Nonlinear Dynamics

, Volume 68, Issue 4, pp 543–554 | Cite as

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

  • Zhouchao WeiEmail author
  • Qigui Yang
Original Paper


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.


Chaotic attractors Degenerate heteroclinic cycles Sil’nikov theorem Lyapunov exponent Coexisting attractors 


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  1. 1.
    Lorenz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141 (1963) CrossRefGoogle Scholar
  2. 2.
    Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57, 397–398 (1976) CrossRefGoogle Scholar
  3. 3.
    Sprott, J.C.: Some simple chaotic flows. Phys. Rev. E 50, 647–650 (1994) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Sprott, J.C.: A new class of chaotic circuit. Phys. Lett. A 266, 19–23 (2000) CrossRefGoogle Scholar
  5. 5.
    Sprott, J.C.: Simplest dissipative chaotic flow. Phys. Lett. A 228, 271–274 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Chen, G.R., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos Appl. Sci. Eng. 9, 1465–1466 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Lü, J.H., Chen, G.R.: A new chaotic attractor coined. Int. J. Bifurc. Chaos Appl. Sci. Eng. 12, 659–661 (2002) zbMATHCrossRefGoogle Scholar
  8. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Shaw, R.: Strange attractor, chaotic behavior and information flow. Z. Naturforsch. 36A, 80–112 (1981) Google Scholar
  11. 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) zbMATHCrossRefGoogle Scholar
  12. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Sparrow, C.: The Lorenz Equations: Bifurcation, Chaos, and Strange Attractor. Springer, New York (1982) CrossRefGoogle Scholar
  14. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 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) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mello, L.F., Coelho, S.F.: Degenerate Hopf bifurcations in the Lü system. Phys. Lett. A 373, 1116–1120 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Wang, Z.: Existence of attractor and control of a 3D differential system. Nonlinear Dyn. 60, 369–373 (2009) CrossRefGoogle Scholar
  22. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Wei, Z.C., Yang, Q.G.: Controlling the diffusionless Lorenz equations with periodic parametric perturbation. Comput. Math. Appl. 58, 1979–1987 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Bao, J.H., Yang, Q.G.: Complex dynamics in the stretch-twist-fold flow. Nonlinear Dyn. 61, 773–781 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 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. 26.
    Sil’nikov, L.P.: A case of the existence of a countable number of periodic motions. Sov. Math., Dokl. 6, 163–166 (1965) zbMATHGoogle Scholar
  27. 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) CrossRefGoogle Scholar
  28. 28.
    Silva, C.P.: Sil’nikov theorem-a tutorial. IEEE Trans. Circuits Syst. I 40, 657–682 (1993) CrossRefGoogle Scholar
  29. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Llibre, J., Messias, M.: Global dynamics of Rikitake system. Physica D 238, 241–252 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Llibre, J., Zhang, X.: Invariant algebraic surfaces of the Rikitake system. J. Phys. A, Math. Gen. 33, 7613–7635 (2000) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsSouth-Central University for NationalitiesWuhanP.R. China
  2. 2.School of Mathematical SciencesSouth China University of TechnologyGuangzhouP.R. China

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