## Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions.(English)Zbl 1239.44001

Summary: Using the Mellin transform approach, it is shown that, in contrast with integer-order derivatives, the fractional-order derivative of a periodic function cannot be a function with the same period. The three most widely used definitions of fractional-order derivatives are taken into account, namely, the Caputo, Riemann-Liouville and Grunwald-Letnikov definitions. As a consequence, the non-existence of exact periodic solutions in a wide class of fractional-order dynamical systems is obtained. As an application, it is emphasized that the limit cycle observed in numerical simulations of a simple fractional-order neural network cannot be an exact periodic solution of the system.

### MSC:

 44A15 Special integral transforms (Legendre, Hilbert, etc.) 34A08 Fractional ordinary differential equations
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### References:

 [1] Reyes-Melo, E.; Martinez-Vega, J.; Guerrero-Salazar, C.; Ortiz-Mendez, U., Application of fractional calculus to the modeling of dielectric relaxation phenomena in polymeric materials, Journal of applied polymer science, 98, 923-935, (2005) [2] Schumer, R.; Benson, D., Eulerian derivative of the fractional advection – dispersion equation, Journal of contaminant, 48, 69-88, (2001) [3] Heymans, N.; Bauwens, J.C., Fractal rheological models and fractional differential equations for viscoelastic behavior, Rheologica acta, 33, 210-219, (1994) [4] Henry, B.; Wearne, S., Existence of Turing instabilities in a two-species fractional reaction – diffusion system, SIAM journal on applied mathematics, 62, 870-887, (2002) · Zbl 1103.35047 [5] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics reports, 339, 1-77, (2000) · Zbl 0984.82032 [6] Glockle, W.; Nonnenmacher, T., A fractional calculus approach to self-similar protein dynamics, Biophysical journal, 68, 46-53, (1995) [7] Picozzi, S.; West, B.J., Fractional Langevin model of memory in financial markets, Physical review E, 66, 46-118, (2002) [8] Ahmad, B.; Nieto, J.J.; Alsaedi, A.; El-Shahed, M., A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear analysis. real world applications, 13, 599-606, (2012) · Zbl 1238.34008 [9] Song, L.; Xu, S.; Yang, J., Dynamical models of happiness with fractional order, Communications in nonlinear science and numerical simulation, 15, 616-628, (2010) · Zbl 1221.93234 [10] Gu, R.; Xu, Y., Chaos in a fractional-order dynamical model of love and its control, (), 349-356 · Zbl 1272.91113 [11] Kilbas, A.; Srivastava, H.; Trujillo, J., Theory and applications of fractional differential equations, (2006), Elsevier · Zbl 1092.45003 [12] Lakshmikantham, V.; Leela, S.; Devi, J.V., Theory of fractional dynamic systems, (2009), Cambridge Scientific Publishers · Zbl 1188.37002 [13] Podlubny, I., Fractional differential equations, (1999), Academic Press · Zbl 0918.34010 [14] Agarwal, R.P.; de Andrade, B.; Cuevas, C., Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations, Nonlinear analysis. real world applications, 11, 3532-3554, (2010) · Zbl 1248.34004 [15] Cao, J.; Yang, Q.; Huang, Z., Optimal mild solutions and weighted pseudo-almost periodic classical solutions of fractional integro-differential equations, Nonlinear analysis, theory, methods and applications, 74, 224-234, (2011) · Zbl 1213.34089 [16] Debbouche, A.; El-Borai, M., Weak almost periodic and optimal mild solutions of fractional evolution equations, Electronic journal of differential equations, 2009, 1-8, (2009) · Zbl 1171.34331 [17] Bai, C., Impulsive periodic boundary value problems for fractional differential equation involving riemann – liouville sequential fractional derivative, Journal of mathematical analysis and applications, 384, 211-231, (2011) · Zbl 1234.34005 [18] Belmekki, M.; Nieto, J.J.; Rodriguez-Lopez, R., Existence of periodic solution for a nonlinear fractional differential equation, Boundary value problems, 2009, 1-18, (2009) · Zbl 1181.34006 [19] Nieto, J.J., Maximum principles for fractional differential equations derived from Mittag-Leffler functions, Applied mathematics letters, 23, 1248-1251, (2010) · Zbl 1202.34019 [20] Wei, Z.; Dong, W.; Che, J., Periodic boundary value problems for fractional differential equations involving a riemann – liouville fractional derivative, Nonlinear analysis. theory, methods & applications, 73, 3232-3238, (2010) · Zbl 1202.26017 [21] Fikioris, G., Mellin-transform method for integral evaluation, (2007), Morgan & Claypool [22] Flajolet, P.; Gourdon, X.; Dumas, P., Mellin transforms and asymptotics: harmonic sums, Theoretical computer science, 144, 3-58, (1995) · Zbl 0869.68057 [23] Tavazoei, M.; Haeri, M., A proof for non existence of periodic solutions in time invariant fractional order systems, Automatica, 45, 1886-1890, (2009) · Zbl 1193.34006 [24] Tavazoei, M., A note on fractional-order derivatives of periodic functions, Automatica, 46, 945-948, (2010) · Zbl 1191.93062 [25] Ishteva, M.; Boyadjiev, L.; Scherer, R., On the Caputo operator of fractional calculus and $$C$$-Laguerre functions, Mathematical sciences research journal, 9, 161-170, (2005) · Zbl 1078.26007 [26] Barbosa, R.S.; Machado, J.T.; Vinagre, B.; Calderón, A., Analysis of the van der Pol oscillator containing derivatives of fractional order, Journal of vibration and control, 13, 1291-1301, (2007) · Zbl 1158.70009 [27] Cafagna, D.; Grassi, G., Bifurcation and chaos in the fractional-order Chen system via a time-domain approach, International journal of bifurcation and chaos, 18, 1845-1863, (2008) · Zbl 1158.34300 [28] Cafagna, D.; Grassi, G., Fractional-order chua’s circuit: time-domain analysis, bifurcation, chaotic behavior and test for chaos, International journal of bifurcation and chaos, 18, 615-639, (2008) · Zbl 1147.34302 [29] Zhang, W.; Zhou, S.; Li, H.; Zhu, H., Chaos in a fractional-order Rössler system, Chaos, solitons and fractals, 42, 1684-1691, (2009) · Zbl 1198.34001 [30] Abd-Elouahab, M.S.; Hamri, N.E.; Wang, J., Chaos control of a fractional-order financial system, Mathematical problems in engineering, 2010, 1-18, (2010) · Zbl 1195.91185 [31] Lundstrom, B.; Higgs, M.; Spain, W.; Fairhall, A., Fractional differentiation by neocortical pyramidal neurons, Nature neuroscience, 11, 1335-1342, (2008) [32] Arena, P.; Fortuna, L.; Porto, D., Chaotic behavior in noninteger-order cellular neural networks, Physical review E, 61, 776-781, (2000) [33] Matsuzaki, T.; Nakagawa, M., A chaos neuron model with fractional differential equation, Journal of the physical society of Japan, 72, 2678-2684, (2003) [34] I. Petras, A note on the fractional-order cellular neural networks, in: IEEE International Conference on Neural Networks, pp. 1021-1024. [35] Boroomand, A.; Menhaj, M., Fractional-order Hopfield neural networks, (), 883-890 [36] Kaslik, E.; Sivasundaram, S., Dynamics of fractional-order neural networks, (), 611-618 [37] Butzer, P.L.; Westphal, U., An introduction to fractional calculus, (), 1-86 · Zbl 0987.26005 [38] Hilfer, R., Threefold introduction to fractional derivatives, (), 17-74
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