Nonlinear Dynamics Time Series Analysis

Much of what is known about physiological systems has been learned using linear system theory. However, many biomedical signals are apparently random or aperiodic in time. Traditionally, the randomness in biological signals has been ascribed to noise or interactions between very large numbers of constituent components. One of the most important mathematical discoveries of the past few decades is that random behavior can arise in deterministic nonlinear systems with just a few degrees of freedom. This discovery gives new hope to providing simple mathematical models for analyzing, and ultimately controlling, physiological systems. The purpose of this chapter is to provide a brief pedagogic survey of the main techniques used in nonlinear time series analysis and to provide a MATLAB tool box for their implementation. Mathematical reviews of techniques in nonlinear modeling and forecasting can be found in Refs. 1-5. Biomedical signals that have been analyzed using these techniques include heart rate [6-8], nerve activity [9], renal flow [10], arterial pressure [11], electroencephalogram [12], and respiratory waveforms [13]. Section 2 provides a brief overview of dynamical systems theory including phase space portraits, Poincare surfaces of section, attractors, chaos, Lyapunov exponents, and fractal dimensions. The forced Duffing-Van der Pol oscillator (a ubiquitous model in engineering problems) is investigated as an illustrative example. Section 3 outlines the theoretical tools for time series analysis using dynamical systems theory. Reliability checks based on forecasting and surrogate data are also described. The time series methods are illustrated using data from the time evolution of one of the dynamical variables of the forced Duffing-Van der Pol oscillator. Section 4 concludes with a discussion of possible future directions for applications of nonlinear time series analysis in biomedical processes.