A Note about Robust Stabilization of Chaotic Hénon System Using Grammatical Evolution

  • Radomil Matousek
  • Ladislav Dobrovsky
  • Petr Minar
  • Katerina Mouralova
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 289)

Abstract

The paper deals with robust stabilization of a well-known deterministic discrete chaotic system denoted as Hénon map. By means of proper utilization of metaheuristic optimization tool, the Grammatical Evolution (GE) can synthesise a new robust control law. As a model of deterministic chaotic system the two-dimensional Hénon map with original definition was used. The Hénon map is an iterated discrete-time system which exhibits chaotic behaviour in two-dimension. Stabilization for the period-2 orbits of the two-dimensional Hénon map is presented. The chaotic system stabilization is based on a time-delay auto-synchronization with its own synthesized control law. This synthesized chaotic controller utilizes own design of advanced GE algorithm with two-level optimization procedures and a proper objective function. The original objective function design considers a low sensitivity dependence on initial conditions and also proper time for stabilisation of the control process. All computing experiments are performed using Matlab/Simulink environment where the double precision floating point arithmetic was used.

Keywords

Hénon map Chaos Control Robust stabilization Metaheuristic optimization Grammatical evolution 

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References

  1. 1.
    Hénon, M.: A two dimensional mapping with a strange attractor. Commun. Math. Phys. 50, 69–77 (1976)CrossRefMATHGoogle Scholar
  2. 2.
    Benedicks, M., Carleson, L.: The dynamics of the Hénon maps. Ann. Math. 133, 1–25 (1991)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Lozi, R.: Un attracteur étrange du type attracteur de Hé non. Journal de Physique. Colloque C5, Supplément au 39(8), 9–10 (1978)Google Scholar
  4. 4.
    May, R.M.: Simple mathematical models with very complicated dynamics. Nature 261(5560), 459–467 (1976), doi:10.1038/s261459a0CrossRefGoogle Scholar
  5. 5.
    Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaotic dynamical systems. In: Campbell, D.K. (ed.) Chaos, Amer. Inst. of Phys., New York, pp. 153–172 (1990)Google Scholar
  7. 7.
    Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Lett. A. 170, 421–428 (1992)CrossRefGoogle Scholar
  8. 8.
    Pyragas, K.: Control of chaos via extended delay feedback. Phys. Lett. A 206, 323–330 (1995)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Richter, H., Reinschke, K.J.: Optimization of local control of chaos by an evolutionary algorithm. Physica D 144, 309–334 (2000)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Richter, H., Reinschke, K.J.: Local control of chaotic systems: a Lyapunov approach. Int. J. Bifurc. Chaos 8, 1565–1573 (1998)CrossRefMATHGoogle Scholar
  11. 11.
    Chen, G.: Control and Synchronization of Chaotic Systems (bibliography). ECE Dept, Univ of Houston, http://www.ee.cityu.edu.hk/~gchen/chaos-bio.html (cited: March 13, 2013)
  12. 12.
    Fradkov, A.L.: Chaos Control Bibliography, Russian Systems and Control Archive, RUSYCON (1997-2000), http://www.rusycon.ru/chaos-control.html (cited: March 13, 2013)
  13. 13.
    Ramaswamy, R., Sinha, R., Gupte, N.: Targeting chaos through adaptive control. Phys. Rev. Lett. 57(3), 2507–2510 (1998)Google Scholar
  14. 14.
    Starrett, J.: Time-optimal chaos control by center manifold targeting. Phys. Rev. Lett. 66(4), 6206–6211 (2002)Google Scholar
  15. 15.
    Bollt, E.J., Kostelich, E.J.: Optimal targeting of chaos. Physics Letters A 245, 399–406 (1998)CrossRefGoogle Scholar
  16. 16.
    Iplikci, S., Denizhan, Y.: An improved neural network based targeting method for chaotic dynamics. Chaos, Solutions & Fractals 17(2-3), 523–529 (2003)CrossRefMATHGoogle Scholar
  17. 17.
    Wang, T., Wang, X., Wang, M.: Chaotic control of Hénon map with feedback and non-feedback methods. Commun. Nonlinear Sci. Numer. Simulat. 16, 3367–3374 (2011)CrossRefMATHGoogle Scholar
  18. 18.
    Zelinka, I.: SOMA–self organizing migrating algorithm. In: Babu, B.V., Onwubolu, G. (eds.) New Optimization Techniques in Engineering. ch. 7, vol. 33. Springer-Verlag (2004)Google Scholar
  19. 19.
