Abstract
In this paper, a new simple 4D smooth autonomous system is proposed, which illustrates two interesting rare phenomena: first, this system can generate a four-wing hyperchaotic and a four-wing chaotic attractor and second, this generation occurs under condition that the system has only one equilibrium point at the origin. The dynamic analysis approach in the paper involves time series, phase portraits, Lyapunov exponents, bifurcation diagram, and Poincaré maps, to investigate some basic dynamical behaviors of the proposed 4D system. The physical existence of the four-wing hyperchaotic attractor is verified by an electronic circuit. Finally, it is shown that the fractional-order form of the system can also generate a chaotic four-wing attractor.
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 nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)
Chakravorty, J., Banerjee, T., Ghatak, R., Bose, A., Sarkar, B.C.: Generating chaos in injection-synchronized Gunn oscillator: an experimental approach. IETE J. Res. 55, 106–111 (2009)
Nana, B., Woafo, P., Domngang, S.: Chaotic synchronization with experimental application to secure communication. Commun. Nonlinear Sci. Numer. Simul. 14, 629–655 (2009)
Coulon, M., Roviras, D.: Multi-user receivers for synchronous and asynchronous transmission for chaos-based multiple-access systems. Signal Process. 89, 583–598 (2009)
Kozic, S., Hasler, M.: Low-density codes based on chaotic systems for simple encoding. IEEE Trans. Circuits Syst. I 56, 405–415 (2009)
Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9, 1465–1466 (1999)
Lü, J., Chen, G.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 12, 659–661 (2002)
Qi, G., Chen, G., Du, S., Chen, Z., Yuan, Z.: Analysis of a new chaotic system. Physica A 352, 295–308 (2005)
Wang, G.Y., Qui, S.S., Li, H.W., Li, C.F., Zheng, Y.: A new chaotic system and its circuit realization. Chin. Phys. 15, 2872–2877 (2006)
Liu, C., Liu, L.: A new three-dimensional autonomous chaotic oscillation system. J. Phys. Conf. Ser. 96, 012173 (2008)
Chen, Z., Yang, Y., Yuan, 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)
Wang, L.: 3-scroll and 4-scroll chaotic attractors generated from a new 3-D quadratic autonomous system. Nonlinear Dyn. 56, 453–462 (2009)
Dadras, S., Momeni, H.R.: A novel three-dimensional autonomous chaotic system generating two-, three- and four-scroll attractors. Phys. Lett. A 373, 3637–3642 (2009)
Baghious, E., Jarry, P.: Lorenz attractor: From differential equations with piecewise-linear terms. Int. J. Bifurc. Chaos 3, 201–210 (1993)
Elwakil, A., Ozoguz, S., Kennedy, M.: Creation of a complex butterfly attractor using a novel Lorenz-type system. IEEE Trans. Circuits Syst. I 49, 527–530 (2002)
Ozoguz, S., Elwakil, A., Kennedy, M.: Experimental verification of the butterfly attractor in a modified Lorenz system. Int. J. Bifurc. Chaos 12, 1627–1632 (2002)
Qi, G., Chen, G., Li, S., Zhang, Y.: Four-wing attractors: From pseudo to real. Int. J. Bifurc. Chaos 16, 859–885 (2006)
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)
Grassi, G.: Novel four-wing and eight-wing attractors using coupled chaotic Lorenz systems. Chin. Phys. B 17, 3247–3251 (2008)
Dadras, S., Momeni, H.R., Qi, G.: Analysis of a new 3D smooth autonomous system with different wing chaotic attractors and transient chaos. Nonlinear Dyn. 62, 391–405 (2010)
Wang, L.: Yet another 3D quadratic autonomous system generating three-wing and four-wing chaotic attractors. Chaos 19, 013107 (2009)
Dadras, S., Momeni, H.R.: Generating one-, two-, three- and four-scroll attractors from a novel four-dimensional smooth autonomous chaotic system. Chin. Phys. B 19, 060506 (2010)
Rössler, O.E.: An equation for hyperchaos. Phys. Lett. A 71, 155–157 (1979)
Thamilmaran, K., Lakshmanan, M., Venkatesan, A.: Hyperchaos in a modified canonical Chua’s circuit. Int. J. Bifurc. Chaos 14, 221–243 (2004)
Li, Y., Tang, S.K., Chen, G.: Generating hyperchaos via state feedback control. Int. J. Bifurc. Chaos 15, 3367–3375 (2005)
Li, Y., Tang, W.K.S., Chen, G.: Hyperchaos evolved from the generalized Lorenz equation. Int. J. Circuit Theory Appl. 33, 235–251 (2005)
Wang, J.Z., Chen, Z.Q., Yuan, Z.Z.: The generation of a hyperchaotic system based on a three-dimensional autonomous chaotic system. Chin. Phys. 15, 1216–1225 (2006)
