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Application of Non-Linear Dynamics Methods for Analysis of Results of Industrial Wastewater Monitoring

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A.P. Karmanov, L.S. Kocheva, T.N. Shchemelinina

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At the complex of measures to reduce the anthropogenic pollution the important role belongs to the monitoring of the enterprises wastewater, implemented by quantifying and analyzing of the indexes of standardized quality wastewater at the treatment plants. This article is devoted to the study of the variability of the composition of wastewater of industrial enterprises in different periods of time with the help of modern methods of mathematical analysis of time-series describing the integral dynamics of the processes of receipt and disposal of pollutants. The main idea of this paper is the reconstruction and quantitative characteristics of the so-called dynamic attractor of the system. To solve this problem we used the results of the daily analytical determinations of COD (chemical oxygen demand), reflecting, as it is known, the total number of substances capable of chemical oxidation. For the quantitative and identifying patterns associated with the dynamics of the system was carried out the mathematical analysis of the geometric image of the dynamic mode - an attractor, which attracts the set of trajectories of the system in the D-dimensional phase space. For this purpose the device of fractal mathematics and chemical synergy involving the methods of the standardized range and Packard’s pseudo-relative space was used. At the research it was carried out the reconstruction of the dynamics of the system at the time-series of COD and established the quantitative characteristics of the dynamic attractor of the system. These results suggest that the studied time-series are characterized by the presence of internal order, which is determined at the moment as a fractal. It means that the time evolution of the system is the stable dynamic state called “a strange attractor”. The estimation of the Hurst exponent and the values of the correlation and fractal dimension of the attractor in the multidimensional spaces pseudo-relative spaces were determined. Thus, it is shown that in this system there are phenomena of nonlinear dynamic self-organization, which is also confirmed by RS-analysis of time-series. The Grassberg-Procaccia algorithm helped to establish the final dimension of the space within the attractor functions. The presented results demonstrate the perspective of using of the modern methods of nonlinear dynamics for the analysis of the functioning of the complex dynamic systems, which include the industrial companies.


A.P. Karmanov1, Doctor of Chemistry, Professor, Senior Research Scientist, L.S. Kocheva2, Doctor of Chemistry, Head of the Laboratory, T.N. Shchemelinina1, Candidate of Biology, Research Assistant


1Institute of Biology, Komi Scientific Center, Ural Division, Russian Academy of Sciences, 2Institute of Geology, Komi Scientific Center, Ural Division, Russian Academy of Sciences ::: 1 Kommunisticheskaya str., 28, Syktyvkar, 167982, Russia; е-mail: 2, Pervomaiskaya str., 54, Syktyvkar, 167982, Russia; е-mail: tatyanakomi@


time-series, correlation dimension, Hurst exponent.


1. Bezruchko B.P., Smirnov D.A. Matematicheskoe modelirovanie i haoticheskie vremennye rjady [Mathematical Modeling and Chaotic Time Evolutions]. Saratov, 2005. 320 p. 2. Malinetsky G.G., Potapov A.B. Sovremennye problemy nelinejnoj dinamiki [Modern Problems of Non-Linear Dynamics]. Moscow, 2000. 336 p. 3. Mun F. Chaotic Variations. Moscow, 1991. 312 p. 4. Rjuel D., Takens F. Strange Attractors. Moscow, 1991. pp. 117-151. 5. Sekushin N.A., Karmanov A.P. Razrabotka novyh podhodov k opisaniju kinetiki polihronnogo tipa [Elaboration of New Approach for Description of Polychronic Type Kinetics]. Lesnoy zhurnal, 2006, no. 4, pp. 120-131.

Application of Non-Linear Dynamics Methods for Analysis of Results of Industrial Wastewater Monitoring


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