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Limitations of frequency domain approximation for detecting chaos in fractional order systems. (English) Zbl 1148.65094

Summary: We analytically study the influences of using frequency domain approximation in numerical simulations of fractional order systems. The number and location of equilibria, and also the stability of these points, are compared between the original system and its frequency based approximated counterpart. It is shown that the original system and its approximation are not necessarily equivalent according to the number, location and stability of the fixed points. This problem can cause erroneous results in special cases.
For instance, to prove the existence of chaos in fractional order systems, numerical simulations have been largely based on frequency domain approximations, but in this paper we show that this method is not always reliable for detecting chaos. This approximation can numerically demonstrate chaos in the non-chaotic fractional order systems, or eliminate chaotic behavior from a chaotic fractional order system.

MSC:

65P20 Numerical chaos
26A33 Fractional derivatives and integrals
65P40 Numerical nonlinear stabilities in dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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[1] Jenson, V. G.; Jeffreys, G. V., Mathematical Methods in Chemical Engineering (1977), Academic: Academic New York · Zbl 0413.00002
[2] Sun, H. H.; Abdelwahad, A. A.; Onaral, B., Linear approximation of transfer function with a pole of fractional order, IEEE Trans. Automat. Control, 29, 441-444 (1984) · Zbl 0532.93025
[3] Ichise, M.; Nagayanagi, Y.; Kojima, T., An analog simulation of non-integer order transfer functions for analysis of electrode process, J. Electro-Anal. Chem., 33, 253-265 (1971)
[4] Heaviside, O., Electromagnetic Theory (1971), Chelsea: Chelsea New York
[5] Bagley, R. L.; Calico, R. A., Fractional order state equations for the control of visco-elastically damped structures, J. Guid. Control Dyn., 14, 304-311 (1991)
[6] Kusnezov, D.; Bulgac, A.; Dang, G. D., Quantum levy processes and fractional kinetics, Phys. Rev. Lett., 82, 1136-1139 (1999)
[7] Laskin, N., Fractional market dynamics, Physica A, 278, 482-492 (2000)
[8] El-Sayed, A. M.A., Fractional order diffusion wave equation, Internat. J. Theoret. Phys., 35, 2, 311-322 (1996) · Zbl 0846.35001
[9] Torvik, P. J.; Bagley, R. L., On the appearance of the fractional derivative in the behavior of real materials, Trans. ASME, 51, 294-298 (1984) · Zbl 1203.74022
[10] Oustaloup, A.; Moreau, X.; Nouillant, M., The CRONE suspension, Control Eng. Pract., 4, 8, 1101-1108 (1996)
[11] B.J. Lurie, Tunable TID controller, US patent 5, 371, 670, December 6, 1994; B.J. Lurie, Tunable TID controller, US patent 5, 371, 670, December 6, 1994
[12] Podlubny, I., Fractional order systems and \(PI^\lambda D^\mu\) controllers, IEEE Trans. Automat. Control, 44, 1, 208-214 (1999) · Zbl 1056.93542
[13] Raynaud, H. F.; Zerga, Inoh A., State space representation for fractional order controllers, Automatica, 36, 1017-1021 (2000) · Zbl 0964.93024
[14] Hartley, T. T.; Lorenzo, C. F.; Qammer, H. K., Chaos in a fractional order Chua’s system, IEEE Trans. CAS-I, 42, 485-490 (1995)
[15] P. Arena, R. Caponetto, L. Fortuna, D. Porto, Chaos in a fractional order Duffing system, in: Proc. ECCTD, Budapest, 1997, pp. 1259-1262; P. Arena, R. Caponetto, L. Fortuna, D. Porto, Chaos in a fractional order Duffing system, in: Proc. ECCTD, Budapest, 1997, pp. 1259-1262
[16] Ahmad, W. M.; Sprott, J. C., Chaos in fractional order autonomous nonlinear systems, Chaos Solitons Fractals, 16, 339-351 (2003) · Zbl 1033.37019
