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fractional-order; multiple delays; Lyapunov stable theorem; synchronization; unknown parameters
In this paper, we propose a new approach of designing a controller and an update rule of unknown parameters for synchronizing fractional-order system with multiple delays and prove the correctness of the approach according to the fractional Lyapunov stable theorem. Based on the proposed approach, synchronizing fractional delayed chaotic system with and without unknown parameters is realized. Numerical simulations are carried out to confirm the effectiveness of the approach.
[1] Chen, L. P., Wei, S. B., Chai, Y.: Adaptive projective synchronization between two different fractional-order chaotic systems with fully unknown parameters. Math. Problems Engrg. 2012 (2012), 1-16. DOI 10.1155/2012/916140 | Zbl 1264.34103
[2] Duarte-Mermoud, M. A., Aguila-Camacho, N., Gallegos, J. A., Castro, R.: Linares using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Comm. Nonlinear Sci. Numer. Simul. 22 (2015), 650-659. DOI 10.1016/j.cnsns.2014.10.008 | MR 3282452
[3] Farivar, F., Shoorehdeli, M. A.: Fault tolerant synchronization of chaotic heavy symmetric gyroscope systems versus external disturbances via Lyapunov rule-based fuzzy control. ISA Trans. 51 (2012), 50-64. DOI 10.1016/j.isatra.2011.07.002
[4] Goldfain, E.: Fractional dynamics and the Standard Model for particle physics. Comm. Nonlinear Sci. Numer. Simul. 13 (2008), 1397-1404. DOI 10.1016/j.cnsns.2006.12.007 | MR 2369469 | Zbl 1221.81175
[5] Gong, Y. B., Lin, X., Wang, L.: Chemical synaptic coupling-induced delay-dependent synchronization transitions in scale-free neuronal networks. Science China - Chemistry 54 (2011), 1498-1503. DOI 10.1007/s11426-011-4363-2
[6] Gutierrez, R. E., Rosario, J. M., Machado, J. T.: Fractional order calculus: Basic concepts and engineering applications. Math. Problems Engrg. 2010 (2010), 1-10. DOI 10.1155/2010/375858 | Zbl 1190.26002
[7] He, J. H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Computer Methods Appl. Mech. Engrg. 167 (1998), 57-68. DOI 10.1016/s0045-7825(98)00108-x | MR 1665221 | Zbl 0942.76077
[8] Li, X. D., Bohner, M.: Exponential synchronization of chaotic neural networks with mixed delays and impulsive effects via output coupling with delay feedback. Math. Computer Modelling 52 (2010), 643-653. DOI 10.1016/j.mcm.2010.04.011 | MR 2661751 | Zbl 1202.34128
[9] Li, C. P., Deng, W. H., Xu, D.: Chaos synchronization of the chua system with a fractional order. Physica A - Statist. Mech. Appl. 360 (2006), 171-185. DOI 10.1016/j.physa.2005.06.078 | MR 2186261
[10] Li, M. D., Li, D. H., Wang, J.: Active disturbance rejection control for fractional-order system. ISA Trans. 52 (2013), 365-374. DOI 10.1016/j.isatra.2013.01.001
[11] Lin, T. C., Kuo, C. H.: H-infinity synchronization of uncertain fractional order chaotic systems: Adaptive fuzzy approach. ISA Trans. 50 (2011), 548-556. DOI 10.1016/j.isatra.2011.06.001
[12] Lu, J. H., Chen, G. R.: A time-varying complex dynamical network model and its controlled synchronization criteria. IEEE Trans. Automat. Control 50 (2005), 841-846. DOI 10.1109/tac.2005.849233 | MR 2142000
[13] Lu, J. H., Chen, G. R.: Generating multiscroll chaotic attractors: Theories, methods and applications. Int. J. Bifurcation Chaos 16 (2006), 775-858. DOI 10.1142/s0218127406015179 | MR 2234259
[14] Merrikh-Bayat, F., Karimi-Ghartemani, M.: An efficient numerical algorithm for stability testing of fractional-delay systems. ISA Trans. 48 (2008), 32-37. DOI 10.1016/j.isatra.2008.10.003
