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Describing function based methods for predicting chaos in a class of fractional order differential equations. (English) Zbl 1176.34051
Summary: This paper deals with two different methods for predicting chaotic dynamics in fractional order differential equations. These methods, which have been previously proposed for detecting chaos in classical integer order systems, are based on using the describing function method. One of these methods is constructed based on Genesio-Tesi conjecture for existence of chaos, and another method is introduced based on Hirai conjecture about occurrence of chaos in a nonlinear system. These methods are restated to use in predicting chaos in a fractional order differential equation of the order between 2 and 3. Numerical simulation results are presented to show the ability of these methods to detect chaos in two fractional order differential equations with quadratic and cubic nonlinearities.

MSC:
34C28 Complex behavior and chaotic systems of ordinary differential equations
26A33 Fractional derivatives and integrals
Software:
FODE
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[1] Westerlund, S.: Dead matter has memory! Phys. Scr. 43(2), 174–179 (1991) · doi:10.1088/0031-8949/43/2/011
[2] Bagley, R.L., Calico, R.A.: Fractional order state equations for the control of visco-elastically damped structures. J. Guid. Control and Dyn. 14, 304–311 (1991) · doi:10.2514/3.20641
[3] Rossikhin, Y.A., Shitikova, M.V.: Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass system. Acta Mech. 120, 109–125 (1997) · Zbl 0901.73030 · doi:10.1007/BF01174319
[4] Engheta, N.: On fractional calculus and fractional multipoles in electromagnetism. IEEE Trans. Antennas Propag. 44(4), 554–566 (1996) · Zbl 0944.78506 · doi:10.1109/8.489308
[5] Podlubny, I.: Fractional order systems and PI \(\lambda\) D \(\mu\) -controllers. IEEE Trans. Autom. Control 44(1), 208–214 (1999) · Zbl 1056.93542 · doi:10.1109/9.739144
[6] Oustaloup, A., Sabatier, J., Lanusse, P.: From fractal robustness to CRONE control. Fract. Calc. Appl. Anal. 2(1), 1–30 (1999) · Zbl 1111.93310
[7] Oustaloup, A., Moreau, X., Nouillant, M.: The CRONE suspension. Control Eng. Pract. 4(8), 1101–1108 (1996) · doi:10.1016/0967-0661(96)00109-8
[8] Calderon, A.J., Vinagre, B.M., Feliu-Batlle, V.: Fractional-order control strategies for power electronic buck converters. Signal Process. 86, 2803–2819 (2006) · Zbl 1172.94377 · doi:10.1016/j.sigpro.2006.02.022
[9] Feliu-Batlle, V., Rivas Perez, R., Sanchez Rodriguez, L.: Fractional robust control of main irrigation canals with variable dynamic parameters. Control Eng. Pract. 15, 673–686 (2007) · doi:10.1016/j.conengprac.2006.11.018
[10] Tavazoei, M.S., Haeri, M., Jafari, S.: Fractional controller to stabilize fixed points of uncertain chaotic systems: Theoretical and experimental study. J. Syst. Control Eng. Part I 222, 175–184 (2008) · doi:10.1243/09596518JSCE481
[11] Wang, Y., Li, C.: Does the fractional Brusselator with efficient dimension less than 1 have a limit cycle? Phys. Lett. A 363, 414–419 (2007) · doi:10.1016/j.physleta.2006.11.038
