Automation and Remote Control

, Volume 78, Issue 1, pp 16–28 | Cite as

Control over chaotic price dynamics in a price competition model

  • E. V. OrlovaEmail author
Nonlinear Systems


The dynamic model of price competition in which processes of strategic interaction between companies on an imperfect competition market are described with the game-theoretic approach and methods of nonlinear dynamics. The pricing dynamics for the companies is modeled with difference equations (mappings). We study the stability of the fixed point of the price mapping. Results of our numerical modeling have shown the existence of periodic and chaotic solutions in the price competition model. We present intra-company adaptation mechanisms based on changing the prices in a way proportional to the rate of change in the companies’ profits; this lets us reduce the prices to a local Nash equilibrium and stabilize the chaotic dynamics of the market.

Key words

nonlinear economic dynamics model of price competition periodic and chaotic dynamic modes nonlinear market dynamics control over pricing bifurcation analysis Lyapunov stability strategic interaction between companies Nash equilibrium 


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  1. 1.
    Orlova, E.V., Efficient Pricing Mechanism for the Product of Industrial Companies, Ekonom. Predprinimatel’stvo, 2013, no. 12–1, pp. 622–626.Google Scholar
  2. 2.
    Kaneman, D., Rational Choice, Values, and Frames, Psikhologich. Zh., 2003, vol. 24, no. 4, pp. 31–42.Google Scholar
  3. 3.
    Orlova, E.V., Modeling the Utility Function with Irrational Factors, Nauch.-Tekhn. Vedomosti S.-Peterburg. Gos. Politekhn. Univ., Ekonom. Nauki, 2012, no. 3, pp. 24–30.Google Scholar
  4. 4.
    Orlova, E., Economical Behavior: A Synthesis of Rational and Irrational, Probl. Teorii Praktiki Upravlen., 2014, no. 3, pp. 127–136.Google Scholar
  5. 5.
    Petrosyan, L.A., Zenkevich, N.A., and Shevkoplyas, E.V., Teoriya igr (Game Theory), St. Petersburg: BKhV-Peterburg, 2012.Google Scholar
  6. 6.
    Zenkevich, N.A., Petrosyan, L.A., and Yang, D.V.K., Dinamicheskie igry i ikh prilozheniya v menedzhmente (Dynamical Games and Their Applications in Management), St. Petersburg: Vysshaya Shkola Menedzhmenta, 2009.Google Scholar
  7. 7.
    Filatov, A.Yu. and Aizenberg, N.I., Matematicheskie modeli nesovershennoi konkurentsii (Mathematical Models of Imperfect Competition), Irkutsk: IGU, 2012.Google Scholar
  8. 8.
    Axelrod, R., The Evolution of Cooperation, New York: Basic Book, 1984.zbMATHGoogle Scholar
  9. 9.
    Vuros, A. and Rozanova, N., Ekonomika otraslevykh rynkov (Economics of Industrial Markets), Moscow: TEIS, 2002.Google Scholar
  10. 10.
    Vasin, A.A. and Morozov, V.V., Teoriya igr i modeli matematicheskoi ekonomiki (Game Theory and Models of Mathematical Economics), Moscow: MAKS Press, 2005.Google Scholar
  11. 11.
    Puu, T., Nonlinear Economic Dynamics, New York: Springer, 1997.CrossRefzbMATHGoogle Scholar
  12. 12.
    Lorenz, H.-W., Nonlinear Dynamical Economics and Chaotic Motion, Berlin: Springer-Verlag, 1989.CrossRefzbMATHGoogle Scholar
  13. 13.
    Kuznetsov, S.P., Dinamicheskii khaos (Dynamical Chaos), Moscow: Fizmatlit, 2006.Google Scholar
  14. 14.