[15] Miao, Q. Y., Fang, J. A., Tang, Y.: Increasing-order projective synchronization of chaotic systems with time delay. Chinese Phys. Lett. 26 (2009), 5, 050501. DOI 10.1088/0256-307x/26/5/050501
[16] Miller, K. S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication, 1993. MR 1219954 | Zbl 0789.26002
[17] Peng, M. S.: Bifurcation and chaotic behavior in the euler method for a ucar prototype delay model. Chaos Solitons and Fractals 22 (2004), 483-493. DOI 10.1016/j.chaos.2004.02.038 | MR 2024872 | Zbl 1061.37022
[18] Podlubny, I.: Fractional Differential Equatons: An Introduction to Fractional Derivatives, Fractional Differential Equations to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego 1999. MR 1658022
[19] Slotine, J. J. E., Li, W.: Applied nonlinear Control. Prentice Hall, 1999. Zbl 0753.93036
[20] Sollund, T., Leib, H.: Feedback communication with reduced delay over noisy time-dispersive channels. IEEE Transa. Commun. 60 (2012), 688-705. DOI 10.1109/tcomm.2012.12.100001
[21] Tan, S. L., Lu, J. H., Yu, X. H.: Adaptive synchronization of an uncertain complex dynamical network. Chinese Sci. Bull. 58 (2013), 28-29. DOI 10.1007/s11434-013-5984-y
[22] Tan, S. L., Lu, J. H., Hill, D. J.: Towards a theoretical framework for analysis and intervention of random drift on general networks. IEEE Trans. Automat. Control 60 (2015), 576-581. DOI 10.1109/tac.2014.2329235 | MR 3310190
[23] Tang, Y., Gao, H., Zou, W., Kurths, J.: Distributed synchronization in networks of agent systems with nonlinearities and random switchings. IEEE Trans. Cybernet. 43 (2013), 358-370. DOI 10.1109/tsmcb.2012.2207718
[24] Tang, Y., Wong, W. K.: Distributed synchronization of coupled neural networks via randomly occurring control. IEEE Trans. Neural Networks Learning Systems 24 (2013), 435-447. DOI 10.1109/tnnls.2012.2236355
[25] Wang, X. Y., Wang, M. J.: Hyperchaotic Lorenz system. Acta Physica Sinica 56 (2007), 5136-5141. MR 2371460 | Zbl 1267.93157
[26] Wang, X. D., Tian, L. X.: Bifurcation analysis and linear control of the Newton-Leipnik system. Chaos Solitions Fractals 27 (2006), 31-38. DOI 10.1016/j.chaos.2005.04.009 | MR 2165262 | Zbl 1091.93031
[27] Wang, S., Yu, Y. G.: Generalized projective synchronization of fractional order chaotic systems with different dimensions. Chinese Phys. Lett. 29 (2012), 2, 020505. DOI 10.1088/0256-307x/29/2/020505
[28] Zhao, L. D., Hu, J. B., al., J. A. Fang et: Adaptive synchronization and parameter identification of chaotic system with unknown parameters and mixed delays based on a special matrix structure. ISA Trans. 52 (2013), 738-743. DOI 10.1016/j.isatra.2013.07.001
[29] Zhang, Y. L., Luo, M. K.: Fractional rayleigh-duffing-like system and its synchronization. Nonlinear Dynamics 70 (2012), 1173-1183. DOI 10.1007/s11071-012-0521-0 | MR 2992124 | Zbl 1268.34089
[30] Zhang, B. T., Pi, Y. G., Luo, Y.: Fractional order sliding-mode control based on parameters auto-tuning for velocity control of permanent magnet synchronous motor. ISA Trans. 51 (2012), 649-656. DOI 10.1016/j.isatra.2012.04.006
[31] Zhou, J., Lu, j. A., Lu, J. H.: Adaptive synchronization of an uncertain complex dynamical network. IEEE Trans. Automat. Control 51 (2006), 652-656. DOI 10.1109/tac.2006.872760 | MR 2228029
[32] Zhu, W., Fang, J. A., Tang, Y.: Identification of fractional-order systems via a switching differential evolution subject to noise perturbations. Physics Lett. A 376 (2012), 3113-3120. DOI 10.1016/j.physleta.2012.09.042
[33] Zhu, H., He, Z. S., Zhou, S. B.: Lag synchronization of the fractional-order system via nonlinear observer. Int. J. Modern Physics B 25 (2011), 3951-3964. DOI 10.1142/s0217979211102253 | Zbl 1247.34099
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