Jia, Q.: Generation and suppression of a new hyperchaotic system with double hyperchaotic attractors. Phys. Lett. A 371, 410–415 (2007)
Jia, Q.: Hyperchaos generated from the Lorenz chaotic system and its control. Phys. Lett. A 366, 217–222 (2007)
Qi, G., Wyk, M.A., Wyk, B.J., Chen, G.: On a new hyperchaotic system. Phys. Lett. A 372, 124–136 (2008)
Liu, L., Liu, C., Zhang, Y.: Analysis of a novel four-dimensional hyperchaotic system. Chin. J. Phys. 46, 386–393 (2008)
Wu, W.J., Chan, Z.Q., Yuan, Z.Z.: Local bifurcation analysis of a four-dimensional hyperchaotic system. Chin. Phys. B 17, 2420–2432 (2008)
Mahmoud, G.M., Al-Kashif, M.A., Farghaly, A.A.: Chaotic and hyperchaotic attractors of a complex nonlinear system. J. Phys. A, Math. Theor. 41, 055104 (2008)
Yujun, N., Xingyuan, W., Mingjun, W., Huaguang, Z.: A new hyperchaotic system and its circuit implementation. Commun. Nonlinear Sci. Numer. Simul. 15, 3518–3524 (2010)
Zheng, S., Dong, G., Bi, Q.: A new hyperchaotic system and its synchronization. Appl. Math. Comput. 215, 3192–3200 (2010)
Mahmoud, G.M., Mahmoud, E.E., Ahmed, M.E.: On the hyperchaotic complex Lü system. Nonlinear Dyn. 58, 725–738 (2009)
Qi, G., Wyk, M.A., Wyk, B.J., Chen, G.: A new hyperchaotic system and its circuit implementation. Chaos Solitons Fractals 40, 2544–2549 (2009)
Yang, Q., Zhang, K., Chen, G.: Hyperchaotic attractors from a linearly controlled Lorenz system. Nonlinear Anal., Real World Appl. 10, 1601–1617 (2009)
Chen, C.H., Sheu, L.J., Chen, H.K., Chen, J.H., Wang, H.C., Chao, Y.C., Lin, Y.K.: A new hyper-chaotic system and its synchronization. Nonlinear Anal., Real World Appl. 10, 2088–2096 (2009)
Chen, Z., Yang, Y., Qi, G., Yuan, Z.: A novel hyperchaos system only with one equilibrium. Phys. Lett. A 360, 696–701 (2007)
Liu, C.: A new hyperchaotic dynamical system. Chin. Phys. 16, 3279–3284 (2007)
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)
Grassi, G., Severance, F.L., Miller, D.A.: Multi-wing hyperchaotic attractors from coupled Lorenz systems. Chaos Solitons Fractals 41, 284–291 (2009)
Makris, N., Constantinou, M.C.: Fractional derivative Maxwell model for viscous dampers. J. Struct. Eng. 117, 2708–2724 (1991)
Scalas, E., Gorenflo, R., Mainardi, F.: Fractional calculus and continuous-time finance. Physica A 284, 376–384 (2000)
Cafagna, D.: Fractional calculus: a mathematical tool from the past for present engineers. IEEE Ind. Electron. Mag. (summer), 35–40 (2007)
Tavazoei, M.S., Haeri, M., Bolouki, S., Siami, M.: Using fractional-order integrator to control chaos in single-input chaotic system. Nonlinear Dyn. 55, 179–190 (2009)
Cafagna, D., Grassi, G.: Bifurcation and chaos in the fractional-order Chen system via a time-domain approach. Int. J. Bifurc. Chaos 18, 1845–1863 (2008)
Cafagna, D., Grassi, G.: Fractional-order Chua’s circuit: time domain analysis, bifurcation, chaotic behavior and test for chaos. Int. J. Bifurc. Chaos 18, 615–639 (2008)
Daftardar-Gejji, V., Bhalekar, S.: Chaos in fractional ordered Liu system. Comput. Math. Appl. 59, 1117–1127 (2010)
Cafagna, D., Grassi, G.: Fractional-order chaos: a novel four-wing attractor in coupled Lorenz systems. Int. J. Bifurc. Chaos 19, 3329–3338 (2009)
Cafagna, D., Grassi, G.: Hyperchaos in the fractional-order Rössler system with lowest order. Int. J. Bifurc. Chaos 19, 339–347 (2009)
Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional order. Electron. Trans. Numer. Anal. 5, 1–6 (1997)
Diethelm, K., Ford, N.J., Freed, A.D.: A predictor–corrector approach for the numerical solution for fractional differential equations. Nonlinear Dyn. 29, 3–22 (2002)
Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)
Charef, A., Sun, H.H., Tsao, Y.Y., Onaral, B.: Fractal systems as represented by singularity function. IEEE Trans. Autom. Control 37, 1465–1470 (1992)
Tavazoei, M.S., Haeri, M.: Limitation of frequency domain approximation for detecting chaos in fractional-order system. Nonlinear Anal. Theory Methods Appl. 69, 1299–1320 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dadras, S., Momeni, H.R., Qi, G. et al. Four-wing hyperchaotic attractor generated from a new 4D system with one equilibrium and its fractional-order form. Nonlinear Dyn 67, 1161–1173 (2012). https://doi.org/10.1007/s11071-011-0060-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-011-0060-0