Finite-time synchronization of chaotic systems with noise perturbation

Wu, Jie ; Ma, Zhi-cai ; Sun, Yong-zheng ; Liu, Feng

Kybernetika, Tome 51 (2015) no. 1, pp. 137-149

Abstract (VO)

In this paper, we investigate the finite-time stochastic synchronization problem of two chaotic systems with noise perturbation. We propose new adaptive controllers, with which we can synchronize two chaotic systems in finite time. Sufficient conditions for the finite-time stochastic synchronization are derived based on the finite-time stability theory of stochastic differential equations. Finally, some numerical examples are examined to demonstrate the effectiveness and feasibility of the theoretical results.

Détail

DOI : 10.14736/kyb-2015-1-0137

Classification : 34F05, 34H10, 93A14, 93C40
Keywords: synchronization; finite-time; noise perturbation; adaptive feedback controller

Bibliographie

[1] Aghababa, M. P., Aghababa, H. P.: A general nonlinear adaptive control scheme for finite-time synchronization of chaotic systems with uncertain parameters and nonlinear inputs. Nonlinear Dyn. 69 (2012), 1903-1914. | DOI | MR | Zbl

[2] Aghababa, M. P., Aghababa, H. P.: A novel finite-time sliding mode controller for synchronization of chaotic systems with input nonlinearity. Arab. J. Sci. Eng. 38 (2013), 3221-3232. | DOI | MR

[3] Aghababa, M. P., Khanmohammadi, S., Alizadeh, G.: Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique. Appl. Math. Model. 35 (2011), 3080-3091. | DOI | MR | Zbl

[4] Alvarez, G., Hernández, L., Muñoz, J., Montoya, F., Li, S. J.: Security analysis of communication system based on the synchronization of different order chaotic systems. Phys. Lett. A 345 (2005), 245-250. | DOI

[5] Argenti, F., DeAngeli, A., DelRe, E., Genesio, R., Pagni, P., Tesi, A.: Secure communications based on discrete time chaotic systems. Kybernetika 33 (1997), 41-50. | MR

[6] Beran, Z.: On characterization of the solution set in case of generalized semiflow. Kybernetika 45 (2009), 701-715. | MR | Zbl

[7] Boccaletti, S., Kurths, J., Osipov, G., Valladares, D. L., Zhou, C. S.: The synchronization of chaotic systems. Phys. Rep. 366 (2002), 1-101. | DOI | MR | Zbl

[8] Cai, N., Li, W. Q., Jing, Y. W.: Finite-time generalized synchronization of chaotic systems with different order. Nonlinear Dyn. 64 (2011), 385-393. | DOI | MR

[9] Cheng, S., Ji, J. C., Zhou, J.: Fast synchronization of directionally coupled chaotic systems. Appl. Math. Model. 37 (2013), 127-136. | DOI | MR

[10] Čelikovský, S.: Observer form of the hyperbolic-type generalized Lorenz system and its use for chaos synchronization. Kybernetika 40 (2004), 649-664. | MR | Zbl

[11] Čelikovský, S., Chen, G. R.: On the generalized Lorenz canonical form. Chaos Solition. Fract. 26 (2005), 1271-1276. | DOI | MR | Zbl

[12] Ding, K., Han, Q. L.: Effects of coupling delays on synchronization in Lur'e complex dynamical networks. Int. J. Bifur. Chaos 20 (2010), 3565-3584. | DOI | MR | Zbl

[13] Ding, K., Han, Q. L.: Master-slave synchronization criteria for horizontal platform systems using time delay feedback control. J. Sound Vibration 330 (2011), 2419-2436. | DOI

[14] Ding, K., Han, Q. L.: Master-slave synchronization of nonautonomous chaotic systems and its application to rotating pendulums. Int. J. Bifur. Chaos 22 (2012), 1250147. | DOI | Zbl

[15] Enjieu, K. H. G., Chabi, O. J. B., Woafo, P.: Synchronization dynamics in a ring of four mutually coupled biological systems. Commun. Nonlinear Sci. Numer. Simul. 13 (2008), 1361-1372. | DOI | MR

[16] Grosu, I., Padmanabanm, E., Roy, P. K., Dana, S. K.: Designing coupling for synchronization and amplification of chaos. Phys. Rev. Lett. 100 (2008), 234102. | DOI

[17] He, W. L., Cao, J. D.: Adaptive synchronization of a class of chaotic neural networks with known or unknown parameters. Phys. Lett. A 372 (2008), 408-416. | DOI | Zbl

[18] He, W. L., Du, W. L., Qian, F., Cao, J. D.: Synchronization analysis of heterogeneous dynamical networks. Neurocomputing 104 (2013), 146-154. | DOI