    Loskutov, A.Yu. and Mikhailov, A.S., Osnovy teorii slozhnykh sistem (Fundamentals of the Theory of Complex Systems), Moscow–Izhevsk: Inst. Komp’yut. Issled., 2007.Google Scholar
  15. 15.
    Neimark, Yu.I., Dinamicheskie sistemy i upravlyaemye protsessy (Dynamical Systems and Controllable Processes), Moscow: Librokom, 2010.Google Scholar
  16. 16.
    Kuznetsov, A.P., Savin, A.V., Sedova, Yu.V., et al., Bifurkatsii otobrazhenii (Bifurcations of Mappings), Saratov: Izd. Tsentr “Nauka,” 2012.Google Scholar
  17. 17.
    Ostrovskii, A.V., On One Class of Competitive Pricing Models in a Market Economy, Differ. Uravn. Prots. Upravlen., 2000, no. 2, pp. 58–77, URL: (accessed at 06.03.2015).Google Scholar
  18. 18.
    Loskutov, A.Yu., Nonlinear Optimization of the Market’s Chaotic Dynamics, Ekonom. Mat. Metod., 2010, vol. 46, no. 3, pp. 58–70.Google Scholar
  19. 19.
    Holyst, J.A. and Urbanowicz, K., Chaos Control in Economical Model by Time-Delayed Feedback Method, Physica A, 2000, vol. 287, pp. 587–598.CrossRefGoogle Scholar
  20. 20.
    Kopel, M., Improving the Performance of an Economic System: Controlling Chaos, J. Evolutionary Econ., 1997, vol. 7, pp. 269–289.CrossRefGoogle Scholar
  21. 21.
    Ahmed, E. and Hassan, S.Z., On Controlling Chaos in Sournot-Games with Two and Three Competitors, Nonlin. Dynamics, Psychology, Life Sci., 2000, vol. 4, no. 2, pp. 189–194.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ahmed, E., Elsadany, A.A., and Puu, T. On Bertrand Duopoly Game with Differentiated Goods, Appl. Math. Comput., 2015, vol. 251, pp. 169–179.MathSciNetzbMATHGoogle Scholar
  23. 23.
    Feichtinger, G., Economic Evolution and Demographic Change, Berlin: Springer, 1992.Google Scholar
  24. 24.
    Agliari, A., Gardini, L., and Puu, T., Global Bifurcations in Duopoly when the Cournot Point is Destabilized through a Subcritical Neimark Bifurcation, Working paper, 2003, URL: digitalAssets/18/18883 cwp 66 03.pdf (accessed at 04.03.2015).Google Scholar
  25. 25.
    Stachurski, D., Economic Dynamics: Theory and Computation, Cambridge: MIT Press, 2009.zbMATHGoogle Scholar
  26. 26.
    Puu, T., Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in Economics, New York: Springer, 2003, 2nd ed.CrossRefzbMATHGoogle Scholar
  27. 27.
    Neimark, Yu.I. and Ostrovskii, A.V., On Certain Pricing Models in a Market Economy, Izv. Yyssh. Uchebn. Zaved., Prikl. Nelin. Dinam., 1999, no. 6, pp. 35–41.Google Scholar
  28. 28.
    Ferguson, B.S. and Lim, G.C., Dynamic Economic Models in Discrete Time: Theory and Empirical Applications, New York: Taylor & Francis e-Library, 2005.Google Scholar
  29. 29.
    Schuster, G., Deterministic Chaos: An Introduction, Weinheim: Physik-Verlag, 1984. Translated under the title Determinirovannyi khaos. Vvedenie, Moscow: Mir, 1988.zbMATHGoogle Scholar
  30. 30.
    Farmer, J.D., Ott, E., and Yorke, J.A., The Dimension of Chaotic Attractors, Physica 7D, 1983, pp. 153–180.Google Scholar
  31. 31.
    Moon, F., Chaotic Vibrations: An Introduction for Applied Scientists and Engineers, New York: Wiley, 1987. Translated under the title Khaoticheskie kolebaniya. Vvodnyi kurs dlya nauchnykh rabotnikov i inzhenerov, Moscow: Mir, 1990.zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Ufa State Aviation Technical UniversityUfaRussia

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