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Applied possibilities of nonlinear dynamics approaches and methods of empirical model reconstruction

Methods for construction of nonlinear dynamical models from complex irregular time series are quite promising tool for investigation of complex systems and solution of different applied problems. Currently, wider application of the proposed techniques in practice with the purpose of their approbation and examination, as well as getting new useful results, is relevant. A small list of problems which can be addressed with the methods of modeling from observed series is reported below.

  1. Restoration of hidden variables and parameters of an object which cannot be measured directly; reconstruction of nonlinear characteristics of an object which cannot be measured ]in another way (it is relevant in applied sense for investigation of objects of different origin).
  2. Better understanding of the mechanism of an object functioning (it is relevant for many fields of science and practice) via validation of different model structures and choice of the most adequate among them. The main idea: of a "good" model of a specified structure cannot be constructed, then the physical or other substantial ideas underlying the structure are not valid.
  3. Forecast of the future behavior of an object and its "response" to different changes of its parameters (relevant for geophysics, astrophysics, economics and finance, etc).
  4. Automatic control (relevant for the technology). If one can voluntarily change parameters of an object, then a model could help to find out in advance which accessible parameters should be changed and in what way to provide a desired regime of an object functioning.
  5. "Characterization" of an object (relevant for the problem of signal classification, medical and technical diagnostics). The values of model parameters and its characteristics (attractor dimension, Lyapunov exponents, etc) can serve as quantities characterizing importnat properties of an object and distinguishing it from others.

Further, we list several practical examples of applications of the time series modeling techniques which are known from the literature. They are systematized according to the amount a priori information about an object:

  • Modeling: parameter estimation;
  • Modeling: reconstruction of nonlinear characteristics;
  • Modeling: "black box" reconstruction.

Our results concerning the use of nonlinear dynamics methods and dynamical modeling to solve applied problems:

  • Reconstruction of characteristics of a nonlinear element of an electric circuit and selection of identical elements.
  • Segmentation of nonstationary EEG recordings.
  • Synchronization of rhythms in cardio-vascular system.
  • Determination of the character of coupling between oscillators.
  • Coupling between EEG channels.
  • Extraction of a signal from under a masking signal of a delayed feedback generator.
  • Climatology.
  • Other trials and examples.

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