THE QUALITATIVE FRACTAL ANALYSIS OF LONG TERM TIME SERIES FOR AGRICULTURAL SOILS’ ELECTRICAL CONDUCTIVITY PARAMETERS: METHODS OF NONLINEAR DYNAMICS, THEORY OF CHAOS, PHASE TRAJECTORIES

Abstract

The procedure of the qualitative fractal analysis of long term time series for agricultural soils’ electrical conductivity parameters, for which the hypothesis of trend existence isn’t confirmed, with application of the methods of nonlinear dynamics, theory of chaos and phase trajectories, is presented. The real time series characterizing mentioned above electrical conductivity parameters of Ukrainian soils are considered. The basis for similar researches is Takens’s theorem. The randomness of the studied dynamical system given by time realizations is established by means of Lyapunov’s indicator. The state stability is estimated by Hausdorff ’s fractal dimension and the fractality index. Visual evaluation of the time series was carried out by means of the phase trajectory restoration procedure. As a result of the analysis of phase points in the phase space the split attractor is indicated, which gives the chaise to speak about its bifurcation. Application of the nonlinear dynamical system theory methods to the time series analysis is based on the hypothesis that the available series describes the behavior of the studied system, and it’s the only available information about this system. According to the well-known Takens’s theorem [1] a single time series suffices for an adequate description of a dynamical system as a whole. The analysis of time series by the methods of nonlinear dynamical system theory is becoming widely applied. In terminology of this theory the process described by time series contains the deterministic chaos, or, in other words, is chaotic. From the linear analysis method point of view they are stochastic processes. The nonlinear analysis demonstrates that neither can these processes be considered as deterministic ones, nor are they absolutely random. In other words, only short-term forecasting of the system condition is possible with certain accuracy.