Nonlinear Dynamics

, Volume 73, Issue 1–2, pp 499–508 | Cite as

Hyperchaos control of the hyperchaotic Chen system by optimal control design

  • S. Effati
  • H. Saberi Nik
  • A. Jajarmi
Original Paper


The aim of this paper is to study the chaos, optimal control, and adaptive control of the hyperchaotic Chen system. In this paper, applying the Pontryagin’s minimum principle (PMP), the optimal control inputs for the interested model are obtained with respect to the selected measure. A piecewise-spectral homotopy analysis method (PSHAM) is used for solving the hyperchaotic Chen system and the extreme conditions obtained from the PMP. Furthermore, an adaptive control approach and a parameter estimation update law are introduced for the hyperchaotic Chen system with completely unknown parameters. The control results are established using the Krasovskii–LaSalle principle. Finally, numerical simulations are included to demonstrate the effectiveness of the proposed control strategy.


Optimal control Hyperchaotic Chen system Lyapunov function Pontryagin’s minimum principle Piecewise-spectral homotopy analysis method 



This research was supported by a grant from Ferdowsi University of Mashhad (No. MA91288SEF).


  1. 1.
    Lakshmanan, M., Murali, K.: Nonlinear Oscillators: Controlling and Synchronization. World Scientific, Singapore (1996) zbMATHGoogle Scholar
  2. 2.
    Han, S.K., Kerrer, C., Kuramoto, Y.: Dephasing and bursting in coupled neural oscillators. Phys. Rev. Lett. 75, 3190–3193 (1995) CrossRefGoogle Scholar
  3. 3.
    Blasius, B., Huppert, A., Stone, L.: Complex dynamics and phase synchronization in spatially extended ecological system. Nature 399, 354–359 (1999) CrossRefGoogle Scholar
  4. 4.
    Cuomo, K.M., Oppenheim, A.V.: Circuit implementation of synchronized chaos with applications to communications. Phys. Rev. Lett. 71, 65–68 (1993) CrossRefGoogle Scholar
  5. 5.
    Rossler, O.E.: An equation for hyperchaos. Phys. Lett. A 71, 155–157 (1979) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Jia, Q.: Hyperchaos generated from the Lorenz chaotic system and its control. Phys. Lett. A 366, 217–222 (2007) zbMATHCrossRefGoogle Scholar
  7. 7.
    Kapitaniak, T., Chua, L.O.: Hyperchaotic attractor of unidirectionally coupled Chua’s circuit. Int. J. Bifurc. Chaos Appl. Sci. Eng. 4, 477–482 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chen, A., Lu, J., Lu, J., Yu, S.: Generating hyperchaotic Lu attractor via state feedback control. Physica A 364, 103–110 (2006) CrossRefGoogle Scholar
  9. 9.
    Lu, J., Chen, G.: A new chaotic attractor coined. Int. J. Bifurc. Chaos Appl. Sci. Eng. 12, 659–661 (2002) CrossRefGoogle Scholar
  10. 10.
    Dadras, S., Momeni, H.R., Qi, G.: Analysis of a new 3D smooth autonomous system with different wing chaotic attractors and transient chaos. Nonlinear Dyn. 62, 391–405 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dadras, S., Momeni, H.R., Qi, G., Wang, Z.l.: Four-wing hyperchaotic attractor generated from a new 4D system with one equilibrium and its fractional-order form. Nonlinear Dyn. 67, 1161–1173 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Huber, A.W.: Adaptive control of chaotic system. Helv. Chim. Acta 62, 343–349 (1989) Google Scholar
  13. 13.
    Aghababa, M.P., Khanmohammadi, S., Alizadeh, G.: Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique. Appl. Math. Model. 35, 3080–3091 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Chen, G.R., Yu, X.H.: On time-delayed feedback control of chaotic systems. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 46, 767–772 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Li, N., Yuan, H., Sun, H., Zhang, Q.: An impulsive multi-delayed feedback control method for stabilizing discrete chaotic systems. Nonlinear Dyn. (2012). doi: 10.1007/s11071-012-0434-y Google Scholar
  16. 16.
