Identification of Complex Processes Based on Analysis of Phase Space Structures

Abstract

Abstract. The problem of investigation of temporal and/or spatial behavior of highly nonlinear or complex natural systems has long been of fundamental scientific interest. At the same time it is presently well understood that identification of dynamics of processes in complex natural systems, through their qualitative description and quantitative evaluation, is far from a purely academic question and has an essential practical importance. This is quite understandable as systems with complex dynamics abound in nature and examples can be found in very different areas such as medicine and biology (rhythms, physiological cycles, epidemics), atmosphere (climate and weather change), geophysics (tides, earthquakes, volcanoes, magnetic field variations), economy (financial markets behavior, exchange rates), engineering (friction, fracturing), communication (electronic networks, internet packet dynamics) etc. The past two decades of research on qualitative and especially quantitative investigations of dynamics of real processes of different origin brought significant progress in the understanding of behavior of natural processes. At the same time serious drawbacks have also been revealed. This is why exhaustive investigation of dynamical properties of complex processes for scientific, engineering or practical purposes is now recognized as one of the main scientific challenges. Much attention is paid to elaboration of appropriate methods aiming to measuring the complexity of both global and local dynamical behaviors from the observed data sets -time series. This chapter presents a short overview of modern methods of qualitative and quantitative evaluation of dynamics of complex natural processes such as calculation of Lyapunov exponents and fractal dimensions, recurrence plots and recurrence quantification analysis. Other related methods are also described. The traditional approach to studying dynamical behavior of complex nonlinear systems is to reconstruct from observation scalar time series state or phase space plot. This graph indicates how the systems behavior changes over the time. We focus on the methods of identification and quantitative evaluation of complex dynamics that are based on the testing of evolutionary and geometric properties of phase space graphs as images of investigated complex dynamics. For practical examples of the application c 2006 Springer. Printed in the Netherlands.of nonlinear methods for identification of complex natural processes, our results on medical, geophysical, hydrological, and stick-slip time series analysis are presented.

Evidence for changes in the dynamics of Earth crust tilts caused by the large dam construction and reservoir filling at the Enguri dam international test area (Georgia)

“…This seems natural and points to the close relationship between local seismicity and geodynamical processes, which seemingly were influenced by anthropogenic impact. The dynamical patterns of seismicity at stages 1-7 are also close to tilt behavior, as was described in earlier reports [12,14].…”

“…Namely, spectral and time frequency properties have been calculated, and long-range properties of datasets have been tested [15,19]. Correlation dimension and maximal Lyapunov exponent calculations were performed as well as recurrence quantitative analysis (RQA) of tilt data series [8,11,12,17] applied. Additionally, a calculation of algorithmic (Lempel-Ziv) complexity measure, as well as entropic measure, based on Tsallis statistics have been performed [3,4,7,17].…”

Scale Features of a Network Echo Mechanism: Case Study for the Different Internet Paths

“…Often for different analysis, purposes stationarity of Internet network is assumed [7]. In fact though, this practically never is the case, like it is for a diverse of time series generated by natural and physical processes [17][18][19][20].…”

Minority Community Development: Testing Landry's (2012) Intergroup Ethno-Linguistic Model

“…The second perspective is obtained by interpreting Landry's cultural autonomy model as a phase diagram. Phase diagrams are concrete demographic maps that help illustrate the spatial or temporal evolution of complex phenomena (Verbitskii & Tereshchenko, 1996;Shone, 2002;Guermond, 2008;Matcharashvili et al, 2007). For example the "X" axis of Figure 1 represents the dynamic dimension of the phase diagram.…”