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Anosov O. L., Butkovskij O. J., Kravtsov Y. A. Reconstruction of dynamical systems from chaotic time series: short review. Izvestiya VUZ. Applied Nonlinear Dynamics, 2000, vol. 8, iss. 1, pp. 29-51. DOI: 10.18500/0869-6632-2000-8-1-29-51

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Reconstruction of dynamical systems from chaotic time series: short review

Anosov Oleg Lvovich, Institute for Nuclear Research
Butkovskij Oleg Jaroslavovich, Federal State Budget Educational Institution of Higher Professional Education "Vladimir Grigorievich Vladimir State University and Nikolai Grigorievich Stoletovykh"
Kravtsov Yury Aleksandrovich, Space Research Institute Russian Academy of Sciences

The brief survey of problems accompanying the reconstruction of dynamical equations from chaotic time series is presented. The most frequently used procedures of reconstruction are described, including estimate for the dimension of a system, choice of the equation type, determination of parameters of nonlincar functions by means оf differential equation fitting to time series, deleting of unreliable coefficients, testing of global stability of the reconstructed system. Several examples of reconstruction are presented. 
New effective criterium for distinguishing dynamical and random processes is suggested, based оп the notion of degree of predictability. Principal limitations are pointed out imposed by the properties of instability in presence of noise оп time-interval of predictable behaviour («horizon of predictability»), on the length of a sample, on amount of coefficient to be determined. So named «discriminant» (two window) approach is outlined, which allows to reveal nonstationarity in dynamical system, and important role of low dimensional models for retrieval of nonstationarities in systems of higher dimension is discussed. In conclusion prospective areas of applicability of reconstruction procedures are pointed out.

Key words: 
The work was supported by the INTAS (grant 96-0305), RFBR (grant 99-02-16625) and Federal Target Program "Integration" (grant А-0030).
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