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Synchronization in fractional-order differential systems. (English) Zbl 1094.34034
Authors’ abstract: An $\omega$-symmetrically coupled system consisting of identical fractional-order differential systems including chaotic and nonchaotic systems is investigated in this paper. Such a coupled system has, in its synchronous state, a mode decomposition by which the linearized equation can be decomposed into motions transverse to and parallel to the synchronous manifold. Furthermore, the decomposition can induce a sufficient condition on synchronization of the overall system, which guarantees, if satisfied, that a group synchronization is achieved. Two typical numerical examples, fractional Brusselators and the fractional Rössler system, are used to verify the theoretical prediction. The theoretical analysis and numerical results show that the lower the order of the fractional system the longer the time for achieving synchronization at a fixed coupling strength.

##### MSC:
 34D05 Asymptotic stability of ODE 26A33 Fractional derivatives and integrals (real functions) 34C14 Symmetries, invariants (ODE)
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##### References:
 [1] Daftardar-Gejji, V.; Babakhani, A.: Analysis of a system of fractional differential euqations. J. math. Anal. appl. 293, 511-522 (2004) · Zbl 1058.34002 [2] Delbosco, D.; Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation. J. math. Anal. appl. 204, 609-625 (1996) · Zbl 0881.34005 [3] Deng, W. H.; Li, C. P.: Synchronization of chaotic fractional Chen system. J. phys. Soc. Japan 74, 1645-1648 (2005) · Zbl 1080.34537 [4] Deng, W. H.; Li, C. P.: Chaos synchronization of the fractional Lü system. Physica A 353, 61-72 (2005) [5] 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 [6] 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 [7] Diethelm, K.; Ford, N. J.: Analysis of fractional differential equations. J. math. Anal. appl. 265, 229-248 (2002) · Zbl 1014.34003 [8] Heagy, J. F.; Carroll, T. L.; Pecora, L. M.: Synchronous chaos in coupled oscillator system. Phys. rev. E 50, 1874-1885 (1994) [9] Heaviside, O.: Electromagnetic theory. (1971) · Zbl 30.0801.03 [10] Koeller, R. C.: Application of fractional calculus to the theory of viscoelasticity. J. appl. Mech. 51, 229-307 (1984) · Zbl 0544.73052 [11] C.P. Li, W.H. Deng, D. Xu, Chaos synchronization of the Chua system with a fractional order, Physica A 360 (2) in press, doi:10.1016/j.physa.2005.06.078 [12] 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 [13] Li, C. P.; Xia, X.: On the bound of the Lyapunov exponents for continuous systems. Chaos 14, No. 3, 557-561 (2004) · Zbl 1080.34038 [14] Lubich, C.: Runge--Kutta theory for Volterra and Abel integral equations of the second kind. Math. comp. 41, 87-102 (1983) · Zbl 0538.65091 [15] Matignon, D.: Stability results for fractional differential equations with applications to control processing. Imacs, ieee-smc 2, 963-968 (1996) [16] Miller, R. K.; Feldstein, A.: Smoothness of solutions of Volterra integral equations with weakly singular kernels. SIAM J. Math. anal. 2, 242-258 (1971) · Zbl 0217.15602 [17] Nicolis, G.; Prigogine, I.: Self-organization in nonequilibrium systems. (1977) · Zbl 0363.93005 [18] Nishimoto, K.: Fractional calculus: integrations and differentiations of arbitrary order. (1984) · Zbl 0605.26006 [19] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008 [20] Rössler, O. E.: An equation for continuous chaos. Phys. lett. A 57, No. 5, 397-398 (1976) [21] Sun, H. H.; Onaral, B.; Tsao, Y.: Application of the positive reality principle to metal electrode linear polarization phenomena. IEEE trans. Biomed. eng. 31, 664-674 (1984) [22] Sun, H. H.; Abdelwahab, 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 [23] Zhang, Z. F.: Theory of differential equations. (1985) · Zbl 0589.34048 [24] Zhou, T. S.: Stability of echo waves in linearly coupled oregonators. Phys. lett. A 320, No. 2--3, 116-126 (2003) · Zbl 1065.74035 [25] Zhou, T. S.; Zhang, S. C.: Dynamical behaviors in linearly coupled oregonators. Physica D 151, 199-216 (2001) · Zbl 1077.34509 [26] Zhou, T. S.; Zhang, S. C.: Echo waves and coexistence phenomena in coupled brusselators. Chaos, solitons fractals 13, 621-632 (2002) · Zbl 1073.34515 [27] Zhou, T. S.; Lü, J. H.; Chen, G. R.; Tang, Y.: Synchronization stability of three chaotic systems with linear coupling. Phys. lett. A 301, 231-240 (2002) · Zbl 0997.37015 [28] Zhou, T. S.; Chen, L. N.; Wang, R. Q.: Excitation functions of coupling. Phys. rev. E 71, 066211 (2005)