Skip to main content
SpringerLink
Log in

A new chaotic system with fractional order and its projective synchronization

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Based on Rikitake system, a new chaotic system is discussed. Some basic dynamical properties, such as equilibrium points, Lyapunov exponents, fractal dimension, Poincaré map, bifurcation diagrams and chaotic dynamical behaviors of the new chaotic system are studied, either numerically or analytically. The obtained results show clearly that the system discussed is a new chaotic system. By utilizing the fractional calculus theory and computer simulations, it is found that chaos exists in the new fractional-order three-dimensional system with order less than 3. The lowest order to yield chaos in this system is 2.733. The results are validated by the existence of one positive Lyapunov exponent and some phase diagrams. Further, based on the stability theory of the fractional-order system, projective synchronization of the new fractional-order chaotic system through designing the suitable nonlinear controller is investigated. The proposed method is rather simple and need not compute the conditional Lyapunov exponents. Numerical results are performed to verify the effectiveness of the presented synchronization scheme.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Lorenz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    Article  Google Scholar 

  2. Sparrow, C.: The Lorenz Equations: Bifurcation, Chaos, and Strange Attractors. Springer, New York (1982)

    Google Scholar 

  3. Stewart, I.: The Lorenz attractor exists. Nature 406, 948–949 (2002)

    Article  Google Scholar 

  4. Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57, 397–398 (1976)

    Article  Google Scholar 

  5. Sprott, J.C.: Some simple chaotic flows. Phys. Rev. E 50, R647 (1994)

    Article  MathSciNet  Google Scholar 

  6. Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9, 1465–1466 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Lü, J.H., Chen, G.R.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 3, 659–661 (2002)

    Google Scholar 

  8. Lü, J.H., Chen, G.R., Cheng, D.Z., Čelikovský, S.: Bridge the gap between the Lorenz and the Chen system. Int. J. Bifurc. Chaos 12, 2917–2926 (2002)

    Article  MATH  Google Scholar 

  9. Liu, C., Liu, T., Liu, L., Liu, K.: A new chaotic attractor. Chaos Solitons Fractals 22, 1031–1038 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Wang, G., Zhang, X., Zheng, Y., Li, X.: A new modified hyperchaotic Lü system. Physica A 371, 260–272 (2006)

    Article  MathSciNet  Google Scholar 

  11. Chen, C.H., Sheu, L.J., Chen, H.K., Chen, J.H., Wang, H.C., Chao, Y.C., Lin, Y.K.: A new hyper-chaotic system and its synchronization. Nonlinear Anal. Real World Appl. 10, 2088–2096 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Rikitake, T.: Oscillations of a system of disk dynamos. Proc. Camb. Philos. Soc. 54, 89–95 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  13. Keisuke, I.: Chaos in the Rikitake two-disk dynamo system. Earth Planet. Sci. Lett. 51, 451–457 (1980)

    Article  Google Scholar 

  14. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  15. Hifer, R.: Applications of Fractional Calculus in Physics. World Scientific, New Jersey (2001)

    Google Scholar 

  16. Koeller, R.C.: Application of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51, 299–307 (1984)

    Article  MathSciNet  Google Scholar 

  17. Sun, H.H., Abdelwahad, A.A., Onaral, B.: Linear approximation of transfer function with a pole of fractional order. IEEE Trans. Automat. Control 29, 441–444 (1984)

    Article  MATH  Google Scholar 

  18. 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)

    Google Scholar 

  19. Heaviside, O.: Electromagnetic Theory. Chelsea, New York (1971)

    Google Scholar 

  20. Podlubny, I.: Fractional-order systems and PIλDμ-controllers. IEEE Trans. Automat. Control 44, 208–214 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. Deng, W., Li, C.P.: Chaos synchronization of the fractional Lü system. Physica A 353, 61–72 (2005)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Li, G.H.: Modified projective synchronization of chaotic system. Chaos Solitons Fractals 32, 1786–1790 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Hung, M.L., Yan, J.J., Liao, T.L.: Generalized projective synchronization of chaotic nonlinear gyros coupled with dead-zone input. Chaos Solitons Fractals 35, 181–187 (2008)

    Article  MathSciNet  Google Scholar 

  27. Li, C.G., Liao, X.F., Yu, J.B.: Synchronization of fractional order chaotic systems. Phys. Rev. E 68, 067203 (2003)

    Article  Google Scholar 

  28. Lu, J.G.: Chaotic dynamics and synchronization of fractional-order Arneodo’s systems. Chaos Solitons Fractals 26, 1125–1133 (2005)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Tavazoei, M.S., Haeri, M.: Synchronization of chaotic fractional-order systems via active sliding mode controller. Physica A 387, 57–70 (2008)

    Article  MathSciNet  Google Scholar 

  31. Hardy, Y., Steeb, W.H.: The Rikitake two-disk dynamo system and domains with periodic orbits. Int. J. Theor. Phys. 38, 2413–2417 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  32. Vaněcěk, A., Čelikovský, S.: Control Systems: From Linear Analysis to Synthesis of Chaos. Prentice-Hall, London (1996)

    MATH  Google Scholar 

  33. Caputo, M.: Linear models of dissipation whose Q is almost frequency independent. Geophys. J. R. Astron. Soc. 13, 529–539 (1967)

    Google Scholar 

  34. Samko, S.G., Klibas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Amsterdam (1993)

    MATH  Google Scholar 

  35. Keil, F., Mackens, W., Werther, J.: Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties. Springer, Heidelberg (1999)

    Google Scholar 

  36. Butzer, P.L., Westphal, U.: An Introduction to Fractional Calculus. World Scientific, Singapore (2000)

    Google Scholar 

  37. Ahmad, W.M., Sprott, J.C.: Chaos in fractional-order autonomous nonlinear systems. Chaos Solitons Fractals 16, 339–351 (2003)

    Article  MATH  Google Scholar 

  38. Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional order. Electron. Trans. Numer. Anal. 5, 1–6 (1997)

    MATH  MathSciNet  Google Scholar 

  39. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3–22 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  41. Diethelm, K., Ford, N.J., Freed, A.D.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36, 31–52 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  42. Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: IEEE-SMC Proceedings, Computational Engineering in Systems and Application Multi-Conference, IMACS, vol. 2, pp. 963–968. Lille, France (1996)

Download references

Author information

Authors and Affiliations

  1. Computing Center, Henan University, Kaifeng, 475004, People’s Republic of China

    Xiangjun Wu & Hui Wang

Authors
  1. Xiangjun Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hui Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiangjun Wu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wu, X., Wang, H. A new chaotic system with fractional order and its projective synchronization. Nonlinear Dyn 61, 407–417 (2010). https://doi.org/10.1007/s11071-010-9658-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-010-9658-x

Keywords

Search

Navigation