Nonlinear Dynamics

, Volume 57, Issue 3, pp 363–373 | Cite as

Describing function based methods for predicting chaos in a class of fractional order differential equations

Original Paper


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.


Chaos Fractional order differential equation Describing function 


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  1. 1.
    Westerlund, S.: Dead matter has memory! Phys. Scr. 43(2), 174–179 (1991) CrossRefGoogle Scholar
  2. 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) CrossRefGoogle Scholar
  3. 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) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Engheta, N.: On fractional calculus and fractional multipoles in electromagnetism. IEEE Trans. Antennas Propag. 44(4), 554–566 (1996) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Podlubny, I.: Fractional order systems and PIλ Dμ-controllers. IEEE Trans. Autom. Control 44(1), 208–214 (1999) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Oustaloup, A., Sabatier, J., Lanusse, P.: From fractal robustness to CRONE control. Fract. Calc. Appl. Anal. 2(1), 1–30 (1999) MATHMathSciNetGoogle Scholar
  7. 7.
    Oustaloup, A., Moreau, X., Nouillant, M.: The CRONE suspension. Control Eng. Pract. 4(8), 1101–1108 (1996) CrossRefGoogle Scholar
  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) CrossRefGoogle Scholar
  9. 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) CrossRefGoogle Scholar
  10. 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) CrossRefGoogle Scholar
  11. 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) CrossRefGoogle Scholar
  12. 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) MATHCrossRefGoogle Scholar
  13. 13.
    Ahmad, W., El-Khazali, R., El-Wakil, A.: Fractional-order Wien-bridge oscillator. Electr. Lett. 37, 1110–1112 (2001) CrossRefGoogle Scholar
  14. 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) CrossRefMathSciNetGoogle Scholar
  15. 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) CrossRefGoogle Scholar
  16. 16.
    Deng, W., Lü, J.: Design of multidirectional multiscroll chaotic attractors based on fractional differential systems via switching control. Chaos 16, 043120 (2006) CrossRefMathSciNetGoogle Scholar
  17. 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) CrossRefGoogle Scholar
  18. 18.
    Arena, P., Fortuna, L., Porto, D.: Chaotic behavior in noninteger-order cellular neural networks. Phys. Rev. E 61(1), 776–781 (2000) CrossRefGoogle Scholar
  19. 19.
    Seredynska, M., Hanyga, A.: A nonlinear differential equation of fractional order with chaotic properties. Int. J. Bifurc. Chaos 14(4), 1291–1304 (2004) MATHCrossRefMathSciNetGoogle Scholar
  20. 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) CrossRefGoogle Scholar
  21. 21.
    Li, C.P., Peng, G.J.: Chaos in Chen’s system with a fractional order. Chaos Solitons Fractals 22(2), 443–450 (2004) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Li, C.P., Deng, W.H., Chen, G.: Scaling attractors of fractional differential systems. Fractals 14(4), 303–314 (2006) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Li, C.P., Deng, W.H.: Chaos synchronization of fractional-order differential system. Int. J. Mod. Phys. B 20(7), 791–803 (2006) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Deng, W.H.: Generalized synchronization in fractional order systems. Phys. Rev. E 75, 0565201-1–0565201-7 (2007) CrossRefGoogle Scholar
  25. 25.
    Linz, S.J.: No-chaos criteria for certain classes of driven nonlinear oscillators. Acta Phys. Pol. B 34(7), 3741–3749 (2003) Google Scholar
  26. 26.
    Ciesielski, K.: On the Poincare–Bendixson theorem. In: Lecture Notes in Nonlinear Analysis, vol. 3, pp. 49–69 (2002) Google Scholar
  27. 27.
    Silva, C.P.: Shil’nikov’s theorem—A tutorial. IEEE Trans. Circuits Syst. I 40, 675–682 (1993) MATHCrossRefGoogle Scholar
  28. 28.
    Tavazoei, M.S., Haeri, M.: Chaotic attractors in incommensurate fractional order systems. Physica D 237(20), 2628–2637 (2008) MATHGoogle Scholar
  29. 29.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) MATHGoogle Scholar
  30. 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) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Diethelm, D., Ford, N.J.: Multi-order fractional differential equations and their numerical solution. Appl. Math. Comput. 154(3), 621–640 (2004) MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Li, C., Deng, W.: Remarks on fractional derivatives. Appl. Math. Comput. 187, 777–784 (2007) MATHCrossRefMathSciNetGoogle Scholar
  33. 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) MATHCrossRefMathSciNetGoogle Scholar
  34. 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) MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Deng, W.H.: Numerical algorithm for the time fractional Fokker–Planck equation. J. Comput. Phys. 227, 1510–1522 (2007) MATHCrossRefMathSciNetGoogle Scholar
  36. 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) CrossRefMathSciNetGoogle Scholar
  37. 37.
    Mees, A.I.: Dynamics of Feedback Systems. Wiley, New York (1981) MATHGoogle Scholar
  38. 38.
    Bonnet, C., Partington, J.R.: Coprime factorizations and stability of fractional differential systems. Syst. Control Lett. 41, 167–174 (2000) MATHCrossRefMathSciNetGoogle Scholar
  39. 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) CrossRefGoogle Scholar
  40. 40.
    Genesio, R., Tesi, A.: Chaos prediction in nonlinear feedback systems. IEE Proc. D 138, 313–320 (1991) MATHGoogle Scholar
  41. 41.
    Genesio, R., Tesi, A.: Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica 28(3), 531–548 (1992) MATHCrossRefGoogle Scholar
  42. 42.
    Genesio, R., Tesi, A.: A harmonic balance approach for chaos prediction: Chua’s Circuit. Int. J. Bifurc. Chaos 2(1), 61–79 (1992) MATHCrossRefMathSciNetGoogle Scholar
  43. 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) MATHCrossRefGoogle Scholar
  44. 44.
    Savaci, F.A., Gunel, S.: Harmonic balance analysis of the generalized Chua’s circuit. Int. J. Bifurc. Chaos 16(8), 2325–2332 (2006) CrossRefMathSciNetGoogle Scholar
  45. 45.
    Gelb, A., Velde, W.E.V.: Multiple-Input Describing Functions and Nonlinear System Design. McGraw Hill, New York (1967) Google Scholar
  46. 46.
    Mees, A.I.: Limit cycle stability. IMA J. Appl. Math. 11(3), 281–295 (1973) MATHCrossRefMathSciNetGoogle Scholar
  47. 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) CrossRefGoogle Scholar
  48. 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) Google Scholar
  49. 49.
    Hirai, K.: Analysis of bifurcation and chaos by describing function method. Chaos Memorial Symposium in Asuka, pp. 13–18 (1997) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Advanced Control System Lab, Electrical Engineering DepartmentSharif University of TechnologyTehranIran

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