zbMATH — the first resource for mathematics

Stability analysis of linear fractional differential system with multiple time delays. (English) Zbl 1185.34115
Summary: We study the stability of a system of linear fractional differential equations with time delays. By using the Laplace transform, we introduce a characteristic equation for the above system with multiple time delays. We discover that if all roots of the characteristic equation have negative parts, then the equilibrium of the above linear system with fractional order is Lyapunov globally asymptotical stable if the equilibrium exist.
We present some examples.

34K37 Functional-differential equations with fractional derivatives
34K20 Stability theory of functional-differential equations
34K06 Linear functional-differential equations
Full Text: DOI
[1] Hale, J.K., Verdugn Lunel, S.M., Introduction to Functional Differential Equations. Springer-Verlag, New York (1991)
[2] Deng, W.H., Wu, Y.J., Li, C.P.: Stability analysis of differential equations with time-dependent delay. Int. J. Bif. Chaos. 16(2), 465–472 (2006) · Zbl 1112.34056 · doi:10.1142/S0218127406014939
[3] Li, C.P., Sun, W.G., Kurths, J.: Synchronization of complex dynamical networks with time delays. Physica A 361, 24–34 (2006) · doi:10.1016/j.physa.2005.07.007
[4] Li, C.P., Sun, W.G., Xu, D.: Synchronization of complex dynamical networks with nonlinear inner-coupling functions and time delays. Prog. Theor. Phys. 114(4), 749–761 (2005) · Zbl 1094.34056 · doi:10.1143/PTP.114.749
[5] Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley-Interscience, New York (1993) · Zbl 0789.26002
[6] Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999) · Zbl 0924.34008
[7] Butzer, P.L., Westphal, U.: An Introduction to Fractional Calculus. World Scientific, Singapore (2000) · Zbl 0987.26005
[8] Agrawal, O.P., Tenreiro Machado, J.A., Sabatier, J.: Introduction. Nonlinear Dyn. 38(1–2), 1–2 (2004)
[9] Kempfle, S., Schäfer, I., Beyer, H.: Fractional calculus: theory and applications. Nonlinear Dyn. 29(1–4), 99–127 (2002) · Zbl 1026.47010
[10] Matignon, D.: Stability result on fractional differential equations with applications to control processing. In: Proceedings of IMACS-SMC, pp. 963–968, Lille, France (1996)
[11] Chen, Y., Moore, K.L.: Analytical stability bound for a class of delayed fractional-order dynamic systems. Nonlinear Dyn. 29, 191–200 (2002) · Zbl 1020.34064 · doi:10.1023/A:1016591006562
[12] Muth, E.J.: Transform Methods with Applications to Engineering and Operations Research. Prentice-Hall, New Jersey (1977) · Zbl 0392.44001
[13] Franklin, P.: Functions of Complex Variables. Prentice-Hall, New Jersey (1958) · Zbl 0095.27702
[14] Chen, Y., Moore, K.L.: Analytical stability bound for delayed second-order systems with repeating poles using Lambert function \(\omega\). Automatica 38, 891–895 (2002) · Zbl 1020.93019 · doi:10.1016/S0005-1098(01)00264-3
[15] Deng, W.H., Li, C.P.: Synchronization of chaotic fractional Chen system. J. Phys. Soc. Jpn. 74, 1645–1648 (2005) · Zbl 1080.34537 · doi:10.1143/JPSJ.74.1645
[16] Deng, W.H., Li, C.P.: Chaos synchronization of the fractional Lü system. Physica A 353, 61–72 (2005) · doi:10.1016/j.physa.2005.01.021
[17] Lu, J.G.: Nonlinear observer design to synchronize fractional-order chaotic system via a scaler transmitted signal. Physica A 359, 107–118 (2005) · doi:10.1016/j.physa.2005.04.040
[18] Lu, J.G.: Synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal. Chaos Solitons Fractals 27(2), 519–525 (2006) · Zbl 1086.94007 · doi:10.1016/j.chaos.2005.04.032
[19] Zhou, T.S., Li, C.P.: Synchronization in fractional-order differential systems. Physica D 212, 111–125 (2005) · Zbl 1094.34034 · doi:10.1016/j.physd.2005.09.012
[20] Li, C.P., Deng, W.H., Xu, D.: Chaos synchronization of the Chua system with a fractional order. Physica A 360, 171–185 (2006) · doi:10.1016/j.physa.2005.06.078
[21] Diethelm, K., Ford, N.J., Freed, A.D.: A predictor–corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1–4), 3–22 (2002) · Zbl 1009.65049
[22] Li, C.P., Peng, G.J.: Chaos in Chen’s system with a fractional order. Chaos Solitons Fractals 22, 443–450 (2004) · Zbl 1060.37026 · doi:10.1016/j.chaos.2004.02.013
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.