×

zbMATH — the first resource for mathematics

Global existence theory and chaos control of fractional differential equations. (English) Zbl 1113.37016
Summary: The initial value problem for a class of fractional differential equations is discussed, which generalizes the existent result to a wide class of fractional differential equations. Also the theoretical result established in the paper ensures the validity of chaos control of fractional differential equations. In particular, feedback control of chaotic fractional differential equation is theoretically investigated and the fractional Lorenz system as a numerical example is further provided to verify the analytical result.

MSC:
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
26A33 Fractional derivatives and integrals
93D15 Stabilization of systems by feedback
93B52 Feedback control
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Oldham, K.B.; Spanier, J., Fractional calculus: theory and applications, differentiation and integration to arbitrary order, (1974), Academic Press New York · Zbl 0292.26011
[2] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), John Wiley and Sons · Zbl 0789.26002
[3] Poinot, T.; Trigeassou, J.C., Identification of fractional systems using an output-error technique, Nonlinear dynam., 38, 133-154, (2004) · Zbl 1134.93324
[4] Heymans, N., Fractional calculus description of non-linear viscoelastic behaviour of polymers, Nonlinear dynam., 38, 221-231, (2004) · Zbl 1142.74312
[5] Hartley, T.T.; Lorenzo, C.F.; Qammer, H.K., Chaos in fractional order Chua’s system, IEEE trans. circuit syst. I, 42, 485-490, (1995)
[6] Oldham, K.B., A signal-independent electroanalytical method, Anal. chem., 44, 196-198, (1972)
[7] Anastasio, T.J., The fractional-order dynamics of bainstem vestibulo-oculomotor neurons, Biological cybernetics, 72, 69-79, (1994)
[8] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[9] Diethelm, K.; Ford, N.J., Analysis of fractional differential equations, J. math. anal. appl., 265, 229-248, (2002) · Zbl 1014.34003
[10] Yu, C.; Gao, G., Existence of fractional differential equations, J. math. anal. appl., 310, 26-29, (2005) · Zbl 1088.34501
[11] Grigorenko, I.; Grigorenko, E., Chaotic dynamics of the fractional Lorenz system, Phys. rev. lett., 91, 034101, (2003)
[12] Y.B. He, W. Lin, J. Ruan, Stability, instability and chaos in fractional dynamical system, in: Proceedings of 1st Shanghai International Symposium on Nonlinear Sciences and Applications, 2003
[13] Li, C.G.; Chen, G., Chaos in the fractional order Chen system and its control, Chaos solitons fractals, 22, 549-554, (2004) · Zbl 1069.37025
[14] Caputo, M., Linear models of dissipation whose Q is almost frequency independent, II, Geophys. J. R. astron. soc., 13, 529-539, (1967)
[15] Ye, Q.X.; Li, Z.Y., An introduction to reaction – diffusion equations, (1994), Science Press
[16] Dyke, P.P.G., An introduction to Laplace transforms and Fourier series, (1999), Springer · Zbl 0933.44001
[17] Pachpatte, B.G., Inequalities for differential and integral equations, (1998), Academic Press San Diego · Zbl 1032.26008
[18] Lin, W., Description of complex dynamics in a class of impulsive differential equations, Chaos solitons fractals, 25, 1007-1017, (2005) · Zbl 1198.34014
[19] Lin, W.; He, Y.B., Complete synchronization of the noise-perturbed Chua’s circuits, Chaos, 15, 023705, (2005)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.