Wavelet and multifractal analysis of the nonlinear structures in chaotic processes for hydroecological systems

Authors: N.G. Serbov, O.Yu. Khetselius, A.A. Svinarenko, O.N. Grushevsky

Year: 2015

Issue: 16

Pages: 171-175


This paper goes on our investigations of the fractal structures in the chaotic and turbulent processes and connected with a great importance the experimental and theoretical studying of the non-linear dynamical systems with aim to discover the fractal features and elements of dynamical chaos. In this paper on the basis of wavelet analysis and multifractal formalism it is carried out an analysis of fractal structures in the chaotic processes (the time series of the nitrates concentrations in the Small Carpathians river’s watersheds Svidnik-Ondrava in the Earthen Slovakia) and the spectrum of the fractal dimensions has been computed. It is carried out numerical modelling and fulfilled a comparison of theoretical data with the earlier received estimates on the basis of other fractal formalism algorithm.

Tags: chaotic processes; fractals structures; hydrological systems; the time series of pollutants concentrations


  1. Svinarenko A.A., Khetselius O.Yu., Mansarliysky V.F., Romanenko S.I. Analysis of the fractal structures in turbulent processes. Ukr. gìdrometeorol. ž – Ulrainian Hydrometeorology Journal, 2014, no. 15, pp. 74-78.
  2. Khetselius O.Yu., Svinarenko A.A. Analysis of the fractal structures in wave processes. Vìsn. Odes. derž. ekol. unìv.– Bulletin of Odessa state environmental university, 2013, vol. 16, pp. 222-226.
  3. Abarbanel H.D.I., Brown R., Sidorowich J.J., Tsimring L.Sh. The Mandelbrot B. Fractal geometry of nature. Moscow: Mir, 2002.
  4. Schertzer D., Lovejoy S. Fractals: Physical Origin and Properites. N.-Y.: Plenum Press, 1990, pp. 71-92. (Ed.: Peitronero L.)
  5. Zaslavsky G.M. Stochasticity of dynamical systems. Moscow: Nauka, 1998.
  6. Zosimov V.V., Lyamshev L.M. Fractals in wave processes. Phys.Uspekhi, 1995, vol.165, pp. 361–402.
  7. Grassberger P., Procaccia I. Measuring the strangeness of strange attractors. Physica D., 1983, vol. 9, pp. 189-208.
  8. Kaplan J.L., Yorke J.A. Chaotic behavior of multidimensional difference equations. Functional differential equations and approximations of fixed points. Lecture Notes in Mathematics. Berlin: Springer, 1979, no. 730, pp. 204-227. (Eds: H.-O. Peitgen, H.-O. Walter)
  9. Packard N.H., Crutchfield J.P., Farmer J.D., Shaw R.S. Geometry from a time series. Phys. Rev. Lett, 1980, vol. 45, pp. 712-716.
  10. Schreiber T. Interdisciplinary application of nonlinear time series methods. Phys. Rep., 1999, vol. 308, pp. 1-64.
  11. Daubechies I. Ten Lectures on Wavelets. Philadelphia: SIAM, 1992.
  12. Morlet J., Arens G., Fourgeau E., Giard D. Wave propagation and sampling theory. Geophysics, 1982, vol.47, pp. 203-236.
  13. Nason G., von Sachs R., Kroisand G. Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. J.Royal Stat.Soc., 2000, vol. B62, pp. 271-292.
  14. Glushkov A.V., Khokhlov V.N., Svinarenko A.A., Bunyakova Yu.Ya., Prepelitsa G.P. Wavelet analysis and sensing the total ozone content in the earth atmosphere: Mycros technology “Geomath”. Sensor Electr. and Microsys.Techn., 2005, vol.2(3), pp. 51-60.
  15. Glushkov A.V., Khokhlov V.N., Tsenenko I.A. Atmospheric teleconnection patterns: wavelet analysis. Nonlin. Proc.in Geophys., 2004, vol. 11, no. 3, pp. 285-293.
  16. Glushkov A.V., Loboda N.S., Khokhlov V.N., Lovett L. Using non-decimated wavelet decomposition to analyse time variations of North Atlantic Oscillation, eddy kinetic energy, and Ukrainian precipitation. Journal of Hydrology. Elsevier, 2006, vol. 322, no. 1-4, pp. 14-24.
  17. Sivakumar B. Chaos theory in geophysics: past, present and future. Chaos, Solitons & Fractals, 2004, vol. 19, pp. 441-462.
  18. Svoboda A., Pekarova P., Miklanek P. Flood hydrology of Danube between Devin and Nagymaros in Slovakia.- Nat. Rep.2000, UNESKO.-Project 4.1. Intern.Water Systems. 2000. 96 p.
  19. Pekarova P., Miklanek P., Konicek A., Pekar J. Water quality in experimental basins. -Nat. Rep.1999 of the UNESKO.-Project 1.1. Intern.Water Systems., 1999. 98 p.
  20. Balan A.K.,Systems Approach in hydrology: Extremal Hydrological Events and Effect of Changes in Hydrospheres State. Proc. Intern. Conf. “Ecology of Siberia, the Far East and the Arctic”. SD RAN, 2001, p. 133.
  21. Glushkov A.V., Balan A.K., Multifractal approach for modeling flow and short-term hydrological forecasts (for example, r. Danube). Meteorology, Climatology and Hydrology, 2004, no. 48, pp. 392-396.
  22. Balan A.K. Method multifactorial system modeling in problems of calculation extremal hydrological phenomena. Meteorology, Climatology and Hydrology, 2002, no. 45, pp. 147-152.
  23. Glushkov A.V. Khokhlov V.N., Serbov N.G., Balan A.K., Bunyakova Y.Y., Balanyuk E.P. Low-dimensional chaos in the time series of concentrations of pollutants in the atmosphere and hydrosphere. Vìsn. Odes. derž. ekol. unìv.– Bulletin of Odessa state environmental university, 2007, no. 4, pp. 337-348.
  24. Glushkov A.V., Khetselius O.Yu., Serbov N.G., Bunyakova Yu.Ya., Balan A.K., Buyadzhi V.V Modelling and forecasting the hydroecological systems pollution dynamics by using a chaos theory methods: I. Advanced data on pollution of the Small Carpathians river’s watersheds. Vìsn. Odes. derž. ekol. unìv.– Bulletin of Odessa state environmental university, 2015, no.19, pp.131-136.
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