Computing the Lyapunov spectrum of a dynamical system from an observed time series

Abstract

We examine the question of accurately determining, from an observed time series, the Lyapunov exponents for the dynamical system generating the data. This includes positive, zero, and some or all of the negative exponents. We show that even with very large data sets, it is clearly advantageous to use local neighborhood-to-neighborhood mappings with higher-order Taylor series, rather than just local linear maps as has been done previously. We give examples using up to fifth-order polynomials. We demonstrate this procedure on two familiar maps and two familiar flows: the H\'enon and Ikeda maps of the plane to itself, the Lorenz system of three ordinary differential equations, and the Mackey-Glass delay differential equation. We stress the importance of maintaining two dimensions for converting the scalar data into time delay vectors: one is a global dimension to ensure proper unfolding of the attractor as a whole, and the other is a local dimension for capturing the local dynamics on the attractor. We show the effects of changing the local and global dimensions, changing the order of the mapping polynomial, and additive (measurement) noise. There will always be some limit to the number of exponents that can be accurately determined from a given finite data set. We discuss a method of determining this limit by numerically obtaining the singularity spectra of the data set and also show how it is often appropriate to make this choice based on the fractal dimension of the attractor. If excessively large dimensions are used, spurious exponents will be generated, and in some cases the accuracy of the true exponents will be affected. We present methods of identifying these spurious exponents by determining the Lyapunov direction vectors at particular points in the data set. We can then use these to identify numerical problems and to associate data-set singularities with particular exponents. The behavior of spurious exponents in the presence of noise is also investigated, and found to be different from that of the true exponents. These provide methods for identifying spurious exponents in the analysis of experimental data where the system dynamics may not be known a priori.