[17] Lu, J. G.; Chen, G., A note on the fractional order Chen system, Chaos Solitons Fractals, 27, 3, 685-688 (2006) · Zbl 1101.37307
[18] Lu, J. G., Chaotic dynamics of the fractional order Lü system and its synchronization, Phys. Lett. A, 354, 4, 305-311 (2006)
[19] Li, C.; Chen, G., Chaos and hyperchaos in the fractional order Rössler equations, Physica A, 341, 55-61 (2004)
[20] Lu, J. G., Chaotic dynamics and synchronization of fractional order Arneodo’s systems, Chaos Solitons Fractals, 26, 4, 1125-1133 (2005) · Zbl 1074.65146
[21] L.J. Sheu, H.K. Chen, J.H. Chen, L.M. Tam, W.C. Chen, K.T. Lin, Y. Kang, Chaos in the Newton-Leipnik system with fractional order, Chaos Solitons Fractals (2006) (in press); L.J. Sheu, H.K. Chen, J.H. Chen, L.M. Tam, W.C. Chen, K.T. Lin, Y. Kang, Chaos in the Newton-Leipnik system with fractional order, Chaos Solitons Fractals (2006) (in press) · Zbl 1152.37319
[22] Diethelm, K.; Ford, N. J.; Freed, A. D., A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29, 3-22 (2002) · Zbl 1009.65049
[23] Diethelm, K.; Ford, N. J.; Freed, A. D., Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36, 31-52 (2004) · Zbl 1055.65098
[24] J.H. Chen, W.C. Chen, Chaotic dynamics of the fractionally damped van der Pol equation, Chaos Solitons Fractals (2006) (in press); J.H. Chen, W.C. Chen, Chaotic dynamics of the fractionally damped van der Pol equation, Chaos Solitons Fractals (2006) (in press)
[25] Sheu, L. J.; Chen, H. K.; Chen, J. H.; Tam, L. M., Chaos in a new system with fractional order, Chaos Solitons Fractals, 31, 2, 1203-1212 (2007)
[26] W.C. Chen, Nonlinear dynamics and chaos in a fractional order financial system, Chaos Solitons Fractals (2006) (in press); W.C. Chen, Nonlinear dynamics and chaos in a fractional order financial system, Chaos Solitons Fractals (2006) (in press)
[27] Vinagre, B. M.; Podlubny, I.; Hernandez, A.; Feliu, V., Some approximations of fractional order operators used in control theory and applications, Fract. Calc. Appl. Anal., 3, 3, 945-950 (2000) · Zbl 1111.93302
[28] Carlson, G. E.; Halijak, C. A., Approximation of fractional capacitors \((1 / s)^{1 / n}\) by a regular Newton process, IRE Trans. Circuit Theory, CT-11, 2, 210-213 (1964)
[29] Matsuda, K.; Fujii, H., \(H_\infty\) optimized wave-absorbing control: Analytical and experimental results, J. Guid. Control Dyn., 16, 6, 1146-1153 (1993) · Zbl 0800.93313
[30] Oustaloup, A., La Commande CRONE (1991), Editions Hermes: Editions Hermes Paris · Zbl 0864.93003
[31] Oustaloup, A.; Levron, F.; Mathieu, B.; Nanot, F. M., Frequency-band complex noninteger differentiator: Characterization and synthesis, IEEE Trans. Circuits Syst. I, 47, 1, 25-39 (2000)
[32] Charef, A.; Sun, H. H.; Tsao, Y. Y.; Onaral, B., Fractal system as represented by singularity function, IEEE Trans. Automat. Control, 37, 9 (1992) · Zbl 0825.58027
[33] Charef, A., Analogue realization of fractional order integrator, differentiator and fractional \(PI^\lambda D^\mu\) controller, IEE Proc. Control Theory Appl., 153, 6, 714-720 (2006)
[34] Tsai, J. S.H.; Chien, T. H.; Guo, S. M.; Chang, Y. P.; Shieh, L. S., State-space self-tuning control for stochastic chaotic fractional-order systems, IEEE Trans. Circuits Syst. I, 54, 3, 632-642 (2007) · Zbl 1374.93324
[35] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[36] Ahmed, E.; El-Sayed, A. M.A.; El-Saka, H. A.A., Equilibrium points, stability and numerical solutions of fractional order predator-prey and rabies models, J. Math. Anal. Appl., 325, 1, 542-553 (2007) · Zbl 1105.65122
