Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Control of finite-time anti-synchronization for variable-order fractional chaotic systems with unknown parameters


Fractional-order chaotic system with variable-order and unknown parameters, as an excellent tool to describe the memory and hereditary characteristics of the complex phenomena in reality, remains important, but nowadays there exist few results about this system. This paper presents a finite-time anti-synchronization of two these systems based on the Mittag-Leffler stable theory and norm theory, in which the order varies with time and the unknown parameters of the systems are estimated. Moreover, a corollary about the monotone effect of variable order on the norm of the error system is deduced. We take different nonlinear variable orders for two identical Lü fractional chaotic systems and for two different Lü and Chen–Lee fractional chaotic systems as examples. The simulations illustrate the effectiveness and feasibility of the proposed control scheme.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9


  1. 1.

    Lin, C.H., Huang, C.H., Du, Y.C., Chen, J.L.: Maximum photovoltaic power tracking for the PV array using the fractional-order incremental conductance method. Appl. Energy 88, 4840–4847 (2011)

  2. 2.

    Trigeassou, J.C., Maamri, N., Sabatier, J., Oustaloup, A.: Transients of fractional-order integrator and derivatives. Signal Image Video P 6, 359–372 (2012)

  3. 3.

    Lonescu, C.M., Desager, K., De Keyser, R.: Fractional order model parameters for the respiratory input impedance in healthy and in asthmatic children. Comput. Methods Prog. Biomed. 101, 315–323 (2011)

  4. 4.

    Eluru, N., Chakour, V., Chamberlain, M., Miranda-Moreno, L.F.: Modeling vehicle operating speed on urban roads in montreal: a panel mixed ordered probit fractional split model. Accid. Anal. Prev. 59, 125–134 (2013)

  5. 5.

    Li, C., Peng, G.: Chaos in chen’s system with a fractional order. Chaos Solitons Fract. 22, 443–450 (2004)

  6. 6.

    Hartley, T.T., Lorenzo, C.F., Qammer, H.K.: Chaos in a fractional order Chua’s system. IEEE T. Circuits I(42), 485–490 (1995)

  7. 7.

    Chen, D.Y., Liu, Y.X., Ma, X.Y., Zhang, R.F.: Control of a class of fractional-order chaotic systems via sliding mode. Nonlinear Dyn. 67, 893–901 (2012)

  8. 8.

    Hu, J., Chen, S., Chen, L.: Adaptive control for anti-synchronization of chua’s chaotic system. Phys. Lett. A 339, 455–460 (2005)

  9. 9.

    Taghvafard, H., Erjaee, G.H.: Phase and anti-phase synchronization of fractional order chaotic systems via active control. Commun. Nonlinear Sci. Numer. Simulat. 16, 4079–4088 (2011)

  10. 10.

    Hegazi, A.S., Ahmed, E., Matouk, A.E.: On chaos control and synchronization of the commensurate fractional order Liu system. Commun. Nonlinear Sci. Numer. Simulat. 18, 1193–1202 (2013)

  11. 11.

    Lei, Y., Xu, W., Zheng, H.: Synchronization of two chaotic nonlinear gyros using active control. Phys. Lett. A 343, 153–158 (2005)

  12. 12.

    Vincent, U.E.: Synchronization of identical and non-identical 4-D chaotic systems using active control. Chaos Solitons Fract. 37, 1065–1075 (2008)

  13. 13.

    Srivastava, M., Ansari, S.P., Agrawal, S.K., Das, S., Leung, A.Y.T.: Anti-synchronization between identical and non-identical fractional-order chaotic systems using active control method. Nonlinear Dyn. 76, 905–914 (2014)

  14. 14.

    Das, S., Srivastava, M., Leung, A.Y.T.: Hybrid phase synchronization between identical and nonidentical three-dimensional chaotic systems using the active control method. Nonlinear Dyn. 73, 2261–2272 (2013)

  15. 15.

    Srivastava, M., Agrawal, S.K., Vishal, K., Das, S.: Chaos control of fractional order Rabinovich-Fabrikant system and synchronization between chaotic and chaos controlled fractional order Rabinovich-Fabrikant system. Appl. Math. Model 38, 3361–3372 (2014)

  16. 16.

    Matouk, A.E.: Chaos, feedback control and synchronization of a fractional-order modified autonomous Van der pol-duffing circuit. Commun. Nonlinear Sci. Numer. Simulat. 16, 975–986 (2011)

  17. 17.

    Faieghi, M.R., Delavari, H.: Chaos in fractional-order Genesio-Tesi system and its synchronization. Commun. Nonlinear Sci. Numer. Simulat. 17, 731–741 (2012)

  18. 18.

    Mankin, R., Laas, K., Lumi, N.: Memory effects for a trapped Brownian particle in viscoelastic shear flows. Phys. Rev. E 88, 042142 (2013)

  19. 19.

    Sun, H.G., Chen, W., Wei, H., Chen, Y.Q.: A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems. Eur. Phys. J. Spec. Top. 193, 185–192 (2011)

  20. 20.

