Advertisement

Nonlinear Dynamics

, Volume 76, Issue 2, pp 905–914 | Cite as

Anti-synchronization between identical and non-identical fractional-order chaotic systems using active control method

  • M. Srivastava
  • S. P. Ansari
  • S. K. Agrawal
  • S. Das
  • A. Y. T. LeungEmail author
Original Paper

Abstract

This article deals with the anti-synchronization between two identical chaotic fractional-order Qi system, Genesio–Tesi system, and also between two different fractional-order Genesio–Tesi and Qi systems using active control method. The chaotic attractors of the systems are found for fractional-order time derivatives described in Caputo sense. Numerical simulation results which are carried out using Adams–Boshforth–Moulton method show that the method is reliable and effective for anti-synchronization of nonlinear dynamical evolutionary systems.

Keywords

Chaos Anti-synchronization Fractional-order derivative Active control method 

Notes

Acknowledgments

The research was supported by Hong Kong Research Council GRF Grant #115712. The second author S.P. Ansari is extending her heartfelt thanks to the TCS, India for selecting her as TCS Research Scholar of the TCS Research Scholar Program.

References

  1. 1.
    Hifer, R.: Applications of Fractional Calculus in Physics, p. 472. World Scientific, Hackensack (2001)Google Scholar
  2. 2.
    Podlubny, I.: Fractional Differential Equations, p. 340. Academic Press, New York (1999)Google Scholar
  3. 3.
    Koeller, R.C.: Application of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51, 299–307 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Sun, H.H., Abdelwahed, A.A., Onaral, B.: Linear approximation for transfer function with a pole of fractional order. IEEE Trans. Autom. Control 29, 441–444 (1984)CrossRefzbMATHGoogle Scholar
  5. 5.
    Ichise, M., Nagayanagi, Y., Kojima, T.: An analog simulation of noninteger order transfer functions for analysis of electrode process. J. Electroanal. Chem. 33, 253–265 (1971)CrossRefGoogle Scholar
  6. 6.
    Heaviside, O.: Electromagnetic Theory, p. 271. Chelsea, New York (1971)Google Scholar
  7. 7.
    Laskin, N.: Fractional market dynamics. Phys. A 287, 482–492 (2000)Google Scholar
  8. 8.
    Kunsezov, D., Bulagc, A., Dang, G.D.: Quantum levy processes and fractional kinetics. Phys. Rev. Lett. 82, 1136–1139 (1999)Google Scholar
  9. 9.
    Hartley, T.T., Lorenzo, C.F.: Dynamics and control of initialized fractional-order systems. Nonlinear Dyn. 29, 201–233 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Daftardar-Gejji, V., Babakhani, A.: Analysis of a system of fractional differential equations. J. Math. Appl. Anal. 293, 511–522 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Matignon, D.: Stability results of fractional differential equations with applications to control processing. In: Proceeding of IMACS, IEEE-SMC, Lille, France, pp. 963–968 (1996)Google Scholar
  12. 12.
    Deng, W., Li, C., Lü, J.: Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn. 48, 409–416 (2007)CrossRefzbMATHGoogle Scholar
  13. 13.
    Ahmed, E., El-Sayed, A.M., El-Saka, H.: Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models. J. Math. Anal. Appl. 325, 542–553 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Blasius, B., Huppert, A., Stone, L.: Complex dynamics and phase synchronization in spatially extended ecological system. Nature 399, 354–359 (1999)CrossRefGoogle Scholar
  16. 16.
    Lakshmanan, M., Murali, K.: Chaos in Nonlinear Oscillators: Controlling and Synchronization, p. 340. World Scientific, Singapore (1996)zbMATHGoogle Scholar
  17. 17.
    Han, S.K., Kerrer, C., Kuramoto, Y.: Dephasing and bursting in coupled neural oscillators. Phys. Rev. Lett. 75, 3190–3193 (1995)CrossRefGoogle Scholar
  18. 18.
