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A proof for non existence of periodic solutions in time invariant fractional order systems. (English) Zbl 1193.34006
Summary: The aim of this note is to highlight one of the basic differences between fractional order and integer order systems. It is analytically shown that a time invariant fractional order system contrary to its integer order counterpart cannot generate exactly periodic signals. As a result, a limit cycle cannot be expected in the solution of these systems. Our investigation is based on Caputo’s definition of the fractional order derivative and includes both the commensurate or incommensurate fractional order systems.

MSC:
34A08Fractional differential equations
34C05Location of integral curves, singular points, limit cycles (ODE)
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References:
[1] Ahmad, W.; El-Khazali, R.; El-Wakil, A.: Fractional-order wien-Bridge oscillator, Electronic letters 37, 1110-1112 (2001)
[2] Andrievsky, B. R.; Fradkov, A. L.: Control of chaos: methods and applications. Part I: Methods, Automation and remote control 64, No. 5, 673-713 (2003) · Zbl 1107.37302 · doi:10.1023/A:1023684619933
[3] Aoun, M.; Malti, R.; Levron, F.; Oustaloup, A.: Synthesis of fractional Laguerre basis for system approximation, Automatica 43, 1640-1648 (2007) · Zbl 1128.93019 · doi:10.1016/j.automatica.2007.02.013
[4] Aoun, M.; Malti, R.; Levron, F.; Oustaloup, A.: Numerical simulations of fractional systems: an overview of existing methods and improvements, Nonlinear dynamics 38, 117-131 (2004) · Zbl 1134.65300 · doi:10.1007/s11071-004-3750-z
[5] Barbosa, R. S.; Machado, J. A. T.; Vinagre, B. M.; Calderon, A. J.: Analysis of the van der Pol oscillator containing derivatives of fractional order, Journal of vibration and control 13, 1291-1301 (2007) · Zbl 1158.70009 · doi:10.1177/1077546307077463
[6] Casas, R. A.; Bitmead, R. R.; Jacobson, C. A.; Johnson, C. R.: Prediction error methods for limit cycle data, Automatica 38, 1753-1760 (2002) · Zbl 1011.93507 · doi:10.1016/S0005-1098(02)00085-7
[7] Chen, G.; Friedman, G.: An RLC interconnect model based on Fourier analysis, IEEE transactions on computer aided design of integrated circuits and systems 24, No. 2, 170-183 (2005)
[8] Daftardar-Gejji, V.; Jafari, H.: Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives, Journal of mathematical analysis and applications 328, 1026-1033 (2007) · Zbl 1115.34006 · doi:10.1016/j.jmaa.2006.06.007
[9] Diethelm, K.; Ford, N. J.: Analysis of fractional differential equations, Journal of mathematical analysis and applications 265, 229-248 (2002) · Zbl 1014.34003 · doi:10.1006/jmaa.2001.7194
[10] Feliu-Batlle, V.; Pérez, R. Rivas; García, F. G. Castillo; Rodríguez, L. Sanchez: Smith predictor based robust fractional order control: application to water distribution in a Main irrigation canal pool, Journal of process control 19, No. 3, 506-519 (2009)
[11] Gafiychuk, V.; Datsko, B.: Stability analysis and limit cycle in fractional system with Brusselator nonlinearities, Physics letters A 372, 4902-4904 (2008) · Zbl 1221.34010 · doi:10.1016/j.physleta.2008.05.045
[12] Hartley, T. T.; Lorenzo, C. F.; Qammer, H. K.: Chaos in a fractional-order Chua’s system, IEEE transactions on circuits and systems I 42, 485-490 (1995)
[13] Ikhouane, F.; Gomis-Bellmunt, O.: A limit cycle approach for the parametric identification of hysteretic systems, Systems & control letters 57, 663-669 (2008) · Zbl 1140.93021
[14] Laskin, N.: Fractional market dynamics, Physica A 287, 482-492 (2000)
[15] Lundstrom, B. N.; Higgs, M. H.; Spain, W. J.; Fairhall, A. L.: Fractional differentiation by neocortical pyramidal neurons, Nature neuroscience 11, No. 11, 1335-1342 (2008)
[16] Monje, C. A.; Vinagre, B. M.; Feliu, V.; Chen, Y. Q.: Tuning and auto-tuning of fractional order controllers for industry applications, Control engineering practice 16, 798-812 (2008)
[17] Ott, E.; Grebogi, C.; Yorke, J. A.: Controlling chaos, Physics review letters 64, 1196-1199 (1990) · Zbl 0964.37501 · doi:10.1103/PhysRevLett.64.1196
[18] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[19] Pyragas, K.: Control of chaos via an unstable delayed feedback controller, Physics review letters 86, 2265-2268 (2001)
[20] Rossikhin, Y. A.; Shitikova, M. V.: Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass system, Acta mechanica 120, 109-125 (1997) · Zbl 0901.73030 · doi:10.1007/BF01174319
[21] Tavazoei, M. S.; Haeri, M.: Unreliability of frequency domain approximation in recognising chaos in fractional order systems, IET signal processing 1, No. 4, 171-181 (2007)
[22] Tavazoei, M. S.; Haeri, M.; Jafari, S.; Bolouki, S.; Siami, M.: Some applications of fractional calculus in suppression of chaotic oscillations, IEEE transactions on industrial electronics 11, 4094-4101 (2008)
[23] Tavazoei, M. S.; Haeri, M.; Nazari, N.: Analysis of undamped oscillations generated by marginally stable fractional order systems, Signal processing 88, 2971-2978 (2008) · Zbl 1151.94415 · doi:10.1016/j.sigpro.2008.07.002
[24] Tavazoei, M. S.; Haeri, M.; Bolouki, S.; Siami, M.: Stability preservation analysis for frequency-based methods in numerical simulation of fractional order systems, SIAM journal on numerical analysis 47, No. 1, 321-338 (2008) · Zbl 1203.26012 · doi:10.1137/080715949
[25] Tavazoei, M. S.; Haeri, M.: Chaotic attractors in incommensurate fractional order systems, Physica D 237, 2628-2637 (2008) · Zbl 1157.26310 · doi:10.1016/j.physd.2008.03.037
[26] Tavazoei, M. S., Haeri, M., Attari, M., Bolouki, S., & Siami, M. (2009). More details on analysis of fractional-order van der Pol oscillator. Journal of Vibration and Control (in press). doi:10.1177/1077546308096101 · Zbl 1273.70037
[27] Tyreus, B. D.; Luyben, W. L.: Tuning of PI controllers for integrator/dead time processes, Industrial & engineering chemistry research 31, 2625-2628 (1992)
[28] Wang, Y.; Li, C.: Does the fractional Brusselator with efficient dimension less than 1 have a limit cycle?, Physics letters A 363, 414-419 (2007)
[29] Westerlund, S.; Ekstam, L.: Capacitor theory, IEEE transactions on dielectrics and electrical insulation 1, No. 5, 826-839 (1994)
[30] Wiggins, S.: Introduction to applied nonlinear dynamical systems and chaos, (2003) · Zbl 1027.37002
[31] Yu, P.: Computation of limit cycles - the second part of Hilbert’s 16th problem, The fields institute communications 49, 151-177 (2006) · Zbl 1286.34050