Skip to main content
SpringerLink
Log in

Dynamical analysis of the generalized Sprott C system with only two stable equilibria

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

Abstract

A generalized Sprott C system with only two stable equilibria is investigated by detailed theoretical analysis as well as dynamic simulation, including some basic dynamical properties, Lyapunov exponent spectra, fractal dimension, bifurcations, and routes to chaos. In the parameter space where the equilibria of the system are both asymptotically stable, chaotic attractors coexist with period attractors and stable equilibria. Moreover, the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated. Periodic solutions and chaotic attractors can be found when these cycles disappear.

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

Access this article

Log in via an institution

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

References

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  3. Sprott, J.C.: Some simple chaotic flows. Phys. Rev. E 50, 647–650 (1994)

    Article  MathSciNet  Google Scholar 

  4. Sprott, J.C.: A new class of chaotic circuit. Phys. Lett. A 266, 19–23 (2000)

    Article  Google Scholar 

  5. Sprott, J.C.: Simplest dissipative chaotic flow. Phys. Lett. A 228, 271–274 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Lü, J.H., Chen, G.R.: A new chaotic attractor coined. Int. J. Bifurc. Chaos Appl. Sci. Eng. 12, 659–661 (2002)

    Article  MATH  Google Scholar 

  8. Yang, Q.G., Chen, G.R., Huang, K.F.: Chaotic attractors of the conjugate Lorenz-type system. Int. J. Bifurc. Chaos Appl. Sci. Eng. 17, 3929–3949 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. van der Schrier, G., Maas, L.R.M.: The diffusionless Lorenz equations: Sil’nikov bifurcations and reduction to an explicit map. Physica D 141, 19–36 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Shaw, R.: Strange attractor, chaotic behavior and information flow. Z. Naturforsch. 36A, 80–112 (1981)

    Google Scholar 

  11. Yang, Q.G., Chen, G.R.: A chaotic system with one saddle and two stable node-foci. Int. J. Bifurc. Chaos Appl. Sci. Eng. 18, 1393–1414 (2008)

    Article  MATH  Google Scholar 

  12. Dias, F.S., Mello, L.F., Zhang, J.G.: Nonlinear analysis in a Lorenz-like system. Nonlinear Anal., Real World Appl. 11, 3491–3500 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

  14. Zhou, T.S., Chen, G.R., Tang, Y.: Complex dynamical behaviors of the chaotic Chen’s system. Int. J. Bifurc. Chaos Appl. Sci. Eng. 13, 2561–2574 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Yang, Q.G., Chen, G.R., Zhou, T.S.: A unified Lorenz-type system and its canonical form. Int. J. Bifurc. Chaos Appl. Sci. Eng. 16, 2855–2871 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kokubu, H., Roussarie, R.: Existence of a singularly degenerate heteroclinic cycle in the Lorenz system and its dynamical consequences: Part 1*. J. Dyn. Differ. Equ. 16, 513–557 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Messias, M.: Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system. J. Phys. A, Math. Theor. 42, 115101 (2009)

    Article  MathSciNet  Google Scholar 

  18. Mello, L.F., Coelho, S.F.: Degenerate Hopf bifurcations in the Lü system. Phys. Lett. A 373, 1116–1120 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mello, L.F., Messias, M., Braga, D.C.: Bifurcation analysis of a new Lorenz-like chaotic system. Chaos Solitons Fractals 37, 1244–1255 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, J., Zhang, J.: New treatment on bifurcation of periodic solutions and homoclinic orbits at high r in the Lorenz equations. SIAM J. Appl. Math. 53, 1059–1071 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang, Z.: Existence of attractor and control of a 3D differential system. Nonlinear Dyn. 60, 369–373 (2009)

    Article  Google Scholar 

  22. Pang, S.Q., Liu, Y.J.: A new hyperchaotic system from the Lü system and its control. J. Comput. Appl. Math. 235, 2775–2789 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wei, Z.C., Yang, Q.G.: Controlling the diffusionless Lorenz equations with periodic parametric perturbation. Comput. Math. Appl. 58, 1979–1987 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Bao, J.H., Yang, Q.G.: Complex dynamics in the stretch-twist-fold flow. Nonlinear Dyn. 61, 773–781 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Chen, D.Y., Wu, C., Liu, C.F., Ma, X.Y., You, Y.J., Zhang, R.F.: Synchronization and circuit simulation of a new double-wing chaos. Nonlinear Dyn. 10, 1–24 (2011)

    Google Scholar 

  26. Sil’nikov, L.P.: A case of the existence of a countable number of periodic motions. Sov. Math., Dokl. 6, 163–166 (1965)

    MATH  Google Scholar 

  27. Sil’nikov, L.P.: A contribution of the problem of the structure of an extended neighborhood of rough equilibrium state of saddle-focus type. Math. USSR. Sbornik 10, 91–102 (1970)

    Article  Google Scholar 

  28. Silva, C.P.: Sil’nikov theorem-a tutorial. IEEE Trans. Circuits Syst. I 40, 657–682 (1993)

    Article  Google Scholar 

  29. 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  MathSciNet  MATH  Google Scholar 

  30. Llibre, J., Messias, M.: Global dynamics of Rikitake system. Physica D 238, 241–252 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Llibre, J., Zhang, X.: Invariant algebraic surfaces of the Rikitake system. J. Phys. A, Math. Gen. 33, 7613–7635 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan, Hubei, 430074, P.R. China

    Zhouchao Wei

  2. School of Mathematical Sciences, South China University of Technology, Guangzhou, Guangdong, 510640, P.R. China

    Qigui Yang

Authors
  1. Zhouchao Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qigui Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhouchao Wei.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wei, Z., Yang, Q. Dynamical analysis of the generalized Sprott C system with only two stable equilibria. Nonlinear Dyn 68, 543–554 (2012). https://doi.org/10.1007/s11071-011-0235-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-011-0235-8

Keywords

Search

Navigation