    Sun, K., Liu, X., Zhu, C., Sprott, J.C.: Hyperchaos and hyperchaos control of the sinusoidally forced simplified Lorenz system. Nonlinear Dyn. 69, 1383–1391 (2012) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Yu, W.G.: Stabilization of three-dimensional chaotic systems via single state feedback controller. Phys. Lett. A 374, 1488–1492 (2010) CrossRefGoogle Scholar
  18. 18.
    Zhao, Y., Zhang, T.Y., Miao, L.H., Deng, W.: Adaptive control for a class of chaotic systems with unknown parameters. In: Control and Decision Conference, pp. 26–28 (2010) Google Scholar
  19. 19.
    Zhang, X., Zhu, H., Yao, H.: Analysis of a new three-dimensional chaotic system. Nonlinear Dyn. 67, 335–343 (2012) zbMATHCrossRefGoogle Scholar
  20. 20.
    Zhang, J., Tang, W.: Control and synchronization for a class of new chaotic systems via linear feedback. Nonlinear Dyn. 58, 675–686 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Sarkar, R., Banerjee, S.: Cancer and self remission and tumor stability, a stochastic approach. Math. Biosci. 169, 65–81 (2005) MathSciNetCrossRefGoogle Scholar
  22. 22.
    El-Gohary, A.: Chaos and optimal control of cancer self-remission and tumor system steady states. Chaos Solitons Fractals 37, 1305–1316 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    El-Gohary, A., Alwasel, I.A.: The chaos and optimal control of cancer model with complete unknown parameters. Chaos Solitons Fractals 42, 2865–2874 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Liao, S.J.: The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. thesis, Shanghai Jiao Tong University (1992) Google Scholar
  25. 25.
    Abbasbandy, S., Zakaria, F.S.: Soliton solutions for the fifth-order KdV equation with the homotopy analysis method. Nonlinear Dyn. 51, 83–87 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Abbasbandy, S.: Solitary wave solutions to the Kuramoto–Sivashinsky equation by means of the homotopy analysis method. Nonlinear Dyn. 52, 35–40 (2008) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Hayat, T., Javed, T., Sajid, M.: Analytic solution for rotating flow and heat transfer analysis of a third-grade fluid. Acta Mech. 191, 219–229 (2007) zbMATHCrossRefGoogle Scholar
  28. 28.
    Herisanu, N., Marinca, V.: Explicit analytical approximation to large-amplitude nonlinear oscillations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia. Meccanica 45, 847–855 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Herisanu, N., Marinca, V.: Optimal homotopy perturbation method for a non-conservative dynamical system of a rotating electrical machine. Z. Naturforsch. 67, 509–516 (2012) Google Scholar
  30. 30.
    Motsa, S.S., Sibanda, P., Shateyi, S.: A new spectral-homotopy analysis method for solving a nonlinear second order BVP. Commun. Nonlinear Sci. Numer. Simul. 15, 2293–2302 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Motsa, S.S., Sibanda, P., Awad, F.G., Shateyi, S.: A new spectral-homotopy analysis method for the MHD Jeffery–Hamel problem. Comput. Fluids 39, 1219–1225 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Liao, S.J.: Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman & Hall/CRC Press, London/Boca Raton (2003) CrossRefGoogle Scholar
  33. 33.
    Li, Y., Tang, S.W., Chen, G.: Generating hyperchaos via state feedback control. Int. J. Bifurc. Chaos Appl. Sci. Eng. 5, 3367–3375 (2005) CrossRefGoogle Scholar
  34. 34.
    Zhenya, Y.: Controlling hyperchaos in the new hyperchaotic Chen system. Appl. Math. Comput. 68, 1239–1250 (2005) Google Scholar
  35. 35.
    Khalil, H.K.: Nonlinear Systems, 3rd. edn. Prentice Hall, New York (2002) zbMATHGoogle Scholar
  36. 36.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988) zbMATHCrossRefGoogle Scholar
  37. 37.
    Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia (2000) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Applied Mathematics, School of Mathematical SciencesFerdowsi University of MashhadMashhadIran
  2. 2.Center of Excellence on Soft Computing and Intelligent Information Processing (SCIIP)Ferdowsi University of MashhadMashhadIran
  3. 3.Department of Electrical EngineeringUniversity of BojnordBojnordIran

Personalised recommendations