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Analysis and synchronization for a new fractional-order chaotic system with absolute value term

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Abstract

A new fractional-order chaotic system with absolute value term is introduced. Some dynamical behaviors are investigated and analyzed. Furthermore, synchronization of this system is achieved by utilizing the drive-response method and the feedback method. The suitable parameters for achieving synchronization are studied. Both the theoretical analysis and numerical simulations show the effectiveness of the two methods.

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References

  1. Li, C., Liao, X.F., Yu, J.B.: Generating chaos by Oja’s rule. Neurocomputing 55, 731–738 (2003)

    Article  Google Scholar 

  2. Liu, Y.J.: Circuit implementation and finite-time synchronization of the 4D Rabinovich hyperchaotic system. Nonlinear Dyn. 67, 89–96 (2012)

    Article  MATH  Google Scholar 

  3. Lu, J.G.: Generating chaos via decentralized linear state feedback and a class of nonlinear functions. Chaos Solitons Fractals 25, 403–413 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Liu, Y.J., Yang, Q.G.: Dynamics of a new Lorenz-like chaotic system. Nonlinear Anal., Real World Appl. 11, 2563–2572 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ross, B.: The development of fractional calculus 1695–1900. Historia Math. 4, 75–89 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  6. He, G.L., Zhou, S.P.: What is the exact condition for fractional integrals and derivatives of Besicovitch functions to have exact box dimension. Chaos Solitons Fractals 26, 867–879 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jumarie, G.: Fractional master equation: non-standard analysis and Liouville–Riemann derivative. Chaos Solitons Fractals 12, 2577–2587 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Elwakil, S.A., Zahran, M.A.: Fractional integral representation of master equation. Chaos Solitons Fractals 10, 1545–1558 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  10. Yu, Y.G., Li, H.X., Wang, S., Yu, J.Z.: Dynamic analysis of a fractional-order Lorenz chaotic system. Chaos Solitons Fractals 42, 1181–1189 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grigorenko, I., Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system. Phys. Rev. Lett. 91, 034101 (2003)

    Article  Google Scholar 

  12. Li, C.G., Chen, G.R.: Chaos and hyperchaos in the fractional-order Rössler equations. Physica A 341, 55–61 (2004)

    Article  MathSciNet  Google Scholar 

  13. Wu, X., Wang, H., Zhu, H.: A new chaotic system with fractional order and its projective synchronization. Nonlinear Dyn. 61, 407–417 (2010)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  15. Li, H., Liao, X., Luo, M.: A novel non-equilibrium fractional-order chaotic system and its complete synchronization by circuit implementation. Nonlinear Dyn. 68, 137–149 (2012)

    Article  MATH  Google Scholar 

  16. Zeng, C., Yang, Q., Wang, J.: Chaos and mixed synchronization of a new fractional-order system with one saddle and two stable node-foci. Nonlinear Dyn. 65, 457–466 (2011)

    Article  MathSciNet  Google Scholar 

  17. Song, L., Yang, J.Y., Xu, S.Y.: Chaos synchronization for a class of nonlinear oscillators with fractional order. Nonlinear Anal.: Theory Methods Appl. 72, 2326–2336 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Matouk, A.E.: Chaos, feedback control and synchronization of a fractional-order modified autonomous van der Pol–Duffing circuit. Commun. Nonlinear Sci. Numer. Simul. 16, 975–986 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mohammad, S.T., Mohammad, H.: Synchronization of chaotic fractional-order systems via active sliding mode controller. Physica A 387, 57–70 (2008)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  21. Li, C., Yan, J.: The synchronization of three fractional differential systems. Chaos Solitons Fractals 32, 751–757 (2007)

    Article  MathSciNet  Google Scholar 

  22. Odibat, Z.M.: Adaptive feedback control and synchronization of non-identical chaotic fractional order systems. Nonlinear Dyn. 60, 479–487 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Deng, W.H.: Smoothness and stability of the solutions for nonlinear fractional differential equations. Nonlinear Anal.: Theory Methods Appl. 72, 1768–1777 (2009)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the reviewers for their constructive and pertinent suggestions for improving the presentation of the work. This wok was supported by the Special Scientific Foundation of Yulin Normal University (No. 2009YJZD06), the Natural Science Foundation of Guangxi Province of China (Grant No. 2012GXNSFAA053014) and the Scientific Research Foundation of the Higher Education Institutions of Guangxi Province Province of China (Grant No. 201202ZD080).

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Correspondence to Lihe Huang.

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Huang, L., Liu, A. Analysis and synchronization for a new fractional-order chaotic system with absolute value term. Nonlinear Dyn 70, 601–608 (2012). https://doi.org/10.1007/s11071-012-0480-5

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  • DOI: https://doi.org/10.1007/s11071-012-0480-5

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