    Zelinka, I., Guanrong, C., Celikovsky, S.: Chaos synthesis by means of evolutionary algorithms. International Journal of Bifurcation and Chaos 18(4), 911–942 (2008)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Zelinka, I., Senkerik, R., Navratil, E.: Investigation on evolutionary optimitazion of chaos control. Chaos, Solitons & Fractals 40, 111–129 (2009)CrossRefMATHGoogle Scholar
  21. 21.
    Lampinen, J., Zelinka, I.: Mechanical engineering design optimization by differential evolution. In: Corne, D., Dorigo, M., Glover, F. (eds.) New Ideas in Optimization, pp. 127–146. McGraw-Hill (1999); 007-709506-5Google Scholar
  22. 22.
    Senkerik, R., Zelinka, I., Oplatkova, Z.: Optimal control of evolutionary synthesized chaotic system. In: Matousek, R. (ed.) 15th International Conference on Soft Computing MENDEL 2009, pp. 220–227 (2009) ISSN:1803-3814, ISBN: 978-80-214-3884-2Google Scholar
  23. 23.
    Senkerik, R., Zelinka, I., Davendra, D., Oplatkova, Z.: Utilization of SOMA and differential evolution for robust stabilization of chaotic Logistic equation. Computers & Mathematics with Applications 60(4), 1026–1037 (2010)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Senkerik, R., Oplatkova, Z., Zelinka, I., Davendra, D.: Evolutionary chaos controller synthesis for stabilizing chaotic Henon maps. Complex Systems 20(3), 205–214 (2012); 0891-2513Google Scholar
  25. 25.
    Senkerik, R., Oplatkova, Z., Zelinka, I., Davendra, D.: Synthesis of feedback controller for three selected chaotic systems by means of evolutionary techniques: analytic programming. Mathematical and Computer Modelling 57(1-2), 57–67 (2013); 0895-7177Google Scholar
  26. 26.
    Kominkova-Oplatkova, Z., Senkerik, R., Zelinka, I., Pluhacek, M.: Analytic programming in the task of evolutionary synthesis of a controller for high order oscillations stabilization of discrete chaotic systems. Computers & Mathematics with Applications (2013) (available online March 5, 2013)Google Scholar
  27. 27.
    Matousek, R.: GAHC: Improved GA with HC mutation. In: WCECS 2007, San Francisco, pp. 915–920 (2007)Google Scholar
  28. 28.
    Matousek, R.: GAHC: Hybrid Genetic Algorithm. In: Advances in Computational Algorithms and Data Analysis. LNEE, vol. 14, pp. 549–562 (2009)Google Scholar
  29. 29.
    Matousek, R.: HC12: The Principle of CUDA Implementation. In: 16th International Conference on Soft Computing, MENDEL 2010, Brno, pp. 303–308 (2010)Google Scholar
  30. 30.
    Matousek, R., Zampachova, E.: Promising GAHC and HC12 algorithms in global optimization tasks. Journal Optimization Methods & Software 26(3), 405–419 (2011)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Matousek, R., Minar, P.: Stabilization of chaotic logistic equation using HC12 and grammatical evolution. In: Zelinka, I., Chen, G., Rössler, O.E., Snasel, V., Abraham, A. (eds.) Nostradamus 2013: Prediction, Model. & Analysis. AISC, vol. 210, pp. 137–146. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  32. 32.
    Goldberg, D.E.: Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley Longman Publishing Co. Inc., Boston (1989)Google Scholar
  33. 33.
    O’Neill, M., Ryan, C.: Grammatical Evolution: Evolutionary Automatic Programming in an Arbitrary Language. Kluwer Academic Publishers (2003)Google Scholar
  34. 34.
    Koza, J.R.: Genetic Programming: On the Programming of Computers by Means of Natural Selection. The MIT Press (1992)Google Scholar
  35. 35.
    Doerner, R., Hübinger, B., Martienssen, W.: Adaptive orbit correction in chaos control. Int. J. of Bifurcation and Chaos 5, 1175–1179 (1995)CrossRefMATHGoogle Scholar
  36. 36.
    Al-shameri, W.F.H.: Dynamical Properties of the Hénon Mapping. Int. Journal of Math. Analysis 6(49), 2419–2430 (2012)MATHMathSciNetGoogle Scholar
  37. 37.
    Sprott, J.C.: High-Dimensional Dynamics in the Delayed Hénon Map. EJTP 3(12), 19–35 (2006)MATHGoogle Scholar
  38. 38.
    Starrett, J.: Time-optimal chaos control by center manifold targeting. Physical Review E 66, 046206 (2002)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Radomil Matousek
    • 1
  • Ladislav Dobrovsky
    • 1
  • Petr Minar
    • 1
  • Katerina Mouralova
    • 1
  1. 1.Department of Applied Computer ScienceBrno University of TechnologyBrnoCzech Republic

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