[19] He, W. L., Qian, F., Han, Q. L., Cao, J. D.: Synchronization error estimation and controller design for delayed Lur'e systems with parameter mismatches. IEEE Trans. Neur. Net. Lear. Systems 23 (2012), 1551-1563. | DOI

[20] Henrion, D.: Semidefinite characterisation of invariant measures for one-dimensional discrete dynamical systems. Kybernetika 48 (2012), 1089-1099. | MR | Zbl

[21] Huang, D. B.: Simple adaptive-feedback controller for identical chaos synchronization. Phys. Rev. E 71 (2005), 037203. | DOI

[22] Lasalle, J. P.: The extend of asymptotic stability. Proc. Natl. Acad. Sci. U. S. A. 46 (1960), 363-365. | DOI | MR

[23] Lasalle, J. P.: Some extensions of Liapunov's second method. IRE Trans. Circuit Theory 7 (1960), 520-527. | DOI | MR

[24] Li, H. Y., Hu, Y. A., Wang, R. Q.: Adaptive finite-time synchronization of cross-strict feedback hyperchaotic systems with parameter uncertainties. Kybernetika 49 (2013), 554-567. | MR

[25] Lin, J. S., Yan, J. J.: Adaptive synchronization for two identical generalized Lorenz chaotic systems via a single controller. Nonlinear Anal. Real. 10 (2009), 1151-1159. | DOI | MR | Zbl

[26] Liu, Y. J.: Circuit implementation and finite-time synchronization of the 4D Rabinovich hyperchaotic system. Nonlinear Dyn. 67 (2012), 89-96. | DOI | Zbl

[27] Lu, W. L., Chen, T. P.: New approach to synchronization analysis of linearly coupled ordinary differential systems. Physica D 213 (2006), 214-230. | DOI | MR | Zbl

[28] Lynnyk, V., Čelikovský, S.: On the anti-synchronization detection for the generalized Lorenz system and its applications to secure encryption. Kybernetika 46 (2010), 1-18. | MR | Zbl

[29] Mao, X.: Stochastic Differential Equations and Applications. Horwood 1997. | Zbl

[30] Ottino, J. M., Muzzio, F. J., Tjahjadi, M., Franjione, J. G., Jana, S. C., Kusch, H. A.: Chaos, symmetry, and self-similarity: exploiting order and disorder in mixing process. Science 257 (1992), 754-760. | DOI

[31] Pecora, L. M., Carroll, T. L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64 (1990), 821-824. | DOI | MR | Zbl

[32] Schiff, S. J., Jerger, K., Duong, D. H., Chang, T., Spano, M. L., Ditto, W. L.: Controlling chaos in the brain. Nature 370 (1994), 615-620. | DOI

[33] Yan, J. J., Hung, M. L., Chiang, T. Y., Yang, Y. Q.: Robust synchronization of chaotic systems via adaptive sliding mode control. Phys. Lett. A 356 (2006), 220-225. | DOI | Zbl

[34] Ma, J., Zhang, A.H., Xia, Y.F., Zhang, L.: Optimize design of adaptive synchronization controllers and parameter observers in different hyperchaotic systems. Appl. Math. Comput. 215 (2010), 3318-3326. | DOI | MR

[35] Vincent, U. E., Guo, R.: Finite-time synchronization for a class of chaotic and hyperchaotic systems via adaptive feedback controller. Phys. Lett. A 375 (2011), 2322-2326. | DOI | MR | Zbl

[36] Wang, H., Han, Z. Z., Xie, Q. Y., Zhang, W.: Finite-time synchronization of uncertain unified chaotic systems based on CLF. Nonlinear Anal. Real. 10 (2009), 2842-2849. | DOI | MR | Zbl

[37] Yang, Y. Q., Wu, X. F.: Global finite-time synchronization of a class of the non-autonomous chaotic systems. Nonlinear Dyn. 70 (2012), 197-208. | DOI | MR | Zbl

[38] Yin, J. L., Khoo, S.: Comments on ``Finite-time stability theorem of stochastic nonlinear systems''. Automatica 47 (2011), 1542-1543. | DOI | MR

[39] Yin, J. L., Khoo, S., Man, Z. H., Yu, X. H.: Finite-time stability and instability of stochastic nonlinear systems. Automatica 47 (2011), 2671-2677. | DOI | MR | Zbl

[40] Zhao, J. K., Wu, Y., Wang, Y. Y.: Generalized finite-time synchronization between coupled chaotic systems of different orders with unknown parameters. Nonlinear Dyn. 74 (2013), 479-485. | DOI | MR | Zbl