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The purpose of this research is to develop and adapt a complex of hybrid mathematical and instrumental methods of analysis and risk management through the prediction of natural time series with memory. The paper poses the problem of developing a constructive method for predictive analysis of time series within the current trend of using so-called “graphical tests” in the process of time series modeling using nonlinear dynamics methods. The main purpose of using graphical tests is to identify both stable and unstable quasiperiodic cycles (quasi-cycles). Modern computer technologies which allow to study in detail complex phenomena and processes were used as a toolkit for the implementation of nonlinear dynamics methods. Authors propose to use for the predictive analysis of time series a modified R=S-analysis algorithm, as well as phase analysis methods for constructing phase portraits in order to identify cycles of the studied time series and confirm the forecast. This approach differs from classical forecasting methods by implementing trends accounting and appears to the authors as a new tool for identifying the cyclical components of the considered time series. Using the proposed hybrid complex, the decision maker has more detailed information that cannot be obtained using classical statistics methods. In this paper, authors analyzed the time series of Kuban mountain river runoffs, revealed the impossibility of using the classical Hurst method for their predictive analysis and also proved the consistency of using the proposed hybrid toolkit to identify the cyclic components of the time series and predict it. The study acquires particular relevance in the light of the absence of any effective methods for predicting natural-economic time series, despite the proven need to study them and their risk-extreme levels.