Recovering The Lyapunov Exponent From Chaotic Time Series

Chaotic time series analysis is currently in wide use as a research tool to recover multidimensional dynamics from univariate experimental time series of chaotic systems. This thesis deals with the methodology of attractor recovery and Lyapunov exponent estimation from chaotic experimental systems. The history of dynamical recovery is reviewed and a consistent approach to accurate attractor reconstruction is advocated through the use of the Karhunen-Loeve coordinate transformation. A procedure for accurately estimating the largest Lyapunov exponent is developed based on the displacement method proposed by Wolf et al. A number of modifications to this method provide greatly improved exponent estimates from short and noisy time series, and a formalism is developed for obtaining exponent estimates in the presence of noise. The central role of replacement vector misalignment has been studied, a point which was underestimated in the original proposal of the method. Misalignment is shown to be dynamically equivalent to a small amplitude random process, and the same model for exponent estimation from noisy time series can be used to deal effectively with vector misalignment. The effect of arbitrary embedding dimension and delay time upon exponent estimates is examined for several model systems.;These procedures were used to recover attractor portraits and Lyapunov exponent estimates from several data sets. Experimental time series from a driven nonlinear pendulum were studied, with the pendulum in periodic and chaotic states. In all cases attractor recovery accurately identified periodic or chaotic motion, and estimated Lyapunov exponents agreed with those predicted by numerical simulation. As a further application, the method was applied to EEG time series from a human epileptic seizure. Direct evidence of chaos within the seizure was found by exponential separation of initially nearby states in the recovered trajectory. The estimated Lyapunov exponent was consistent with the mean information dissipation rate measured by the autocorrelation function, with estimated Lyapunov exponents being numerically stable over a range of embedding dimensions consistent with the measured correlation dimension. This is the first consistent indication of chaos within the seizure state to be found using careful cross-testing of different aspects of the dynamics