    Zhang, L., Liu, S.T., Yu, C.: Control of Weierstrass-Mandelbrot function model with morlet wavelets. Int. J. Bifurcat. Chaos 24, 1450121 (2014)

  21. 21.

    Zhang, L., Liu, S.T., Yu, C.: Chaotic behaviour of nonlinear coupled reaction-diffusion system in four-dimensional space. Pramana-J. Phys. 82, 995–1009 (2014)

  22. 22.

    Samko, S.G., Ross, B.: Integration and differentiation to a variable fractional order. Integral Transforms Spec. Funct. 1, 277–300 (1993)

  23. 23.

    Lorenzo, C.F., Hartley, T.T.: Variable order and distributed order fractional operators. Nonlinear Dyn. 29, 57–98 (2002)

  24. 24.

    Lorenzo, C.F., Hartley, T.T.: Preface special issue: initialization, conceptualization, and application in the generalized (fractional) calculus. CRC. Crit. Rev. Biomed. Eng. 35, 447–553 (2007)

  25. 25.

    Sun, H.G., Chen, W., Chen, Y.Q.: Variable-order fractional differential operators in anomalous diffusion modeling. Physica. A 388, 4586–4592 (2009)

  26. 26.

    Sun, H.G., Zhang, Y., Chen, W., Reeves, D.M.: Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media. J. Contam. Hydrol. 157, 47–58 (2014)

  27. 27.

    Rapaić, M.R., Pisano, A.: Variable-order fractional operators for adaptive order and parameter estimation. IEEE T. Autom. Control 45, 1–1 (1995)

  28. 28.

    Atangana, A.: On the stability and convergence of the time-fractional variable order telegraph equation. J. Comput. Phys. 293, 104–114 (2015)

  29. 29.

    Sun, H.G., Chen, W., Chen, Y.Q.: Variable-order fractional differential operators in anomalous diffusion modelling. Physica A. 388, 4586–4592 (2009)

  30. 30.

    Xu, Y., He, Z.: Synchronization of variable-order fractional financial system via active control method. Cent. Eur. J. Phys. 11, 824–835 (2013)

  31. 31.

    Rehan, M.: Synchronization and anti-synchronization of chaotic oscillators under input saturation. Appl. Math. Model. 13, 6829–6837 (2013)

  32. 32.

    Ingman, D., Suzdalnitsky, J.: Control of dampling oscillations by fractional differential operator with time-dependent order. Comput. Methods Appl. Mech. Eng. 193, 5585–5595 (2002)

  33. 33.

    Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)

  34. 34.

    Odibat, Z.M., Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. 7, 27–34 (2006)

  35. 35.

    Zhang, R.X., Yang, S.P.: Adaptive synchronization of fractional-order chaotic systems via a single driving variable. Nonlinear Dyn. 66, 831–837 (2011)

  36. 36.

    Zhang, S.: Existence and uniqueness result of solutions to initial value problems of fractional differential equations of variable-order. J. Fract. Calc. Appl. 4, 1–17 (2013)

  37. 37.

    Li, Y., Chen, Y.Q., Podlubny, I.: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized mittag-leffler stability. Comput. Math. Appl. 59, 1810–1821 (2010)

  38. 38.

    Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of the difference methods for the space-time fractional advection-diffusion. Appl. Math. Comput. 191, 12–20 (2007)

  39. 39.

    Butkovskii, A.G., Postnov, S.S., Postnova, E.A.: Fractional integro-differential calculus and its control-theoretical applications. II. Fractional dynamic systems: modeling and hardware implementation. Automat. Rem. Control 74, 543–574 (2013)

  40. 40.

    Li, Y., Chen, Y.Q., Ahn, H.S., Tian, G.: A survey on fractional-order iterative learning control. J. Optimiz. Theory Appl. 156, 127–140 (2013)

  41. 41.

    Wilson, F.W.: The structure of the level surfaces of a Lyapunov function. J. Differ. Equ. 3, 323–329 (1967)

  42. 42.

    Aguila-Camacho, N., Duarte-Mermoud, M.A., Gallegos, J.A.: Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simulat. 19, 2951–2957 (2014)

Download references


The authors are very grateful to editors and referees for valuable suggestions. This work is supported by the Foundation for the China Postdoctoral Science Foundation funded project (No. 2015M572033), the Foundation for the National Natural Science Foundation of China (No. 61273088; No. 61533011), Social Science Planning Fund Program of Shandong Province (No. 15CJJJ34) and development of Shandong University of Political Science and Law (No.2015Z01B).

Author information

Correspondence to Li Zhang.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhang, L., Yu, C. & Liu, T. Control of finite-time anti-synchronization for variable-order fractional chaotic systems with unknown parameters. Nonlinear Dyn 86, 1967–1980 (2016).

Download citation


  • Fractional chaotic system
  • Anti-synchronization
  • Variable-order
  • Unknown parameters