Four-wing hyperchaotic attractor generated from a new 4D system with one equilibrium and its fractional-order form. (English) Zbl 1245.34054

Summary: 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.


34C28 Complex behavior and chaotic systems of ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34D45 Attractors of solutions to ordinary differential equations
34A08 Fractional ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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[1] Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963) · Zbl 1417.37129
[2] 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)
[3] Nana, B., Woafo, P., Domngang, S.: Chaotic synchronization with experimental application to secure communication. Commun. Nonlinear Sci. Numer. Simul. 14, 629–655 (2009)
[4] Coulon, M., Roviras, D.: Multi-user receivers for synchronous and asynchronous transmission for chaos-based multiple-access systems. Signal Process. 89, 583–598 (2009) · Zbl 1157.94326
[5] Kozic, S., Hasler, M.: Low-density codes based on chaotic systems for simple encoding. IEEE Trans. Circuits Syst. I 56, 405–415 (2009)
[6] Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9, 1465–1466 (1999) · Zbl 0962.37013
[7] Lü, J., Chen, G.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 12, 659–661 (2002) · Zbl 1063.34510
[8] Qi, G., Chen, G., Du, S., Chen, Z., Yuan, Z.: Analysis of a new chaotic system. Physica A 352, 295–308 (2005)
[9] 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)
[10] Liu, C., Liu, L.: A new three-dimensional autonomous chaotic oscillation system. J. Phys. Conf. Ser. 96, 012173 (2008)
[11] 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) · Zbl 1152.37312
[12] Wang, L.: 3-scroll and 4-scroll chaotic attractors generated from a new 3-D quadratic autonomous system. Nonlinear Dyn. 56, 453–462 (2009) · Zbl 1204.70021
[13] 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) · Zbl 1233.37022
[14] Baghious, E., Jarry, P.: Lorenz attractor: From differential equations with piecewise-linear terms. Int. J. Bifurc. Chaos 3, 201–210 (1993) · Zbl 0873.34045
[15] 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) · Zbl 1368.37040
[16] 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)
[17] Qi, G., Chen, G., Li, S., Zhang, Y.: Four-wing attractors: From pseudo to real. Int. J. Bifurc. Chaos 16, 859–885 (2006) · Zbl 1111.37025
[18] 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) · Zbl 1156.37006
[19] Grassi, G.: Novel four-wing and eight-wing attractors using coupled chaotic Lorenz systems. Chin. Phys. B 17, 3247–3251 (2008)
[20] 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) · Zbl 1242.37025
[21] Wang, L.: Yet another 3D quadratic autonomous system generating three-wing and four-wing chaotic attractors. Chaos 19, 013107 (2009) · Zbl 1311.37028
[22] 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) · Zbl 1236.34016
[23] Rössler, O.E.: An equation for hyperchaos. Phys. Lett. A 71, 155–157 (1979) · Zbl 0996.37502
[24] Thamilmaran, K., Lakshmanan, M., Venkatesan, A.: Hyperchaos in a modified canonical Chua’s circuit. Int. J. Bifurc. Chaos 14, 221–243 (2004) · Zbl 1067.94597
[25] Li, Y., Tang, S.K., Chen, G.: Generating hyperchaos via state feedback control. Int. J. Bifurc. Chaos 15, 3367–3375 (2005)
[26] Li, Y., Tang, W.K.S., Chen, G.: Hyperchaos evolved from the generalized Lorenz equation. Int. J. Circuit Theory Appl. 33, 235–251 (2005) · Zbl 1079.34032
[27] 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)
[28] Jia, Q.: Generation and suppression of a new hyperchaotic system with double hyperchaotic attractors. Phys. Lett. A 371, 410–415 (2007) · Zbl 1209.37033
[29] Jia, Q.: Hyperchaos generated from the Lorenz chaotic system and its control. Phys. Lett. A 366, 217–222 (2007) · Zbl 1203.93086
[30] Qi, G., Wyk, M.A., Wyk, B.J., Chen, G.: On a new hyperchaotic system. Phys. Lett. A 372, 124–136 (2008) · Zbl 1217.37032
[31] Liu, L., Liu, C., Zhang, Y.: Analysis of a novel four-dimensional hyperchaotic system. Chin. J. Phys. 46, 386–393 (2008)
[32] 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)
[33] 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) · Zbl 1131.37036
[34] 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) · Zbl 05958109
[35] Zheng, S., Dong, G., Bi, Q.: A new hyperchaotic system and its synchronization. Appl. Math. Comput. 215, 3192–3200 (2010) · Zbl 1188.34051
[36] Mahmoud, G.M., Mahmoud, E.E., Ahmed, M.E.: On the hyperchaotic complex Lü system. Nonlinear Dyn. 58, 725–738 (2009) · Zbl 1183.70053
[37] 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)
[38] Yang, Q., Zhang, K., Chen, G.: Hyperchaotic attractors from a linearly controlled Lorenz system. Nonlinear Anal., Real World Appl. 10, 1601–1617 (2009) · Zbl 1175.37041
[39] 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) · Zbl 1163.65337
[40] Chen, Z., Yang, Y., Qi, G., Yuan, Z.: A novel hyperchaos system only with one equilibrium. Phys. Lett. A 360, 696–701 (2007) · Zbl 1236.37022
[41] Liu, C.: A new hyperchaotic dynamical system. Chin. Phys. 16, 3279–3284 (2007)
[42] 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) · Zbl 1183.70049
[43] Grassi, G., Severance, F.L., Miller, D.A.: Multi-wing hyperchaotic attractors from coupled Lorenz systems. Chaos Solitons Fractals 41, 284–291 (2009) · Zbl 1198.37045
[44] Makris, N., Constantinou, M.C.: Fractional derivative Maxwell model for viscous dampers. J. Struct. Eng. 117, 2708–2724 (1991)
[45] Scalas, E., Gorenflo, R., Mainardi, F.: Fractional calculus and continuous-time finance. Physica A 284, 376–384 (2000)
[46] Cafagna, D.: Fractional calculus: a mathematical tool from the past for present engineers. IEEE Ind. Electron. Mag. (summer), 35–40 (2007)
[47] 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) · Zbl 1220.70025
[48] 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) · Zbl 1158.34300
[49] 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) · Zbl 1147.34302
[50] Daftardar-Gejji, V., Bhalekar, S.: Chaos in fractional ordered Liu system. Comput. Math. Appl. 59, 1117–1127 (2010) · Zbl 1189.34081
[51] Cafagna, D., Grassi, G.: Fractional-order chaos: a novel four-wing attractor in coupled Lorenz systems. Int. J. Bifurc. Chaos 19, 3329–3338 (2009) · Zbl 1182.34003
[52] Cafagna, D., Grassi, G.: Hyperchaos in the fractional-order Rössler system with lowest order. Int. J. Bifurc. Chaos 19, 339–347 (2009) · Zbl 1170.34327
[53] Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional order. Electron. Trans. Numer. Anal. 5, 1–6 (1997) · Zbl 0890.65071
[54] 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) · Zbl 1009.65049
[55] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002) · Zbl 1014.34003
[56] 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) · Zbl 0825.58027
[57] 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) · Zbl 1148.65094
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