[12] Barbosa, R.S., Machado, J.A.T., Vingare, B.M., Calderon, A.J.: Analysis of the Van der Pol oscillator containing derivatives of fractional order. J. Vib. Control 13(9–10), 1291–1301 (2007) · Zbl 1158.70009 · doi:10.1177/1077546307077463
[13] Ahmad, W., El-Khazali, R., El-Wakil, A.: Fractional-order Wien-bridge oscillator. Electr. Lett. 37, 1110–1112 (2001) · doi:10.1049/el:20010756
[14] Tavazoei, M.S., Haeri, M.: Regular oscillations or chaos in a fractional order system with any effective dimension. Nonlinear Dyn. 54(3), 213–222 (2008) · Zbl 1187.70043 · doi:10.1007/s11071-007-9323-1
[15] Hartley, T.T., Lorenzo, C.F., Qammer, H.K.: Chaos in a fractional order Chua’s system. IEEE Trans. Circuits Syst. I 42, 485–490 (1995) · doi:10.1109/81.404062
[16] Deng, W., Lü, J.: Design of multidirectional multiscroll chaotic attractors based on fractional differential systems via switching control. Chaos 16, 043120 (2006) · Zbl 1146.37316 · doi:10.1063/1.2401061
[17] Tavazoei, M.S., Haeri, M.: A necessary condition for double scroll attractor existence in fractional order systems. Phys. Lett. A 367(1–2), 102–113 (2007) · Zbl 1209.37037 · doi:10.1016/j.physleta.2007.05.081
[18] Arena, P., Fortuna, L., Porto, D.: Chaotic behavior in noninteger-order cellular neural networks. Phys. Rev. E 61(1), 776–781 (2000) · doi:10.1103/PhysRevE.61.776
[19] Seredynska, M., Hanyga, A.: A nonlinear differential equation of fractional order with chaotic properties. Int. J. Bifurc. Chaos 14(4), 1291–1304 (2004) · Zbl 1062.34002 · doi:10.1142/S0218127404009818
[20] Wu, Z.M., Lu, J.G., Xie, J.Y.: Analysing chaos in fractional-order systems with the harmonic balance method. Chin. Phys. 15(6), 1201–1207 (2006) · doi:10.1088/1009-1963/15/6/013
[21] Li, C.P., Peng, G.J.: Chaos in Chen’s system with a fractional order. Chaos Solitons Fractals 22(2), 443–450 (2004) · Zbl 1060.37026 · doi:10.1016/j.chaos.2004.02.013
[22] Li, C.P., Deng, W.H., Chen, G.: Scaling attractors of fractional differential systems. Fractals 14(4), 303–314 (2006) · Zbl 1147.34042 · doi:10.1142/S0218348X06003337
[23] Li, C.P., Deng, W.H.: Chaos synchronization of fractional-order differential system. Int. J. Mod. Phys. B 20(7), 791–803 (2006) · Zbl 1101.37025 · doi:10.1142/S0217979206033620
[24] Deng, W.H.: Generalized synchronization in fractional order systems. Phys. Rev. E 75, 0565201-1–0565201-7 (2007) · doi:10.1103/PhysRevE.75.056201
[25] Linz, S.J.: No-chaos criteria for certain classes of driven nonlinear oscillators. Acta Phys. Pol. B 34(7), 3741–3749 (2003)
[26] Ciesielski, K.: On the Poincare–Bendixson theorem. In: Lecture Notes in Nonlinear Analysis, vol. 3, pp. 49–69 (2002) · Zbl 1097.34501
[27] Silva, C.P.: Shil’nikov’s theorem–A tutorial. IEEE Trans. Circuits Syst. I 40, 675–682 (1993) · Zbl 0850.93352 · doi:10.1109/81.246142
[28] Tavazoei, M.S., Haeri, M.: Chaotic attractors in incommensurate fractional order systems. Physica D 237(20), 2628–2637 (2008) · Zbl 1157.26310 · doi:10.1016/j.physd.2008.03.037
[29] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[30] Diethelm, K., Ford, N.J., Freed, A.D.: A predictor–corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3–22 (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341
[31] Diethelm, D., Ford, N.J.: Multi-order fractional differential equations and their numerical solution. Appl. Math. Comput. 154(3), 621–640 (2004) · Zbl 1060.65070 · doi:10.1016/S0096-3003(03)00739-2
