Chaotic attractors in incommensurate fractional order systems

Physica D: Nonlinear Phenomena

Volumn 237, Issue 20, 2008, Pages 2628-2637

  Tavazoei, Mohammad Saleh ^a   Haeri, Mohammad ^a  

^a SHARIF UNIVERSITY OF TECHNOLOGY   (Iran)

Author keywords

Chaos; Chaotic attractor; Fractional order system; Incommensurate order systems

Indexed keywords

CHAOS THEORY; CHAOTIC SYSTEMS; DIFFERENTIAL EQUATIONS;

CHAOTIC ATTRACTORS; FRACTIONAL DIFFERENTIAL EQUATIONS; FRACTIONAL-ORDER SYSTEMS; INCOMMENSURATE ORDERING; MULTISCROLL CHAOTIC ATTRACTOR; NUMERICAL RESULTS; STABILITY THEOREMS;

ALGEBRA;

EID: 51249088596     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2008.03.037     Document Type: Article

References (46)

1. Bagley R.L., and Calico R.A. Fractional order state equations for the control of visco-elastically damped structures. J. Guid. Control Dyn. 14 (1991) 304-311
2. Rossikhin Y.A., and Shitikova M.V. Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass system. Acta Mech. 120 (1997) 109-125
3. Engheta N. On fractional calculus and fractional multipoles in electromagnetism. IEEE Trans. Antennas Propag. 44 4 (1996) 554-566
4. Arkhincheev V.E. Anomalous diffusion in inhomogeneous media: Some exact results. Modeling Meas. Control A 26 2 (1993) 11-29
5. El-Sayed A.M.A. Fractional order diffusion wave equation. Int. J. Theoret. Phys. 35 2 (1996) 311-322
6. Le Méhauté A., and Crepy G. Introduction to transfer and motion in fractal media: The geometry of kinetics. Solid State Ion. 9-10 (1983) 17-30
7. Nakagawa M., and Sorimachi K. Basic characteristics of a fractance device. IEICE Trans. Fundam. E75-A 12 (1992) 1814-1819
8. Oldham K.B., and Zoski C.G. Analogue instrumentation for processing polarographic data. J. Electroanal. Chem. 157 (1983) 27-51
9. Chen G., and Friedman G. An RLC interconnect model based on Fourier analysis. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 24 2 (2005) 170-183
10. Jenson V.G., and Jeffreys G.V. Mathematical Methods in Chemical Engineering. 2nd ed. (1977), Academic Press, New York
11. K.S. Cole, Electric conductance of biological systems, in: Proc. Cold Spring Harbor Symp. Quant. Biol., Cold Spring Harbor, New York, 1993, pp. 107-116
12. Laskin N. Fractional market dynamics. Physica A 287 (2000) 482-492
13. In: Hilfer R. (Ed). Applications of Fractional Calculus in Physics (2000), World Scientific Pub. Co., Singapore
14. Podlubny I. Fractional Differential Equations (1999), Academic Press, New York
15. Ahmad W., El-Khazali R., and El-Wakil A. Fractional order Wien bridge oscillator. Electron. Lett. 37 (2001) 1110-1112
16. S. Ramiro, J.A. Barbosa, T. Machado, M.I. Ferreira, K.J. Tar, Dynamics of the fractional order Van der Pol oscillator, in: Proceedings of the 2nd IEEE International Conference on Computational Cybernetics, ICCC'04, Aug. 30-Sep. 1, Vienna University of Technology, Austria, 2004, pp. 373-378
17. Wang Y., and Li C. Does the fractional Brusselator with efficient dimension less than 1 have a limit cycle?. Phys. Lett. A 363 (2007) 414-419
18. Hartley T.T., Lorenzo C.F., and Qammer H.K. Chaos in a fractional order Chua's system. IEEE Trans. Circuits Syst. I 42 (1995) 485-490
19. P. Arena, R. Caponetto, L. Fortuna, D. Porto, Chaos in a fractional order Duffing system, in: Proceedings ECCTD, Budapest, Hungry, 1997, pp. 1259-1262
20. Ahmad W.M., and Sprott J.C. Chaos in fractional order autonomous nonlinear systems. Chaos Solitons Fractals 16 (2003) 339-351
21. Grigorenko I., and Grigorenko E. Chaotic dynamics of the fractional Lorenz system. Phys. Rev. Lett. 91 (2003) 034101
