Estimating Lyapunov Exponents from Approximate Return Maps

Abstract

Phase portrait reconstructions from experimental time series are widely used for the calculation of dimensions, Lyapunov exponents and entropy, all of which can be used to characterise a dynamical system and investigate the effect of varying external parameters (Schuster 1984). Often it is desirable to reduce the dimensionality of the system by the method of Poincaré sections (Hirsch and Smale 1974). The reduction in the amount of data available for calculating quantities such as correlation dimension is compensated for by the improvement in quantitative and qualitative deductions that can be made (Brandstater and Swinney 1987). An alternative approach, sometimes possible in low dimensions, is to consider an approximate one-dimensional return map of the flow, for example by plotting successive minima of some component (Sparrow 1982, Chapman et.al. 1989). If the trajectories end up near an attractor with Hausdorff dimension slightly greater than two, then the resulting return map is close to one dimensional and the comprehensive theory of one-dimensional maps can be used to characterise bifurcations in the system as a parameter is varied. In some cases this approach can be made rigorous and when it works it is simpler than using the reconstructed flow. In this report the two approaches are related: the first produces a ‘Poincaré section of a Reconstruction’ while the second produces a ‘Reconstruction of a Poincaré section’. The one-dimensional approximate return map can then be used to estimate the largest Lyapunov exponent of a Poincaré map of the flow. Only the case in ℝ³ is considered and when the measured signal is the first component of the state vector x(t).