Nonlinear Dynamics

, Volume 59, Issue 3, pp 515–527 | Cite as

A four-wing hyper-chaotic attractor and transient chaos generated from a new 4-D quadratic autonomous system

Original Paper

Abstract

This paper presents a new four-dimensional (4-D) smooth quadratic autonomous chaotic system, which can present periodic orbit, chaos, and hyper-chaos under the conditions on different parameters. Importantly, the system can generate a four-wing hyper-chaotic attractor and a pair of coexistent double-wing hyper-chaotic attractors with two symmetrical initial conditions. Furthermore, a four-wing transient chaos occurs in the system. The dynamic analysis approach- in the paper involves time series, phase portraits, Poincaré maps, bifurcation diagrams, and Lyapunov exponents, to investigate some basic dynamical behaviors of the proposed 4-D system.

Keywords

Chaos Hyper-chaos Four-wing attractor Transient chaos 

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References

  1. 1.
    Lorenz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141 (1963) CrossRefGoogle Scholar
  2. 2.
    Weiss, N., Garfinkel, A., Spano, M.L., Ditto, W.L.: Chaos and chaos control in biology. J. Clin. Invest. 93, 1355–1360 (1994) CrossRefGoogle Scholar
  3. 3.
    Goedgebuer, J.P., Larger, L., Port, H.: Optical cryptosystem based on synchronization of hyper-chaos generated by a delayed feedback laser diode. Phys. Rev. Lett. 80, 2249–2254 (1998) CrossRefGoogle Scholar
  4. 4.
    Goedgebuer, J.P., Larger, L., Chen, C.C., Rhodes, W.T.: Optical Communications with synchronized hyper-chaos generated electro-optical. IEEE J. Quantum Electron. 38, 1178–1183 (2002) CrossRefGoogle Scholar
  5. 5.
    Udaltsov, V.S., Goedgebuer, J.P., Larger, L., Cuenot, J.B., Levy, P., Rhodes, W.T.: Communicating with hyper-chaos: the dynamics of a DNLF emitter and recovery of transmitted information. Opt. Spectrosc. 95, 114–118 (2003) CrossRefGoogle Scholar
  6. 6.
    Yu, S.M., 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) MATHCrossRefGoogle Scholar
  7. 7.
    Lü, J.H., Chen, G.R.: Generating multi-scroll chaotic attractors: theories, methods and applications. Int. J. Bifurc. Chaos 16, 775–858 (2006) MATHCrossRefGoogle Scholar
  8. 8.
    Yalcin, M.E., Ozoguz, S., Suykens, J.A.K., Vandewalle, J.: n-Scroll chaos generators: a simple circuit model. Electron. Lett. 37, 147–148 (2001) CrossRefGoogle Scholar
  9. 9.
    Qi, G.Y., Chen, G.R., Li, S.W., Zhang, Y.H.: Four-wing attractors: from pseudo to real. Int. J. Bifurc. Chaos 16, 859–885 (2006) MATHCrossRefMathSciNetGoogle 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, 705–721 (2008) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Chen, Z.Q., Yong, 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) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Giuseppe, G., Frank, L.S., Emil, D.M., Bradley, J.B., Damon, A.M.: Generation of a four-wing chaotic attractor by two weakly-coupled Lorenz systems. Int. J. Bifurc. Chaos 18, 2089–2094 (2008) MATHCrossRefGoogle Scholar
  13. 13.
    Giuseppe, G.: Novel four-wing and eight-wing attractors using coupled chaotic Lorenz systems. Chin. Phys. 17, 3247–3251 (2008) CrossRefGoogle Scholar
  14. 14.
    Matsumoto, T.: A chaotic attractor from Chua’s circuit. IEEE Trans. Circuits Syst. I 31, 1055–1058 (1984) MATHCrossRefGoogle Scholar
  15. 15.
    Yan, L.Z., Jie, Z., Chen, G.R.: Adaptive control of chaotic n-scroll Chua’s circuit. Int. J. Bifurc. Chaos 16, 1089–1096 (2006) MATHCrossRefGoogle Scholar
  16. 16.
    Yalçin, M.E.: Increasing the entropy of a random Number generator using n-scroll chaotic attractors. Int. J. Bifurc. Chaos 17, 4471–4479 (2007) CrossRefGoogle Scholar
  17. 17.
    Suykens, J.A.K., Chua, L.O.: n-double scroll hyper-cubes in 1-D CNNs. Int. J. Bifurc. Chaos. 7, 1873–1885 (1997) Google Scholar
  18. 18.