[32] Li, C., Deng, W.: Remarks on fractional derivatives. Appl. Math. Comput. 187, 777–784 (2007) · Zbl 1125.26009 · doi:10.1016/j.amc.2006.08.163
[33] Nimmo, S., Evans, A.K.: The effects of continuously varying the fractional differential order of a chaotic nonlinear system. Chaos Solitons Fractals 10, 1111–1118 (1999) · Zbl 0980.34032 · doi:10.1016/S0960-0779(98)00088-5
[34] Deng, W.H.: Short memory principle and a predictor–corrector approach for fractional differential equations. J. Comput. Appl. Math. 206(1), 174–188 (2007) · Zbl 1121.65128 · doi:10.1016/j.cam.2006.06.008
[35] Deng, W.H.: Numerical algorithm for the time fractional Fokker–Planck equation. J. Comput. Phys. 227, 1510–1522 (2007) · Zbl 1388.35095 · doi:10.1016/j.jcp.2007.09.015
[36] Tavazoei, M.S., Haeri, M.: Unreliability of frequency-domain approximation in recognizing chaos in fractional-order systems. IET Signal Process. 1(4), 171–181 (2007) · doi:10.1049/iet-spr:20070053
[37] Mees, A.I.: Dynamics of Feedback Systems. Wiley, New York (1981) · Zbl 0454.93003
[38] Bonnet, C., Partington, J.R.: Coprime factorizations and stability of fractional differential systems. Syst. Control Lett. 41, 167–174 (2000) · Zbl 0985.93048 · doi:10.1016/S0167-6911(00)00050-5
[39] Tavazoei, M.S., Haeri, M., Jafari, S., Bolouki, S., Siami, M.: Some applications of fractional calculus in suppression of chaotic oscillations. IEEE Trans. Ind. Electron. 55(11), 4094–4101 (2008) · doi:10.1109/TIE.2008.925774
[40] Genesio, R., Tesi, A.: Chaos prediction in nonlinear feedback systems. IEE Proc. D 138, 313–320 (1991) · Zbl 0754.93024
[41] Genesio, R., Tesi, A.: Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica 28(3), 531–548 (1992) · Zbl 0765.93030 · doi:10.1016/0005-1098(92)90177-H
[42] Genesio, R., Tesi, A.: A harmonic balance approach for chaos prediction: Chua’s Circuit. Int. J. Bifurc. Chaos 2(1), 61–79 (1992) · Zbl 0874.94042 · doi:10.1142/S0218127492000070
[43] Genesio, R., Tesi, A., Villoresi, F.: A frequency approach for analyzing and controlling chaos in nonlinear circuits. IEEE Trans. Circuits Syst. I 40(11), 819–828 (1993) · Zbl 0848.93029 · doi:10.1109/81.251820
[44] Savaci, F.A., Gunel, S.: Harmonic balance analysis of the generalized Chua’s circuit. Int. J. Bifurc. Chaos 16(8), 2325–2332 (2006) · Zbl 1192.93048 · doi:10.1142/S0218127406016082
[45] Gelb, A., Velde, W.E.V.: Multiple-Input Describing Functions and Nonlinear System Design. McGraw Hill, New York (1967) · Zbl 0177.12602
[46] Mees, A.I.: Limit cycle stability. IMA J. Appl. Math. 11(3), 281–295 (1973) · Zbl 0263.93042 · doi:10.1093/imamat/11.3.281
[47] Hirai, K.: A simple criterion for the occurrence of chaos in nonlinear feedback systems. Electron. Commun. Jpn. Part 3 82(2), 11–19 (1999) · doi:10.1002/(SICI)1520-6440(199902)82:2<11::AID-ECJC2>3.0.CO;2-X
[48] Hirai, K.: A simple criterion for the occurrence of chaos. In: Proc. Int. Conf. on Nonlinearity, Bifurcation and Chaos 96 (ICNBC 96), Lodz, Poland, pp. 133–136 (1996)
[49] Hirai, K.: Analysis of bifurcation and chaos by describing function method. Chaos Memorial Symposium in Asuka, pp. 13–18 (1997)
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