Nonlinear Dynamics

, Volume 75, Issue 4, pp 745–760 | Cite as

Analytical and numerical investigation of a new Lorenz-like chaotic attractor with compound structures

  • Shijian Cang
  • Zenghui Wang
  • Zengqiang Chen
  • Hongyan Jia
Original Paper


This paper presents a three-dimensional autonomous Lorenz-like system formed by only five terms with a butterfly chaotic attractor. The dynamics of this new system is completely different from that in the Lorenz system family. This new chaotic system can display different dynamic behaviors such as periodic orbits, intermittency and chaos, which are numerically verified through investigating phase trajectories, Lyapunov exponents, bifurcation diagrams and Poincaré sections. Furthermore, this new system with compound structures is also proved by the presence of Hopf bifurcation at the equilibria and the crisis-induced intermittency.


Lorenz-like system Chaotic attractor Hopf bifurcation Crisis-induced intermittency Compound structures 



This work was supported in part by the National Natural Science Foundation of China (Nos. 6117-4094 and 11202148), Specialized Research Fund for the Doctoral Program of Higher Education (No. 20090031110029), the China/South Africa Research Cooperation Programme (Nos. 78673 and CS06-L02) and the South African National Research Foundation Incentive Grant (No. 81705).


  1. 1.
    Cafagna, D., Grassi, G.: Generation of chaotic beats in a modified Chua’s circuit, Part I: Dynamic behaviour. Nonlinear Dyn. 44, 91–99 (2006). doi: 10.1007/s11071-006-1948-y CrossRefMATHGoogle Scholar
  2. 2.
    Cafagna, D., Grassi, G.: Generation of chaotic beats in a modified Chua’s circuit, Part II: Circuit design. Nonlinear Dyn. 44, 101–108 (2006). doi: 10.1007/s11071-006-1949-x CrossRefMATHGoogle Scholar
  3. 3.
    Gaspard, P.: Microscopic chaos and chemical reactions. Physica D 263, 315–328 (1999). doi: 10.1016/S0378-4371(98)00504-4 CrossRefMathSciNetGoogle Scholar
  4. 4.
    Shabunin, A., Astakhov, V., Demidov, V., Provata, A., Baras, F., Nicolis, G., Anishchenko, V.: Modeling chemical reactions by forced limit-cycle oscillator: synchronization phenomena and transition to chaos. Chaos Solitons Fractals 15, 395–405 (2003). doi: 10.1016/S0960-0779(02)00106-6 CrossRefMATHGoogle Scholar
  5. 5.
    Kahan, S., Montagne, R.: Controlling spatiotemporal chaos in optical systems. Int. J. Bifurc. Chaos 11, 2853–2861 (2001). doi: 10.1142/S0218127401003917 CrossRefGoogle Scholar
  6. 6.
    Liu, J.M., Chen, H.F., Tang, S.: Optical-communication systems based on chaos in semiconductor lasers. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 48, 1475–1483 (2001). doi: 10.1109/Tcsi.2001.972854 CrossRefGoogle Scholar
  7. 7.
    Hentschel, M., Richter, K.: Quantum chaos in optical systems: the annular billiard. Phys. Rev. E 66, 056207 (2002). doi: 10.1103/Physreve.66.056207 CrossRefMathSciNetGoogle Scholar
  8. 8.
    Aref, H.: Fluid dynamics—order in chaos. Nature 401, 756–758 (1999). doi: 10.1038/44495 CrossRefGoogle Scholar
  9. 9.
    Duane, G.S., Tribbia, J.J.: Synchronized chaos in geophysical fluid dynamics. Phys. Rev. Lett. 86, 4298–4301 (2001). doi: 10.1103/PhysRevLett.86.4298 CrossRefGoogle Scholar
  10. 10.
    Weiss, H.L., Szeri, A.J.: Nested invariant 3-tori embedded in a sea of chaos in a quasiperiodic fluid flow. Int. J. Bifurc. Chaos 19, 2181–2191 (2009). doi: 10.1142/S0218127409024001 CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963). doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 CrossRefGoogle Scholar
  12. 12.
    Montangero, S., Fronzoni, L., Girgolini, P.: The complexity of the logistic map at the chaos threshold. Phys. Lett. A 285, 81–87 (2001). doi: 10.1016/S0375-9601(01)00332-2 CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Borges, E.P., Tirnakli, U.: Mixing and relaxation dynamics of the Henon map at the edge of chaos. Physica D 193, 148–152 (2004). doi: 10.1016/j.physd.2004.01.015 CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57, 397–398 (1976). doi: 10.1016/0375-9601(76)90101-8 CrossRefGoogle Scholar
  15. 15.
    Sprott, J.C.: Simplest dissipative chaotic flow. Phys. Lett. A 228, 271–274 (1997). doi: 10.1016/S0375-9601(97)00088-1 CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Muthuswamy, B., Chua, L.O.: Simplest chaotic circuit. Int. J. Bifurc. Chaos 20, 1567–1580 (2010). doi: 10.1142/S0218127410027076 CrossRefGoogle Scholar
  17. 17.
    Chen, G.R., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9, 1465–1466 (1999). doi: 10.1142/S0218127499001024 CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Lü, J.H., Chen, G.R.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 12, 659–661 (2002). doi: 10.1142/S0218127402004620 CrossRefMATHGoogle Scholar
