Nonlinear Dynamics

, Volume 56, Issue 4, pp 453–462 | Cite as

3-scroll and 4-scroll chaotic attractors generated from a new 3-D quadratic autonomous system

Original Paper


This article introduces a new chaotic system of 3-D quadratic autonomous ordinary differential equations, which can display 2-scroll chaotic attractors. Some basic dynamical behaviors of the new 3-D system are investigated. Of particular interest is that the chaotic system can generate complex 3-scroll and 4-scroll chaotic attractors. Finally, bifurcation analysis shows that the system can display extremely rich dynamics. The obtained results clearly show that this is a new chaotic system which deserves further detailed investigation.


Chaotic attractor 3-D quadratic autonomous system 3-scroll 4-scroll Bifurcation 


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  1. 1.
    Chen, G., Dong, X. (eds.): From Chaos to Order: Methodologies, Perspectives and Applications. World Scientific, Singapore (1998) zbMATHGoogle Scholar
  2. 2.
    Chen, G., Yu, X. (eds.): Chaos Control: Theory and Applications. Springer, Berlin (2003) zbMATHGoogle Scholar
  3. 3.
    Lü, J., Lu, J., Chen, S. (eds.): Chaotic Time Series Analysis and Its Applications. Wuhan University Press, Wuhan (2002) (in Chinese) Google Scholar
  4. 4.
    Lü, J.H., Chen, G.R., Cheng, D.Z.: A new chaotic system and beyond: the generalized Lorenz-like system. Int. J. Bifurc. Chaos 14(5), 1507–1537 (2004). doi: 10.1142/S021812740401014X zbMATHCrossRefGoogle Scholar
  5. 5.
    Lü, J.H., Chen, G.R.: Generating multiscroll chaotic attractors: theories methods and applications. Int. J. Bifurc. Chaos 16(4), 775–858 (2006). doi: 10.1142/S0218127406015179 zbMATHCrossRefGoogle Scholar
  6. 6.
    Chua, L.O. (ed.): CNN: A Paradigm for Complexity. World Scientific, Singapore (1998) zbMATHGoogle Scholar
  7. 7.
    Chua, L.O., Komuro, M., Matsumoto, T.: The double scroll family. IEEE Trans. Circuits Syst. 33, 1072–1118 (1986). doi: 10.1109/TCS.1986.1085869 zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kennedy, M.P.: Three steps to chaos, part II: a Chua’s circuit primer. IEEE Trans. Circuits Syst.-I 40, 657–674 (1993) zbMATHCrossRefGoogle Scholar
  9. 9.
    Qin, Q., Lin, W., Qiao, N.: Dynamical behaviours of Liu system with time delayed feedbacks. Chin. Phys. B 17(2), 569–572 (2008). doi: 10.1088/1674-1056/17/2/035 CrossRefGoogle Scholar
  10. 10.
    Qi, G.Y., Chen, G.R, van Wyk, M.A., van Wyk, B.J., Zhang, Y.H.: A four-wing chaotic attractor generated from a new 3-D quadratic autonomous system. Chaos Solitons Fractals 38(3), 705–721 (2008). doi: 10.1016/j.chaos.2007.01.029 zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Sprott, J.C.: Simplest dissipative chaotic flow. Phys. Lett. A 228(4–5), 271–274 (1997). doi: 10.1016/S0375-9601(97)00088-1 zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lü, J.H., Chen, G.R.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 12(3), 659–661 (2002) zbMATHCrossRefGoogle Scholar
  13. 13.
    Li, T.C., Chen, G.R., Tang, Y., Yang, L.J.: Hopf bifurcation of the generalized Lorenz canonical form. Nonlinear Dyn. 47(4), 367–375 (2007). doi: 10.1007/s11071-006-9036-x CrossRefMathSciNetGoogle Scholar
  14. 14.
    Zhou, T.S., Chen, G.R., Celikovský, S.: Si’lnikov chaos in the generalized Lorenz canonical form of dynamical systems. Nonlinear Dyn. 39(4), 319–334 (2005). doi: 10.1007/s11071-005-4195-8 zbMATHCrossRefGoogle Scholar
  15. 15.
    Wang, L., Ni, Q., Liu, P., Huang, Y.Y.: Chaos and its forming mechanism of a new Lorenz-like system. J. Dyn. Control 3, 1–6 (2005) (in Chinese) zbMATHGoogle Scholar
  16. 16.
    Liu, W.B., Chen, G.R.: Can a three-dimensional smooth autonomous quadratic chaotic system generate a single four-scroll attractor? Int. J. Bifurc. Chaos 14(4), 1395–1403 (2004). doi: 10.1142/S0218127404009880 zbMATHCrossRefGoogle Scholar
  17. 17.
    Li, D.Q.: A three-scroll chaotic attractor. Phys. Lett. A 372(4), 387–393 (2008) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MechanicsHuazhong University of Science and TechnologyWuhanChina

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