Flicker noise spectroscopy and its application: Information hidden in chaotic signals (review)

Abstract

Fundamentals of flicker noise spectroscopy (FNS), a general phenomenological approach to analyzing dynamics and structure of complex nonlinear systems by extracting information contained in chaotic signals of diverse nature generated by such systems, are presented. The primary idea of FNS is to disclose information hidden in correlation links which are present in a sequence of various irregularities (spikes, jumps, discontinuities in derivatives of different orders) that occur in the measured dynamic variables at all levels of spatiotemporal hierarchy of systems under study. The information is derived from power spectra S(f) (f, frequency) and transient difference moments Φ(p)(τ)(τ, time delay parameter) of different orders p. The procedures of averaging over time interval T, which are introduced in FNS when computing S(f) and Φ(p)(τ), differ from the procedures of averaging in the Gibbs approach. In the latter case, due to the adoption of the ergodic hypothesis, the average values of dynamic quantities over time are replaced by the average values of the same quantities over a statistical ensemble. It came to pass that the Φ(p)(τ) functions are formed exclusively by jumps of a dynamic variable on different spatiotemporal levels of the system’s hierarchy, whereas the formation of S(f) is contributed to by spikes and jumps. The informative parameters extracted from S(f) and Φ(p)(τ) describe correlation times and characterize loss of “memory” (correlation links) in these correlation time intervals for the “spike” and “jump” irregularities. These parameters can be determined using the expressions derived for the case of steady-state evolution. Here the “steady state” implies an evolution state that is characterized by the same values of informative parameters on every level of the system’s hierarchy. In contrast to the scaling self-similarity in theory of fractals and renormgroup, FNS introduces a multiparametric self-similarity for the S(f) and Φ(p)(τ) functions, which is generally characterized by a set of parameters rather than one scaling factor. The S(f) and Φ(2)(τ) functions which are related to different types of information may be viewed, in the steady-state case, as fluctuation-dissipation relations that complement each other informatively. Examples of such generalized relations are presented for fluctuations of electric voltage under open-circuit conditions (the Nyquist theorem), the Levy diffusion, the hydrodynamic fluctuations at fully developed turbulence, and the flicker noise fluctuations of the electric current density in electron-conducting systems. To analyze the dynamics of non-steady processes, formulas are presented for calculating nonstationarity indicators (factors) and estimating time instants when most noticeable changes occur in systems under study, including those preceding catastrophic evolution changes. The studies carried out to date show that the FNS method can be used in solving problems of three types. The first is the determination of parameters or patterns that characterize the dynamics or specific features of structural organization of open complex systems. The second involves search for precursors of sharpest changes in the states of various open dissipative systems on the basis of available information conceming the dynamics of such systems. And the third problem concerns the dynamics of redistribution of perturbations in distributed systems. It is solved by analyzing dynamic correlations in chaotic signals that are measured simultaneously at different points in space. The review demonstrates some applications of the FNS methodology. In particular, it considers some physicochemical and natural processes the data for which were obtained from electrochemical and biological measurements.