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.

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