Estimating High-Order Lyapunov Exponents From A Time Series And Its Application On EEG High-Dimensional Chaos

Chaos is a kind of anomalous movement, seemingly stochastic, generated from nonlinear determined systems and widely exist in nature, economy, society and so on. Many chaos systems take on the properties of high dimension, especially in the field of life science. So researches of high-dimensional chaos have been the major subject in the research of chaos theory. At present, reducing dimension is the leading method to study high-dimensional chaos, but such methods is limited due to losing some information.In this research, one of our objectives is to study high-dimensional chaos based on high-order Lyapunov exponents and provide the algorithm and program to estimate the first three Lyapunov exponents, λ1, λ2 and λ3 from time series. The feasibility and validity of this arithmetic is tested on 4x3 Lorenz high-dimensional chaos system and HyperRossler chaos system. The other purpose of this thesis is to answer whether or not epilepsy EEG is derived from high-dimensional chaos.Lyapunov exponents describe the average change rate of radiation and convergence of two adjacent orbits in phase space. We can know the movement properties of orbits in each direction of the phase space. Furthermore, the Y-K formula offers the relation between the dimension of chaos systems and Lyapunov exponents. Theoretically, it is possible to estimate all Lyapunov exponents of chaos systems from one-dimensional time series. This paper determines the algorithm and program to estimate the first three Lyapunov exponents, λ1, λ2 and λ3 from time series and employs the re-conformation of the phase space and applies 21, λ2 and λ3 to seven cases of epilepsy EEG compared by the normal EEG. Furthermore, it is possible to judge the dimension range of epilepsy EEG.The results are as follows: (1)The algorithm and program of this thesis to estimate λ1,λ2 and λ3 is feasibility and validity for 4x3 Lorenz high-dimensional chaos system and HyperRossler chaos system and the error is in 5%. (2)The epilepsy EEG have two or three positive Lyapunov exponents and the dimension is in the range between three and four. The chaos dimension of epilepsy EEG can be classified as follows: (1)λ3 is equal to zero in λ1, λ2 and A3, and the dimension of corresponding abstractor is less than four and the system belongs to the low-dimensional chaos; (2)λ1, λ2 and λ3 are more than zero, but very small and the system belongs to the medial state between high dimension and low dimension; (3)λ1, λ2 and λ3 are far more than zero and the sysem belongs to the high-dimensional chaos. (3)The first three Lyapunov exponents of normal EEG are all far more than zero and the dimension is more than that of epilepsy EEG.The extended research of this thesis is to estimate all Lyapunov exponents of high-dimensional chaos from time series and use the Y-K formula to calculate the fractal dimension of high-dimensional chaos. So we can give the entire results about the dimension of epilepsy EEG..