    Cuomo, K.M., Oppenheim, A.V.: Circuit implementation of synchronized chaos with application to communication. Phys. Rev. Lett. 71, 65–68 (1993)CrossRefGoogle Scholar
  19. 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. 20.
    Balasubramaniam, P., Vembarasan, V.: Synchronization of recurrent neural networks with mixed time-delays via output coupling with delayed feedback. Nonlinear Dyn. 70, 677–691 (2012)Google Scholar
  21. 21.
    Vembarasan, V., Balasubramaniam, P.: Chaotic synchronization of Rikitake system based on T–S fuzzy control techniques. Nonlinear Dyn. 74, 31–44 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Theesar, S.J.S., Banerjee, S., Balasubramaniam, P.: Synchronization of chaotic systems under sampled-data control. Nonlinear Dyn. 70, 1977–1987 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Yang, T., Chua, L.: Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 44, 976–988 (1997)Google Scholar
  24. 24.
    Yang, T., Chua, L.: Control of chaos using sampled-data feedback control. Int. J. Bifurcation Chaos 8, 2433–2438 (1998)CrossRefzbMATHGoogle Scholar
  25. 25.
    Sahaverdiev, E.M., Shore, K.A.: Lag synchronization in time-delayed systems. Phys. Lett. A 292, 320–324 (2002)CrossRefGoogle Scholar
  26. 26.
    Sahaverdiev, E.M., Sivaprakasam, S., Shore, K.A.: Inverse anticipating chaos synchronization. Phys. Rev. E 66, 017204 (2002)CrossRefGoogle Scholar
  27. 27.
    Zhan, M., Wang, X., Gong, X., Wei, G.W., Lai, C.H.: Complete synchronization and generalized synchronization of one-way coupled time-delay systems. Phys. Rev. E 68, 036208 (2003)Google Scholar
  28. 28.
    Senthilkumar, D.V., Lakshmanan, M., Kurths, J.: Phase synchronization in time-delay systems. Phys. Rev. E 74, 035205R (2006)CrossRefGoogle Scholar
  29. 29.
    Theesar, S.J.S., Banerjee, S., Balasubramaniam, P.: Adaptive synchronization in noise perturbed chaotic systems. Phys. Scripta 85, 065010 (2012)CrossRefGoogle Scholar
  30. 30.
    Huang, L., Feng, R., Wang, M.: Synchronization of chaotic systems via nonlinear control. Phys. Lett. A 320, 271–275 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Park, J.H., Kwon, O.M.: A novel criterion for delayed feedback control of time-delay chaotic systems. Chaos Solitons Fractals 23, 495–501 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Chen, S.H., Lu, J.: Synchronization of an uncertain unified chaotic system via adaptive control. Chaos Solitons Fractals 14, 643–647 (2002)CrossRefzbMATHGoogle Scholar
  33. 33.
    Al-sawalha, M.M., Noorani, M.S.M.: Anti-synchronization of chaotic systems with uncertain parameters via adaptive control. Phys. Lett. A 373, 2852–2857 (2009)Google Scholar
  34. 34.
    Yassen, M.T.: Chaos synchronization between two different chaotic systems using active control. Chaos Solitons Fractals 23, 131–140 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Wu, X., Lü, J.: Parameter identification and backstepping control of uncertain Lü system. Chaos Solitons Fractals 18, 721–729 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Fang, L., Li, T., Li, Z., Li, R.: Adaptive terminal sliding mode control for anti-synchronization of uncertain chaotic systems. Nonlinear Dyn. 74, 991–1002 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Yau, H.T.: Design of adaptive sliding mode controller for chaos synchronization with uncertainties. Chaos Solitons Fractals 22, 341–347 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Yang, S.S., Duan, C.K.: Generalized synchronization in chaotic systems. Chaos Solitons Fractals 9, 1703–1707 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Yu, H.J., Liu, Y.Z.: Chaotic synchronization based on stability criterion of linear systems. Phys. Lett. A 314, 292–298 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Rosenblum, M.G., Pikovsky, A.S., Kurths, J.: From phase to lag synchronization in coupled chaotic oscillators. Phys. Rev. Lett. 78, 4193–4196 (1997)CrossRefGoogle Scholar
