Skip to main content
SpringerLink
Log in

Global Chaos Synchronization of n-Scroll Chua Circuit and Lur’e System using Backstepping Control Design with Recursive Feedback

  • Research Article - Systems Engineering
  • Published:
Arabian Journal for Science and Engineering Aims and scope Submit manuscript

Abstract

In this paper, global chaos synchronization is investigated for n-scroll Chua (Tang et al. in IEEE Trans Circ Syst I Fundam Theory Appl 48:1369–1372, 2001) and Lur’e (Suykens and Vandewalle in Int J Bifurc Chaos 7:1323–1325, 1997) chaotic systems using backstepping control with recursive feedback. Our theorems on synchronization for n-scroll Chua and Lur’e chaotic systems are established using Lyapunov stability theory. The backstepping scheme is a recursive procedure that links the choice of Lyapunov function with the design of a feedback controller and guarantees global stability performance of strict-feedback chaotic systems. Mainly the backstepping technique gives flexibility in designing a feedback control law. Numerical simulations are also given to illustrate and validate the synchronization results derived in this paper.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Fujisaka H, Yamada T: Stability theory of synchronized motion in coupled-oscillator systems. Prog. Theor. Phys. 69, 32–47 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  2. Pecora L.M, Carroll T.L: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  3. Pecora L.M, Carroll T.L: Synchronizing chaotic circuits. IEEE Trans. Circ. Syst. 38, 453–456 (1991)

    Article  Google Scholar 

  4. Alligood K.T, Sauer T, Yorke J.A: Chaos: An Introduction to Dynamical Systems. Springer, Berlin (1997)

    Book  Google Scholar 

  5. Ott, E.: Chaos in Dynamical Systems, Cambridge University Press, Cambridge (2002)

  6. Murali, K., Lakshmanan, M.: Secure communication using a compound signal using sampled-data feedback, Applied Mathematics and Mechanics,11, 1309–1315 (2003)

    Google Scholar 

  7. Yang T, Chua L.O: Generalized synchronization of chaos via linear transformations. Int. J. Bifurc. Chaos 9, 215–219 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  8. Lakshmanan, M.; Murali, K.: Chaos in Nonlinear Oscillators: Controlling and Synchronization. World Scientific, Singapore (1996)

  9. Khan, Y.; Askari, K.H.; Saadatnia, Z.: Dynamics analysis of generalized conservative nonlinear oscillators via frequency amplitude formulation. Arab. J. Sci. Eng. doi:10.007/s13369-011-0035-y

  10. Hayat T, Alsaedi A: On thermal radiation and joule heating effects in MHD flow of an oldroyd-B fluid with thermophoresis. Arab. J. Sci. Eng. 36, 1113–1124 (2011)

    Article  Google Scholar 

  11. Han S.K, Kerrer C, Kuramoto Y: D-phasing and bursting in coupled neural oscillators. Phys. Rev. Lett. 75, 3190–3193 (1995)

    Article  Google Scholar 

  12. Blasius B, Huppert A, Stone L: Complex dynamics and phase synchronization in spatially extended ecological system. Nature 399, 354–359 (1999)

    Article  Google Scholar 

  13. Yan J.: Global attractivity of almost periodic sequence solutions of delayed discrete-time neural network. Arab. J. Sci. Eng. 36, 1447–1459 (2011)

    Article  MathSciNet  Google Scholar 

  14. Kocarev L, Parlitz U: General approach for chaotic synchronization with applications to communications. Phys. Rev. Lett. 74, 5028–5030 (1995)

    Article  Google Scholar 

  15. Yazdi M, Yazdi M: A new 2-D fractal dimension estimation based on contourlet transform for texture segmentation. Arab. J. Sci. Eng. 35, 293–317 (2010)

    Google Scholar 

  16. Maher R.A, Mohammed I.A, Ibraheem I.K: State space based H ∞ robust controller design for boiler-turbine system. Arab. J. Sci. Eng. 36, 1447–1459 (2011)

    Article  MathSciNet  Google Scholar 

  17. Eker I.: Second order sliding mode control with PI sliding surface and experimental application to an electromechanical plant. Arab. J. Sci. Eng. 37, 1969–1986 (2012)

    Article  Google Scholar 

  18. Ott E, Grebogi C, Yorke J.A: Controlling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  19. Murali K, Lakshmanan M: Secure communication using a compound signal using sampled-data feedback. Appl. Math. Mech. 11, 1309–1315 (2003)

    Google Scholar 

  20. Lu J, Wu X, Han X, Lu J: Adaptive feedback synchronization of a unified chaotic system. Phys. Lett. A 329, 327–333 (2004)

    Article  MATH  Google Scholar 

  21. Park J.H, Lee S.M, Kwon O.M: Adaptive synchronization of Genesio–Tesi chaotic system via a novel feedback control. Phys. Lett. A 371, 263–270 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Park J.H.: Adaptive control for modified projective synchronization of a four-dimensional chaotic system with uncertain parameters. J. Comput. Appl. Math. 213, 288–293 (2008)

    Article  MATH  Google Scholar 

  23. Yau, H.T.: Design of adaptive sliding mode controller for chaos synchronization with uncertainties. Chaos Solitons Fractals 22 341–347 (2004)

    Google Scholar 

  24. Sundarapandian V.: Global chaos synchronization of the Pehlivan systems by sliding mode control. Int. J. Comput. Sci. Eng. 3, 2163–2169 (2004)

    Google Scholar 

  25. Sundarapandian V, Suresh R: Global chaos synchronization for Rossler and Arneodo chaotic systems by nonlinear control. Far East J. Appl. Math. 44, 137–148 (2010)

