and allows to compute the KS-entropy:

next up previous
Next: Dimensions and entropies Up: Lyapunov exponents Previous: The maximal exponent

Thomas Schreiber
Wed Jan 6 15:38:27 CET 1999