Wavelet transformation of chaotic biological signals

Wavelet analysis is a recently developed mathematical theory and computational method for decomposing a nonstationary signal into components that have good localization properties both in time and frequency and hierarchical structures. Wavelet transform provides local information and multiresolution decomposition on a signal that cannot be obtained using traditional methods such as Fourier transforms and statistical estimation theory. Hence the change in complex biological signals can be detected. We use wavelet analysis as an innovative method for identifying and characterizing chaotic biological signals in this paper. We usually do not know the underlying mechanism that determine the behavior of a biosystem. We are instead presented with nothing more than a phenomenological time series signal of the behavior, and must infer the mechanism from simple measurements of that time series. Data we used are simulated chaotic signals from the logistic equation. Using wavelet transformation we extract instantaneous frequencies of the signal varying in time across scales. The results under different parameters and initial conditions show that the phase maps of their wavelet transforms are different between period doubling bifurcation and chaos. This information could be used as a diagnostic for detecting different nonlinear dynamic responses. This may lead to a better understanding of the system, that may allow us to predict the onset of lethal arrhythmias and to intervene prior to the development of catastrophic clinical events.