ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


For citation:

Shabunin A. V. A neural network as a predictor of the discrete map. Izvestiya VUZ. Applied Nonlinear Dynamics, 2014, vol. 22, iss. 5, pp. 58-72. DOI: 10.18500/0869-6632-2014-22-5-58-72

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 397)
Language: 
Russian
Article type: 
Article
UDC: 
517.9, 621.372

A neural network as a predictor of the discrete map

Autors: 
Shabunin Aleksej Vladimirovich, Saratov State University
Abstract: 

The possibility of predicting the regular and chaotic dynamics of a discrete map by using artificial neural network is studied. The method of error back­propagation is used for calculation the coefficients of the multilayer network. The predicting properties of the neural network are explored in a wide region of the system parameter for both regular and chaotic behaviors. The dependance of the prediction accuracy from the degree of chaos and from the number of layers of the network is studied.

Reference: 
  1. Packard NH, Crutchield JP, Farmer JD, Shaw RS. Geometry from a time series. Phys. Rev. Lett. 1980;45(9):712—716. DOI: 10.1103/PhysRevLett.45.712.
  2. Takens F. Detecting strange attractors in turbulence. In: Rand D, Young LS. editors. Dynamical Systems and Turbulence. Vol. 898 of Lect. Notes in Math. Berlin: Springer; 1980. P. 366—381. DOI: 10.1007/BFb0091924.
  3. Farmer JD, Sidorowich JJ. Predicting chaotic time series. Phys. Rev. Lett. 1987;59(8):845—848. DOI: 10.1103/PhysRevLett.59.845.
  4. Casdagli M. Nonlinear prediction of chaotic time series. Physica D. 1989;35(3):335—356. DOI: 10.1016/0167-2789(89)90074-2.
  5. Abarbanel HDI, Brown R, Kadtke JB. Prediction in chaotic nonlinear systems: Methods for time series with broadband Fourier spectra. Phys. Rev. A. 1990;41(4):1782—1807. DOI: 10.1103/PhysRevA.41.1782.
  6. Grassberger P, Procaccia I. Characterization of strange attractors. Phys. Rev. Lett. 1983;50(5):346—349. DOI: 10.1103/PhysRevLett.50.346.
  7. Cremers J, Hubler A. Construction of differential equations from experimental data. Z. Naturforschung A. 1987;42(8):797—802. DOI: 10.1515/zna-1987-0805.
  8. Crutchfield JP, McNamara BS. Equations of motion from a data series. Complex Systems. 1987;1(3):417—452.
  9. Pisarenko VF, Sornette D. Statistical methods of parameter estimation for deterministically chaotic time series. Phys. Rev. E. 2004;69(3):036122. DOI: 10.1103/PhysRevE.69.036122.
  10. Smirnov DA, Vlaskin VS, Ponomarenko VI. Estimation of parameters in one dimentional maps from noisy chaotic time series. Phys. Lett. A. 2005;336(6):448—458. DOI: 10.1016/j.physleta.2004.12.092.
  11. Mukhin DN, Feigin AM, Loskutov EM, Molkov YI. Modified Bayesian approach for the reconstruction of dynamical systems from time series. Phys. Rev. E. 2006;73(3):036211. DOI: 10.1103/PhysRevE.73.036211.
  12. Rosenblum MG, Pikovsky AS. Detecting direction of coupling in interacting oscillators. Phys. Rev. E. 2001;64(4):045202. DOI: 10.1103/PhysRevE.64.045202.
  13. Smirnov D, Bezruchko B. Estimation of interaction strength and direction from short and noisy time series. Phys. Rev. E. 2003;68(4):046209. DOI: 10.1103/PhysRevE.68.046209.
  14. Smirnov D, Bezruchko B. Detecting of couplings in ensembles of stochastic oscillators. Phys. Rev. E. 2009;79(4):046204. DOI: 10.1103/PhysRevE.79.046204.
  15. Pavlov AN, Janson NB. Application of the method of reconstruction of the mathematical model to the cardiogram. Izvestiya VUZ. Applied Nonlinear Dynamics. 1997;5(1):93 (in Russian).
  16. Bezruchko BP, Smirnov DA, Zborovsky AV, Sidak AV, Ivanov RN, Bespyatov AB. Reconstruction from time series and problems of diagnostics. Technologies of Living Systems. 2007;4(3):49—56 (in Russian).
  17. Anishchenko VS, Pavlov AN. Global reconstruction in application to multichannel communication. Phys. Rev. E. 1998;57(2):2455—2457. DOI: 10.1103/PhysRevE.57.2455.
  18. Prokhorov MD, Ponomarenko VI, Karavaev AS, Bezruchko BP. Recovery of dynamical models of time-delay systems from time series: Application to chaotic communication. In: Wang CW, editor. Nonlinear Phenomena Research Perspectives. New York: Nova Science Publishers; 2007. P. 7—55.
  19. Kadtke J. Classification of highly noisy signals using global dynamical models. Phys. Lett. A. 1995;203(4):196—202. DOI: 10.1016/0375-9601(95)00375-D.