  41. 41.
    Erjaee, G.H., Taghvafard, H.: Phase and anti-phase synchronization of fractional order chaotic systems via active control. Commun. Nonlinear Sci. Numer. Simul. 16, 4079–4088 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Liu, W.Q.: Anti-phase synchronization in coupled chaotic oscillators. Phys. Rev. E 73, 57203–57207 (2006)CrossRefGoogle Scholar
  43. 43.
    Liu, J.B., Ye, C.F., Zhang, S.J., Song, W.T.: Anti-phase synchronization in coupled map lattices. Phys. Lett. A 274, 27–29 (2000)Google Scholar
  44. 44.
    Hu, J., Chen, S., Chen, L.: Adaptive control for anti-synchronization of chua’s chaotic system. Phys. Lett. A 339, 455–460 (2005)CrossRefzbMATHGoogle Scholar
  45. 45.
    Zhang, Y., Sun, J.: Chaotic synchronization and anti-synchronization based on suitable separation. Phys. Lett. A 330, 442–447 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    Mossa, M., Sawalha, A., Noorani, M.S.M.: Anti-synchronization between two different hyperchaotic systems. J. Uncertain Syst. 3, 192–200 (2009)Google Scholar
  47. 47.
    Al-sawalha, M.M., Noorani, M.S.M.: Chaos reduced-order anti-synchronization of chaotic systems with fully unknown parameters. Commun. Nonlinear Sci. Numer. Simul. 17, 1908–1920 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  48. 48.
    Bai, E.W., Lonngren, K.E.: Synchronization of two Lorenz systems using active control. Chaos Solitons Fractals 8, 51–58 (1997)Google Scholar
  49. 49.
    Agrawal, S.K., Srivastava, M., Das, S.: Synchronization of fractional order chaotic systems using active control method. Chaos Solitons Fractals 45, 737–752 (2012)CrossRefGoogle Scholar
  50. 50.
    Agrawal, S.K., Srivastava, M., Das, S.: Synchronization between fractional-order Ravinovich–Fabrikant and Lotka–Volterra systems. Nonlinear Dyn. 69, 2277–2288 (2012)CrossRefMathSciNetGoogle Scholar
  51. 51.
    Agrawal, S.K., Srivastava, M., Das, S.: Hybrid synchronization between different fractional order hyperchaotic systems using active control method. J. Nonlinear Syst. Appl. 4, 70–76 (2013)Google Scholar
  52. 52.
    Wang, Z.L., Shi, X.R.: Anti-synchronization of Liu system and Lorenz system with known or unknown parameters. Nonlinear Dyn. 57, 425–430 (2009)CrossRefzbMATHGoogle Scholar
  53. 53.
    Al-sawalha, M.M., Alomari, A.K., Goh, S.M., Noorani, M.S.M.: Active anti-synchronization of two identical and different fractional-order chaotic systems. Int. J. Nonlinear Sci. 11, 267–274 (2011)zbMATHMathSciNetGoogle Scholar
  54. 54.
    Bhalekar, S., Gejji, V.D.: Anti-synchronization of non-identical fractional order chaotic systems using active control. Int. J. Diff. Equ. (2011). doi: 10.1155/2011/250763
  55. 55.
    Diethelm, K., Ford, J., Freed, A.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36, 31–52 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  56. 56.
    Diethelm, K., Ford, J.: Multi-order fractional differential equations and their numerical solution. Appl. Math. Comput. 154, 621–640 (2004)Google Scholar
  57. 57.
    Wu, X., Yang, Y.: Chaos in the fractional-order Qi system and its synchronization using active control. In: International Conference on Intelligent Computing and Cognitive Informatics, Kuala Lumpur, Malaysia (2010) Google Scholar
  58. 58.
    Faieghi, M.R., Delavari, H.: Chaos in fractional-order Genesio–Tesi system and its synchronization. Commun. Nonlinear Sci. Numer. Simul. 17, 731–734 (2011)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • M. Srivastava
    • 1
  • S. P. Ansari
    • 1
  • S. K. Agrawal
    • 1
  • S. Das
    • 1
  • A. Y. T. Leung
    • 2
    Email author
  1. 1.Department of Mathematical SciencesIndian Institute of Technology (BHU)VaranasiIndia
  2. 2.Department of Civil and Architectural Engineering City University of Hong KongHong KongChina

Personalised recommendations