The mathematician H. Poincaré discovered in 1892 that certain deterministic mechanical systems could display chaotic behaviour. The word `chaotic' denotes disorder and irregularity. But only in 1963 the meteorologist E.N. Lorenz discovered a simple set of three first-order nonlinear differential equations whose state variables may display chaotic behaviour.
Since we encounter many phenomena with irregular motion, e.g. the weather, turbulence, carbon resistor noise, chemical reactions and biological signals, we are tempted to investigate whether we could model the dynamics with nonlinear differential equations.
Our aim is to find order within the chaos; to find evidence that the irregular behaviour is governed by a small set of deterministic equations, using experimental time series. We might be successful in particular when the state variables of the system are strongly coupled.
In this report, we will restrict ourselves to the determination of two properties that describe a chaotic system: the dimension and entropy spectra. Loosely speaking, the dimension is a measure for the number of differential equations needed to describe the system, while the entropy is a measure for the loss of information about the state of the system in the course of time. A positive but finite entropy is a hall-mark of chaos.
This research started with the investigation of the applicability of nearest neighbour methods of dimension estimation to clinical electroencephalograms [van Erp, 1988,Olofsen, 1987]. There was a need for improvement of this method, for comparing the results with the correlation integral method, and for obtaining a measure for the errors of the estimates.
In the following chapter we will define the dimension and entropy spectra and show that it is possible, in principle, to determine them from a time series. In Chapters 3 and 4 we describe the two types of dimension and entropy estimators. Expressions for the variances of the (correlation) dimension and entropy are derived and supported by Monte Carlo simulations in Chapter 5. In Chapter 6 we discuss practical aspects of the identification of chaotic systems, and in Chapter 7 we present some results obtained using signals from biological origin. Finally, in Chapter 8, we discuss the performance of the estimators studied.