    MATH  MathSciNet  Google Scholar 

  26. Zhang X, Zhu H: Anti-synchronization of two different hyperchaotic systems via active and adaptive control. Int. J. Nonlinear Sci. 6, 216–223 (2008)

    MATH  MathSciNet  Google Scholar 

  27. Chiang T, Lin J, Liao T, Yan J: Anti-synchronization of uncertain unified chaotic systems with dead-zone nonlinearity. Nonlinear Anal. 68, 2629–2637 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  28. Sundarapandian V.: Anti-synchronization of Arneodo and Coullet systems by active nonlinear control. Int. J. Control Theory Appl. 4, 25–36 (2011)

    Google Scholar 

  29. Xu X, Chen Z, Si G, Hu X, Luo P: A novel definition of generalized synchronization on networks and a numerical simulation example. Comput. Math. Appl. 56, 2789–2794 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  30. Liu H, Chen J, Lu J, Cao M: Generalized synchronization in complex dynamical networks via adaptive couplings. Physica A 389, 1759–1770 (2010)

    Article  Google Scholar 

  31. Wu Y, Li C, Wu Y, Kurths J: Generalized synchronization between two different complex networks. Commun. Nonlinear Sci. Numer. Simul. 17, 349–355 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  32. Hu M, Yang Y, Xu Z, Zhang R, Guo L: Projective synchronization in drive-response dynamical networks. Physica A 381, 457–466 (2007)

    Article  Google Scholar 

  33. Hu M, Yang Y, Xu Z: Impulsive control of projective synchronization in chaotic systems. Phys. Lett. A 372, 3228–3233 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  34. Cao J, Ho D.W.C, Yang Y: Projective synchronization of a class of delayed chaotic systems via impulsive control. Phys. Lett. A 373, 3128–3133 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  35. Li W, Chen X: Projective synchronization of Chua’s chaotic system with dead-zone in the control input. Commun. Nonlinear Sci. Numer. Simul. 14, 3100–3107 (2009)

    Article  Google Scholar 

  36. Li R.H, Xu W, Li S: Adaptive generalized projective synchronization in different chaotic systems based on parameter identification. Phys. Lett. A 367, 199–206 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  37. Yau H.T.: Generalized projective chaos synchronization of gyroscope systems subjected to dead-zone nonlinear inputs. Phys. Lett. A 372, 2380–2385 (2008)

    Article  MATH  Google Scholar 

  38. Sarasu P, Sundarapandian V: Active controller design for generalized projective synchronization of four-scroll chaotic systems. Int. J. Syst. Signal Control Eng. Appl. 4, 26–33 (2011)

    Google Scholar 

  39. Sarasu P, Sundarapandian V: The generalized projective synchronization of hyperchaotic Lorenz and hyperchaotic Qi systems via active control. Int. J. Soft Comput. 6, 216–223 (2011)

    Article  Google Scholar 

  40. Sarasu P, Sundarapandian V: Global chaos synchronization of three-scroll chaotic systems via adaptive control. Eur. J. Sci. Res. 74, 504–522 (2012)

    Google Scholar 

  41. Lu X.B, Qin B.Z: Adaptive cluster synchronization in complex dynamical networks. Phys. Lett. A 373, 3650–3658 (2009)

    Article  MATH  Google Scholar 

  42. Wu X, Lu H: Cluster synchronization in the adaptive complex dynamical networks via a novel approach. Phys. Lett. A 375, 1559–1565 (2011)

    Article  MATH  Google Scholar 

  43. Wu Z, Fu X: Cluster mixed synchronization via pinning control and adaptive coupling strength in community networks with nonidentical nodes. Commun. Nonlinear Sci. Numer. Simul. 17, 1628–1636 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  44. Wang S, Yao H, Zheng S, Xie Y: A novel criterion for cluster synchronization of complex dynamical networks with coupling time-varying delays. Commun. Nonlinear Sci. Numer. Simul. 17, 2997–3004 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  45. Park J.H, Kwon O.M: A novel criterion for delayed feedback control of time-delay chaotic systems. Chaos Solitons Fractals 17, 709–716 (2003)

    Article  MathSciNet  Google Scholar 

  46. Wu X, Lü J: Parameter identification and backstepping control of uncertain Lu system. Chaos Solitons Fractals 18, 721–729 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  47. Yu Y.G, Zhang S.C: Adaptive backstepping synchronization of uncertain chaotic systems. Chaos Solitons Fractals 27, 1369–1375 (2006)

    Article  MathSciNet  Google Scholar 

  48. Tang W.K.S, Zhong Q, Chen G, Man K.F: Generation of n-scroll attractors via sine function. IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 48, 1369–1372 (2001)

    Article  MathSciNet  Google Scholar 

  49. Suykens J, Vandewalle C.L: Nonlinear H ∞ synchronization of chaotic Lur’e system. Int. J. Bifurc. Chaos 7, 1323–1325 (1997)

    Article  MATH  Google Scholar 

  50. Hahn, W.: The Stability of Motion. Springer, Berlin (1967)

Download references

Author information

Authors and Affiliations

  1. Research and Development Center, Vel Tech Dr. RR & Dr. SR Technical University, Avadi, Chennai, 600062, India

    Sundarapandian Vaidyanathan

  2. Department of Mathematics, Vel Tech Dr. RR & Dr. SR Technical University, Avadi, Chennai, 600062, India

    Suresh Rasappan

Authors
  1. Sundarapandian Vaidyanathan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Suresh Rasappan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Suresh Rasappan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vaidyanathan, S., Rasappan, S. Global Chaos Synchronization of n-Scroll Chua Circuit and Lur’e System using Backstepping Control Design with Recursive Feedback. Arab J Sci Eng 39, 3351–3364 (2014). https://doi.org/10.1007/s13369-013-0929-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13369-013-0929-y

Keywords

Search

Navigation