Chaos, Fractal And The Application To Biomedicine

This paper concerns studies of chaos and fractal of nonlinear theory, including analysis on chaotic and general features of different dimensional nonlinear mappings; discussion of generalized M-J sets and analysis on EEG signals by using chaos theory.(1) By using phase space reconstruct technique from a time series and the quantitative criterion and rule of system chaos, different nonlinear mappings are studied.At the base of calculation and anaylize by using phase graphics, bifurcation graphics, power spectra, the computation of the fractal dimension and the Lyapunov exponent, the general features of chaos and "approach to chaos" are discussed.(2) The method constructing the J-M sets from a simple nonanalytic mapping developed by Michelitsch and Rossler was expanded. According to the complex mapping expanded by the author, a series of the generalized J-M sets for real index number were constructed. Using the experimental mathematics method combining the theory of analytic function of one complex variable with computer aided drawing, the fractal features and evolutions of the generalized J-M sets are studied. The results show-. (1) the geometry structure of the generalized Julia sets depends on the parameters of α, R and c;and the Mandelbrot sets depends on a and R (2) the generalized J-sets and the generalized Mandelbrot sets for integer index number have symmetry and fractal feature; (3) the generalized J-M sets for decimal index number have discontinuity and collapse, and their evolutions depend on the choice of the principal range of the phase angle.(3) the author analyses EEG (Electroencephalogram) signals of piglets in the HAI (Hypoxic-Asphyxic Injury) experiments, the following conclusions are shown: (1)The analyses reflect the whole dynamic characteristics of the brains, and they may become a new method of researching EEG quantitatively to early diagnose of brain disease. (2)Under normal physiological conditions, the EEG signals are chaotic, while under injury conditions the signals approach regularity. Analyses and computations are conducted on EEG dynamics model, the following conclusions are shown: (1) Chaotic patterns of the dynamics model may emerge out of Pomeau-Manneville route, and relevant to double-periodic bifurcation, Hopf bifurcation, and reverse bifurcation; (2) To further support the view that chaos exist in EEG signals.