  20. Mokhov II, Smirnov DA. Diagnostics of a cause-effect relation between solar activity and the Earth’s global surface temperature. Izv. Atmos. Ocean. Phys. 2008;44(3):263—272. DOI: 10.1134/S0001433808030018.
  21. Mokhov II, Smirnov DA. Empirical estimates of the influence of natural and anthropogenic factors on the global surface temperature. Doklady Earth Sciences. 2009;427(1):798—803. DOI: 10.1134/S1028334X09050201.
  22. Markel JD, Gray AH. Linear Prediction of Speech. Berlin: Springer; 1976. 290 p. DOI: 10.1007/978-3-642-66286-7.
  23. Djigan VI. Adaptive filters and their applications in radio engineering and communications. Part 1. Modern Electronics. 2009;(9):56—63 (in Russian).
  24. Djigan VI. Adaptive Signal Filtering: Theory and Algorithms. Moscow: Tehnosphera; 2013. 528 p. (in Russian).
  25. Rosenblatt F. Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms. Washington DC: Spartan Books; 1962. 616 p.
  26. Minsky M, Papert S. Perceptrons: An Introduction to Computational Geometry. Cambridge: MIT Press; 1969. 258 p.
  27. Callan R. The Essence of Neural Networks. Prentice Hall Europe; 1999. 232 p.
  28. Haykin S. Neural Networks and Learning Machines. Prentice Hall Europe; 1999. 938 p.
  29. Galushkin AI. Neural Networks. Fundamentals of Theory. Moscow: «Telekom»; 2012. 496 p. (in Russian).
  30. Savchenko AV. Image recognition on the basis of probabilistic neural network with homogeneity testing. Computer Optics. 2013;37(2):254—261 (in Russian).
  31. Dumskij DV, Pavlov AN, Tupicyn AN, Makarov VV. Classification of neuronal action potentials using wavelet-transform. Izvestiya VUZ. Applied Nonlinear Dynamics. 2005;13(6):77—98 (in Russian). DOI: 10.18500/0869-6632-2005-13-5-77-98.
  32. Tupitsyn AN, Nazlmov AI, Pavlov AN. Identification of Action Potentials of Small Neuron Ensembles Using Wavelet-Analysis and Neural Networks Method. Izvestiya of Sarat. Univ. Physics. 2009;9(2):57—65 (in Russian). DOI: 10.18500/1817-3020-2009-9-2-57-65.
  33. Uma Maheswari N, Kabilan AP, Venkatesh R. Speaker independent speech recognition system using neural networks. Journal of Radio Electronics. 2009;(7):1.
  34. Kuzovnikov AV. Classification of radio signals in neural networks. Journal Neurocomputers. 2012;(8):52—56 (in Russian).
  35. Tereshonok MV. Classification and recognition of signals from radio communication systems using self-organizing Kohonen maps with various output layer topologies and learning algorithms. T-Comm. 2008;(6):28—31 (in Russian).
  36. Govorova YS, Soldatov YA. Application of neural networks to the solution of the phase problem. Vestnik of Lobachevsky University of Nizhni Novgorod. 2004;(1):185—190 (in Russian).
  37. Kondratenko YP, Korobko AV, Sviridov AI. Interference filtering and analysis of filter characteristics based on adaptive algorithms and neural network ADALINE. Herald of the National Technical University «KhPI». Series «Informatics and Modeling». 2012;(38):102—115 (in Russian).
  38. Smirnov AA, Smirnov AS, Kostornova SV, Streker EN. Development and study of a parallel digital matched filter with a neural network. Design and Technology of Electronic Means. 2008;(3):18—21 (in Russian).
  39. Xianjun N. Research of data mining based on neural networks. World Academy of Science, Engineering and Technology. 2008;15(39):381—384.
  40. Manzhula VG, Fedyashev VS. Kohonen neural networks and fuzzy neural networks in data mining. Fundamental Research. 2011;(4):108—115 (in Russian).
  41. Kulkarni DR, Parikh JC, Pandya AS. Dynamic predictions from time series data – an artificial neural network approach. International Journal of Modern Physics C. 1997;8(6):1345—1360. DOI: 10.1142/S0129183197001193.
  42. de Oliveira KA, Vannucci A, da Silva EC. Using artificial neural networks to forecast chaotic time series. Physica A. 2000;284(1—4):393—404.
  43. Antipov OI, Neganov VA. Neural network prediction and fractal analysis of the chaotic processes in discrete nonlinear systems. Doklady Physics. 2011;56(1):7—9. DOI: 10.1134/S1028335811010034.
  44. Rumelhart DE, Hinton GE, Williams RJ. Learning representations of back-propagation errors. Nature. 1986;323:533—536. DOI: 10.1038/323533a0.
Received: 
21.11.2014
Accepted: 
19.12.2014
Published: 
31.03.2015
Short text (in English):
(downloads: 343)