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

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Abstract

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.

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References

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  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)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

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).

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Authors and Affiliations

  1. Business School, Shandong University of Political Science and Law, Jinan, 250014, China

    Li Zhang & Tao Liu

  2. College of Control Science and Engineering, Shandong University, Jinan, 250061, China

    Li Zhang

  3. Mind and Brain Theme, South Australian Health and Medical Research Institute, North Terrace, Adelaide, SA, 5000, Australia

    Chenglong Yu

  4. School of Medicine, Flinders University, Adelaide, SA, 5001, Australia

    Chenglong Yu

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  1. Li Zhang
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Correspondence to Li Zhang.

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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). https://doi.org/10.1007/s11071-016-3008-6

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