    Chen, G.R., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9, 1465–1466 (1999) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lü, J.H., Chen, G.R.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 12, 659–661 (2002) MATHCrossRefGoogle Scholar
  20. 20.
    Lü, J.H., Chen, G.R., Cheng, D.Z., Čelikovský, S.: Bridge the gap between the Lorenz system and the Chen system. Int. J. Bifurc. Chaos 12, 2917–2926 (2002) MATHCrossRefGoogle Scholar
  21. 21.
    Zhong, G.: Implementation of Chua’s circuit with a cubic nonlinearity. IEEE Trans. Circuits Syst. I 41, 934–941 (1994) CrossRefGoogle Scholar
  22. 22.
    Tang, W.K.S., Zhong, G.Q., Chen, G.R., Man, K.F.: Generation of n-scroll attractors via sine function. IEEE Trans. Circuits Syst. I 48, 1369–1372 (2001) CrossRefMathSciNetGoogle Scholar
  23. 23.
    Elwakil, A.S., Salama, K.N., Kennedy, M.P.: A system for chaos generation and its implementation in monolithic form. In: Proc. IEEE Int. Symp. Circuits Syst., vol. 5, pp. 217–220 (2000) Google Scholar
  24. 24.
    Čelikovský, S., Chen, G.R.: On the generalized Lorenz canonical form. Chaos Solitons Fractals 26, 1271–1276 (2005) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Baghious, E.H., Jarry, P.: Lorenz attractor from differential equations with piecewise-linear terms. Int. J. Bifurc. Chaos 3, 201–210 (1993) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Elwakil, A.S., Kennedy, M.P.: Construction of classes of circuit-independent chaotic oscillators using passive-only nonlinear devices. IEEE Trans. Circuits Syst. I 48, 289–307 (2001) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Elwakil, A.S., Özoĝuz, S., Kennedy, M.P.: A four-wing butterfly attractor from a fully autonomous system. Int. J. Bifurc. Chaos 13, 3093–3098 (2003) MATHCrossRefGoogle Scholar
  28. 28.
    Liu, W.B., Chen, G.R.: A new chaotic system and its generation. Int. J. Bifurc. Chaos 13, 261–266 (2003) MATHCrossRefGoogle Scholar
  29. 29.
    Rössler, O.E.: An equation for hyperchaos. Phys. Lett. A 71, 155–157 (1979) CrossRefMathSciNetGoogle Scholar
  30. 30.
    Cang, S.J., Chen, Z.Q., Yuan, Z.Z.: Analysis and circuit implementation of a new four-dimensional non-autonomous hyper-chaotic system. Acta Phys. Sin. 57, 1493–1501 (2008) MATHGoogle Scholar
  31. 31.
    Qi, G.Y., Van Wyk, M.A., Van Wyk, B.J., Chen, G.R.: On a new hyperchaotic system. Phys. Lett. A 372, 124–136 (2008) CrossRefMathSciNetGoogle Scholar
  32. 32.
    Mesquita, A., Rempel, E.L., Kienitz, K.H.: Bifurcation analysis of attitude control systems with switching-constrained actuators. Nonlinear Dyn. 51, 207–216 (2008) MATHCrossRefGoogle Scholar
  33. 33.
    Liu, X.L., Han, M.A.: Bifurcation of periodic solutions and invariant tori for a four-dimensional system. Nonlinear Dyn. 57, 75–83 (2009) MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Li, R.H., Xu, W., Li, S.: Chaos control and synchronization of the Φ6-Van der Pol system driven by external and parametric excitations. Nonlinear Dyn. 53, 261–271 (2008) MATHCrossRefGoogle Scholar
  35. 35.
    Woltering, M., Markus, M.: Riddled-like basins of transient chaos. Phys. Rev. Lett. 84, 630–633 (2000) CrossRefGoogle Scholar
  36. 36.
    Dhamala, M., Lai, Y.C., Kostelich, E.J.: Analyses of transient chaotic time series. Phys. Rev. E 61, 056207 (2003) Google Scholar
  37. 37.
    Yorke, J.A., Yorke, E.D.: The transition to sustained chaotic behavior in the Lorenz model. J. Stat. Phys. 21, 263–277 (1979) CrossRefMathSciNetGoogle Scholar
  38. 38.
    Astaf’ev, G.B., Koronovskii, A.A., Hramov, A.E.: Behavior of dynamical systems in the regime of transient chaos. Tech. Phys. Lett. 29, 923–926 (2003) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.F’SATIE and Department of Electrical EngineeringTshwane University of TechnologyPretoriaSouth Africa
  2. 2.Department of Industry DesignTianjin University of Science and TechnologyTianjinP.R. China
  3. 3.Department of AutomationNankai UniversityTianjinP.R. China

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