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.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Lorenz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141 (1963)
Weiss, N., Garfinkel, A., Spano, M.L., Ditto, W.L.: Chaos and chaos control in biology. J. Clin. Invest. 93, 1355–1360 (1994)
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)
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)
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)
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)
Lü, J.H., Chen, G.R.: Generating multi-scroll chaotic attractors: theories, methods and applications. Int. J. Bifurc. Chaos 16, 775–858 (2006)
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)
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)
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)
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)
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)
Giuseppe, G.: Novel four-wing and eight-wing attractors using coupled chaotic Lorenz systems. Chin. Phys. 17, 3247–3251 (2008)
Matsumoto, T.: A chaotic attractor from Chua’s circuit. IEEE Trans. Circuits Syst. I 31, 1055–1058 (1984)
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)
Yalçin, M.E.: Increasing the entropy of a random Number generator using n-scroll chaotic attractors. Int. J. Bifurc. Chaos 17, 4471–4479 (2007)
Suykens, J.A.K., Chua, L.O.: n-double scroll hyper-cubes in 1-D CNNs. Int. J. Bifurc. Chaos. 7, 1873–1885 (1997)
Chen, G.R., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9, 1465–1466 (1999)
Lü, J.H., Chen, G.R.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 12, 659–661 (2002)
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)
Zhong, G.: Implementation of Chua’s circuit with a cubic nonlinearity. IEEE Trans. Circuits Syst. I 41, 934–941 (1994)
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)
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)
Čelikovský, S., Chen, G.R.: On the generalized Lorenz canonical form. Chaos Solitons Fractals 26, 1271–1276 (2005)
Baghious, E.H., Jarry, P.: Lorenz attractor from differential equations with piecewise-linear terms. Int. J. Bifurc. Chaos 3, 201–210 (1993)
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)
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)
Liu, W.B., Chen, G.R.: A new chaotic system and its generation. Int. J. Bifurc. Chaos 13, 261–266 (2003)
Rössler, O.E.: An equation for hyperchaos. Phys. Lett. A 71, 155–157 (1979)
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)
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)
Mesquita, A., Rempel, E.L., Kienitz, K.H.: Bifurcation analysis of attitude control systems with switching-constrained actuators. Nonlinear Dyn. 51, 207–216 (2008)
Liu, X.L., Han, M.A.: Bifurcation of periodic solutions and invariant tori for a four-dimensional system. Nonlinear Dyn. 57, 75–83 (2009)
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)
Woltering, M., Markus, M.: Riddled-like basins of transient chaos. Phys. Rev. Lett. 84, 630–633 (2000)
Dhamala, M., Lai, Y.C., Kostelich, E.J.: Analyses of transient chaotic time series. Phys. Rev. E 61, 056207 (2003)
Yorke, J.A., Yorke, E.D.: The transition to sustained chaotic behavior in the Lorenz model. J. Stat. Phys. 21, 263–277 (1979)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cang, S., Qi, G. & Chen, Z. A four-wing hyper-chaotic attractor and transient chaos generated from a new 4-D quadratic autonomous system. Nonlinear Dyn 59, 515–527 (2010). https://doi.org/10.1007/s11071-009-9558-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-009-9558-0