  19. 19.
    Lü, J.H., Chen, G.R., Zhang, S.C.: Dynamical analysis of a new chaotic attractor. Int. J. Bifurc. Chaos 12, 1001–1015 (2002). doi: 10.1142/S0218127402004851 CrossRefMATHGoogle Scholar
  20. 20.
    Yang, Q.G., Chen, G.R.: A chaotic system with one saddle and two stable node-foci. Int. J. Bifurc. Chaos 18, 1393–1414 (2008). doi: 10.1142/S0218127408021063 CrossRefMATHGoogle Scholar
  21. 21.
    Chen, Z.Q., Yang, Y., Yuan, Z.Z.: A single three-wing or four-wing chaotic attractor generated from a three-dimensional smooth quadratic autonomous system. Chaos Solitons Fractals 38, 1187–1196 (2008). doi: 10.1016/j.chaos.2007.01.058 CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Grassi, G., Severance, F.L., Mashev, E.D., Bazuin, B.J., Miller, D.A.: Generation of a four-wing chaotic attractor by two weakly-coupled Lorenz systems. Int. J. Bifurc. Chaos 18, 2089–2094 (2008). doi: 10.1142/S0218127408021580 CrossRefMATHGoogle Scholar
  23. 23.
    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, 705–721 (2008). doi: 10.1016/j.chaos.2007.01.029 CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Cang, S.J., Qi, G.Y., Chen, Z.Q.: A four-wing hyper-chaotic attractor and transient chaos generated from a new 4-D quadratic autonomous system. Nonlinear Dyn. 59, 515–527 (2010). doi: 10.1007/s11071-009-9558-0 CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Yalcin, M.E., Suykens, J.A.K., Vandewalle, J.: Experimental confirmation of 3- and 5-scroll attractors from a generalized Chua’s circuit. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47, 425–429 (2000). doi: 10.1109/81.841929 CrossRefGoogle Scholar
  26. 26.
    Lü, J.H., Yu, X.H., Chen, G.R.: Generating chaotic attractors with multiple merged basins of attraction: a switching piecewise-linear control approach. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 50, 198–207 (2003). doi: 10.1109/Tcsi.2002.808241 CrossRefGoogle Scholar
  27. 27.
    Yu, S., Tang, W.K.S., Chen, G.R.: Generation of n × m-scroll attractors under a Chua-circuit framework. Int. J. Bifurc. Chaos 17, 3951–3964 (2007). doi: 10.1142/S0218127407019809 CrossRefMATHGoogle Scholar
  28. 28.
    Wang, L.: 3-scroll and 4-scroll chaotic attractors generated from a new 3-D quadratic autonomous system. Nonlinear Dyn. 56, 453–462 (2009). doi: 10.1007/s11071-008-9417-4 CrossRefMATHGoogle Scholar
  29. 29.
    C̆elikovský, S., Chen, G.R.: On the generalized Lorenz canonical form. Chaos Solitons Fractals 26, 1271–1276 (2005). doi: 10.1016/j.chaos.2005.02.040 CrossRefMathSciNetGoogle Scholar
  30. 30.
    Lu, J.H., Chen, G.R., Cheng, D.Z., Celikovsky, S.: Bridge the gap between the Lorenz system and the Chen system. Int. J. Bifurc. Chaos 12, 2917–2926 (2002). doi: 10.1142/S021812740200631x CrossRefMathSciNetGoogle Scholar
  31. 31.
    Sprott, J.C.: A proposed standard for the publication of new chaotic systems. Int. J. Bifurc. Chaos 21, 2391–2394 (2011). doi: 10.1142/S021812741103009x CrossRefMathSciNetGoogle Scholar
  32. 32.
    Zhou, T.S., Chen, G.R., C̆elikovský, S.: Sil’nikov chaos in the generalized Lorenz canonical form of dynamical systems. Nonlinear Dyn. 39, 319–334 (2005). doi: 10.1007/s11071-005-4195-8 CrossRefMATHGoogle Scholar
  33. 33.
    Yang, Q.G., Chen, G.R., Zhou, T.S.: A unified Lorenz-type system and its canonical form. Int. J. Bifurc. Chaos 16, 2855–2871 (2006). doi: 10.1142/S0218127406016501 CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Yang, Q.G., Zhang, K.M., Chen, G.R.: A modified generalized Lorenz-type system and its canonical form. Int. J. Bifurc. Chaos 19, 1931–1949 (2009). doi: 10.1142/S0218127409023834 CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Sprott, J.C.: Some simple chaotic flows. Phys. Rev. E 50, R647–R650 (1994). doi: 10.1103/PhysRevE.50.R647 CrossRefMathSciNetGoogle Scholar
  36. 36.
    Moloia, J.L., Chen, G.R.: Hopf Bifurcation Analysis: A Frequency Domain Approach, pp. 20–28. World Scientific, Singapore (1996) Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Shijian Cang
    • 1
  • Zenghui Wang
    • 2
  • Zengqiang Chen
    • 3
  • Hongyan Jia
    • 4
  1. 1.Department of Industry DesignTianjin University of Science and TechnologyTianjinP.R. China
  2. 2.Department of Electrical and Mining EngineeringUniversity of South AfricaFloridaSouth Africa
  3. 3.Department of AutomationNankai UniversityTianjinP.R. China
  4. 4.Department of AutomationTianjin University of Science and TechnologyTianjinP.R. China

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