[37] D. Matignon, Stability results for fractional differential equations with applications to control processing, in: Computational Engineering in Systems Applications, Lille, France, IMACS, IEEE-SMC, vol. 2, July 1996, pp. 963-968; D. Matignon, Stability results for fractional differential equations with applications to control processing, in: Computational Engineering in Systems Applications, Lille, France, IMACS, IEEE-SMC, vol. 2, July 1996, pp. 963-968
[38] G.R. Duan, Parametric control systems design: Theory and applications, SICE-ICASE International Join Conference, Korea, October 18-21, 2006, pp. I-15: I-23; G.R. Duan, Parametric control systems design: Theory and applications, SICE-ICASE International Join Conference, Korea, October 18-21, 2006, pp. I-15: I-23
[39] Roman, S., The Formula of Faa di Bruno, Amer. Math. Monthly, 87, 805-809 (1980) · Zbl 0513.05009
[40] Johnson, W. P., The curious history of Faà di Bruno’s formula, Amer. Math. Monthly, 109, 217-234 (2002) · Zbl 1024.01010
[41] Mishkov, R. L., Generalization of the formula of Faa di Bruno for a composite function with a vector argument, Internat. J. Math. Math. Sci., 24, 7, 481-491 (2000) · Zbl 0967.46031
[42] Chua, L. O.; Komuro, M.; Matsumoto, T., The double scroll family, IEEE Trans. Circuits Syst., 33, 1072-1118 (1986) · Zbl 0634.58015
[43] Silva, C. P., Shil’nikov’s theorem—A tutorial, IEEE Trans. Circuits Syst. I, 40, 675-682 (1993) · Zbl 0850.93352
[44] Cafagna, D.; Grassi, G., New 3-D-scroll attractors in hyperchaotic Chua’s circuit forming a ring, Internat. J. Bifur. Chaos, 13, 10, 2889-2903 (2003) · Zbl 1057.37026
[45] Lü, J.; Chen, G.; Yu, X.; Leung, H., Design and analysis of multi-scroll chaotic attractors from saturated function series, IEEE Trans. Circuits Syst. I, 51, 12, 2476-2490 (2004) · Zbl 1371.37060
[46] Rössler, O. E., An equation for continuous chaos, Phys. Lett. A, 57, 5, 397-398 (1976) · Zbl 1371.37062
[47] Chen, G.; Ueta, T., Yet another attractor, Internat. J. Bifur. Chaos, 9, 1465-1466 (1999) · Zbl 0962.37013
[48] Lü, J.; Chen, G., A new chaotic attractor coined, Internat. J. Bifur. Chaos, 12, 3, 659-661 (2002) · Zbl 1063.34510
[49] Yu, S. M.; Lu, J. H.; Tang, W. K.S.; Chen, G., A general multiscroll Lorenz system family and its DSP realization, Chaos, 16, 1-10 (2006) · Zbl 1151.94432
[50] Lü, J.; Chen, G., Generating multiscroll chaotic attractors: Theories, methods and applications, Internat. J. Bifur. Chaos, 16, 775-858 (2006) · Zbl 1097.94038
[51] Lu, J.; Yu, S.; Leung, H.; Chen, G., Experimental verification of multi-directional multi-scroll chaotic attractors, IEEE Trans. Circuits Syst.-I, 53, 1, 149-165 (2006)
[52] Ahmad, W. M., Generation and control of multi-scroll chaotic attractors in fractional order systems, Chaos Solitons Fractals, 25, 727-735 (2005) · Zbl 1092.37509
[53] I. Petras, I. Podlubny, P. O’Leary, L. Dorcak, B. Vinagre, Analogue Realization of Fractional Order Controllers, FBERG, Technical University of Kosice, Kosice, 2002; I. Petras, I. Podlubny, P. O’Leary, L. Dorcak, B. Vinagre, Analogue Realization of Fractional Order Controllers, FBERG, Technical University of Kosice, Kosice, 2002 · Zbl 1041.93022
[54] I. Petras, A note on the fractional-order Chua’s system, Chaos Solitons Fractals (2006) (in press); I. Petras, A note on the fractional-order Chua’s system, Chaos Solitons Fractals (2006) (in press)
[55] Kou, S.; Han, K., Limitations of the describing function method, IEEE Trans. Automat. Control, 20, 2, 291-292 (1975)
[56] Engelberg, S., Limitations of the describing function for limit cycle prediction, IEEE Trans. Automat. Control, 47, 11, 1887-1890 (2002) · Zbl 1364.93334
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