22. Li C., and Chen G. Chaos in the fractional order Chen system and its control. Chaos Solitons Fractals 22 3 (2004) 549-554
23. Lu J.G. Chaotic dynamics of the fractional order Lü system and its synchronization. Phys. Lett. A 354 4 (2006) 305-311
24. Li C., and Chen G. Chaos and hyperchaos in the fractional order Rössler equations. Physica A 341 (2004) 55-61
25. Lu J.G. Chaotic dynamics and synchronization of fractional order Arneodo's systems. Chaos Solitons Fractals 26 4 (2005) 1125-1133
26. Sheu L.J., Chen H.K., Chen J.H., Tam L.M., Chen W.C., Lin K.T., and Kang Y. Chaos in the Newton-Leipnik system with fractional order. Chaos Solitons Fractals 36 (2008) 98-103
27. Lu J.G. Chaotic dynamics and synchronization of fractional order Genesio-Tesi systems. Chinese J. Phys. 14 (2005) 1517-1521
28. Lu J.G. Chaotic dynamics of the fractional order Ikeda delay system and its synchronization. Chinese J. Phys. 15 (2006) 301-305
29. Arena P., Caponetto R., Fortuna L., and Porto D. Bifurcation and chaos in noninteger order cellular neural networks. Internat. J. Bifur. Chaos 8 7 (1998) 1527-1539
30. Tavazoei M.S., and Haeri M. A necessary condition for double scroll attractor existence in fractional order systems. Phys. Lett. A 367 1-2 (2007) 102-113
31. Tavazoei M.S., and Haeri M. Unreliability of frequency-domain approximation in recognizing chaos in fractional-order systems. IET Signal Process. 1 4 (2007) 171-181
32. D. Matignon, Stability results for fractional differential equations with applications to control processing, in: Computational Engineering in Systems and Application multi-conference, vol. 2, IMACS, IEEE-SMC Proceedings, Lille, France, July 1996, pp. 963-968
33. Ahmed E., El-Sayed A.M.A., and El-Saka H.A.A. Equilibrium points, stability and numerical solutions of fractional order predator-prey and rabies models. J. Math. Anal. Appl. 325 (2007) 542-553
34. Deng W., Li C., and Lü J. Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynam. 48 (2007) 409-416
35. Chua L.O., Komuro M., and Matsumoto T. The double scroll family. IEEE Trans. Circuits Syst. 33 (1986) 1072-1118
36. Silva C.P. Shil'nikov's theorem-A tutorial. IEEE Trans. Circuits Syst. I 40 (1993) 675-682
37. Cafagna D., and Grassi G. New 3-D-scroll attractors in hyperchaotic Chua's circuit forming a ring. Internat. J. Bifurc. Chaos 13 10 (2003) 2889-2903
38. Lü J., Chen G., Yu X., and Leung H. Design and analysis of multi-scroll chaotic attractors from saturated function series. IEEE Trans. Circuits Syst. I 51 12 (2004) 2476-2490
39. Deng W., and Lü J. Design of multidirectional multi-scroll chaotic attractors based on fractional differential systems via switching control. Chaos 16 (2006) 043120
40. Deng W., and Lü J. Generating multidirectional multi-scroll chaotic attractors via a fractional differential hysteresis system. Phys. Lett. A 369 (2007) 438-443
41. Yorke J.A., and Yorke E.D. Metastable chaos: The transition to sustained chaotic oscillation in a model of Lorenz. J. Statist. Phys. 21 (1979) 263-277
42. Diethelm K., Ford N.J., and Freed A.D. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynam. 29 (2002) 3-22
43. Rössler O.E. An equation for continuous chaos. Phys. Lett. A 57 5 (1976) 397-398
44. Chen G., and Ueta T. Yet another attractor. Internat. J. Bifurc. Chaos 9 (1999) 1465-1466
45. Lü J., Chen G., Cheng D., and Celikovsky S. Bridge the gap between the Lorenz system and the Chen system. Internat. J. Bifurc. Chaos 12 12 (2002) 2917-2926
46. Chen W.C. Nonlinear dynamics and chaos in a fractional order financial system. Chaos Solitons Fractals 36 (2008) 1305-1314