Can the analytic techniques of nonlinear dynamics distinguish periodic, random and chaotic signals?

Recent advances in the mathematical discipline of nonlinear dynamics have led to its use in the analysis of many biologic processes. But the ability of the tools of nonlinear dynamic analysis to identify chaotic behavior has not been determined. We analyzed a series of signals--periodic, chaotic and random--with five tools of nonlinear dynamics. Periodic signals were sine, square, triangular, sawtooth, modulated sine waves and quasiperiodic, generated at multiple amplitudes and frequencies. Chaotic signals were generated by solving sets of nonlinear equations including the logistic map, Duffing's equation, Lorenz equations and the Silnikov attractor. Random signals were both discontinuous and continuous. Gaussian noise was added to some signals at magnitudes of 1, 2, 5, 10 and 20% of the signal's amplitude. Each signal was then subjected to tools of nonlinear dynamics (phase plane plot, return map, Poincaré section, correlation dimension and spectral analysis) to determine the relative ability of each to characterize the underlying system as periodic, chaotic or random. In the absence of noise, phase plane plots and return maps were the most sensitive detectors of chaotic and periodic processes. Spectral analysis could determine if a process was periodic or quasiperiodic, but could not distinguish between chaotic and random signals. Correlation dimension was useful to determine the overall complexity of a signal, but could not be used in isolation to identify a chaotic process. Noise at any level effaced the structure of the phase plane plot. Return maps were relatively immune to noise at levels of up to 5%. Spectral analysis and correlation dimension were insensitive to noise. Accordingly, we recommend that unknown signals be subjected to all of the techniques to increase the accuracy of identification of the underlying process. Based on these data, we conclude that no single test is sufficiently sensitive or specific to categorize an unknown signal as chaotic.