Mean square function synchronization of chaotic systems with stochastic effects. (English) Zbl 1267.93080
Summary: In this paper, we give the definition of mean square function synchronization. Secondly, we investigate mean square function synchronization of chaotic systems with stochastic perturbation and unknown parameters. Based on the Lyapunov stability theory, inequality techniques, and the properties of the Weiner process, the controller, and adaptive laws are designed to ensure achieving stochastic synchronization of chaotic systems. A sufficient synchronization condition is given to ensure the chaotic systems to be mean-square stable. Furthermore, a numerical simulation is also given to demonstrate the effectiveness of the proposed scheme.
MSC:
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |
| 34D06 | Synchronization of solutions to ordinary differential equations |
| 34H10 | Chaos control for problems involving ordinary differential equations |
References:
| [1] | Pecora, L., Carroll, T.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990) · Zbl 0938.37019 · doi:10.1103/PhysRevLett.64.821 |
| [2] | Zhu, C.: Adaptive synchronization of two novel different hyperchaotic systems with partly uncertain parameters. Appl. Math. Comput. 215, 557–561 (2009) · Zbl 1182.37028 · doi:10.1016/j.amc.2009.05.026 |
| [3] | Du, H., Zeng, Q., Wang, C., Ling, M.: Function projective synchronization in coupled chaotic systems. Nonlinear Anal. 11, 705–712 (2010) · Zbl 1181.37039 · doi:10.1016/j.nonrwa.2009.01.016 |
| [4] | An, H., Chen, Y.: The function cascade synchronization method and applications. Commun. Nonlinear Sci. Numer. Simul. 13, 2246–2255 (2008) · Zbl 1221.93124 · doi:10.1016/j.cnsns.2007.05.029 |
| [5] | Zhang, R., Yang, Y., Xu, Z., Hu, M.: Function projective synchronization in drive–response dynamical network. Phys. Lett. A 374, 3025–3028 (2010) · Zbl 1237.34034 · doi:10.1016/j.physleta.2010.05.041 |
| [6] | Xu, Y., Zhou, W., Fang, J., Sun, W.: Adaptive bidirectionally coupled synchronization of chaotic systems with unknown parameters. Nonlinear Dyn. 66, 67–76 (2011) · Zbl 1280.93043 · doi:10.1007/s11071-010-9911-3 |
| [7] | Yu, Y., Li, H.: Adaptive generalized function projective synchronization of uncertain chaotic systems. Nonlinear Anal. 11, 2456–2464 (2010) · Zbl 1202.34095 · doi:10.1016/j.nonrwa.2009.08.002 |
| [8] | Yang, W., Sun, J.: Function projective synchronization of two-cell quantum-CNN chaotic oscillators by nonlinear adaptive controller. Phys. Lett. A 374, 557–561 (2010) · Zbl 1235.34161 · doi:10.1016/j.physleta.2009.11.050 |
| [9] | Du, H., Zeng, Q., Wang, C.: Function projective synchronization of different chaotic systems with uncertain parameters. Phys. Lett. A 372, 5402–5410 (2008) · Zbl 1223.34077 · doi:10.1016/j.physleta.2008.06.036 |
| [10] | Tang, Y., Fang, J.: General methods for modified projective synchronization of hyperchaotic systems with known or unknown parameters. Phys. Lett. A 372, 1816–1826 (2008) · Zbl 1220.37027 · doi:10.1016/j.physleta.2007.10.043 |
| [11] | Xu, Y., Zhou, W., Fang, J.: Topology identification of the modified complex dynamical network with non-delayed and delayed coupling. Nonlinear Dyn. 68, 195–205 (2012) · Zbl 1243.93025 · doi:10.1007/s11071-011-0217-x |
| [12] | Lu, J., Wu, X., Han, X., Lü, J.: Adaptive feedback synchronization of a unified chaotic system. Phys. Lett. A 329, 327–333 (2004) · Zbl 1209.93119 · doi:10.1016/j.physleta.2004.07.024 |
| [13] | Bowong, S.: Adaptive synchronization between two different chaotic dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 12, 976–985 (2007) · Zbl 1115.37030 · doi:10.1016/j.cnsns.2005.10.003 |
| [14] | Yu, W., Cao, J.: Adaptive synchronization and lag synchronization of uncertain dynamical system with time delay based on parameter identification. Physica A 375, 467–482 (2007) · doi:10.1016/j.physa.2006.09.020 |
| [15] | Luo, R.: Impulsive control and synchronization of a new chaotic system. Phys. Lett. A 372, 648–653 (2008) · Zbl 1217.37033 · doi:10.1016/j.physleta.2007.08.010 |
| [16] | Zhou, J., Lu, J., Lü, J.: Adaptive synchronization of an uncertain complex dynamical network. IEEE Trans. Autom. Control 51, 652–656 (2006) · Zbl 1366.93544 · doi:10.1109/TAC.2006.872760 |
| [17] | Lu, J., Cao, J.: Adaptive synchronization of uncertain dynamical networks with delayed coupling. Nonlinear Dyn. 53, 107–115 (2008) · Zbl 1182.92007 · doi:10.1007/s11071-007-9299-x |
| [18] | Xiao, M., Ho, D., Cao, J.: Time-delayed feedback control of dynamical small-world networks at Hopf bifurcation. Nonlinear Dyn. 58, 319–344 (2009) · Zbl 1183.93073 · doi:10.1007/s11071-009-9485-0 |
| [19] | Lü, L., Li, C.: Generalized synchronization of spatiotemporal chaos in a weighted complex network. Nonlinear Dyn. 63, 699–710 (2011) · doi:10.1007/s11071-010-9831-2 |
| [20] | Yang, C.: Adaptive control and synchronization of identical new chaotic flows with unknown parameters via single input. Appl. Math. Comput. 216, 1316–1324 (2010) · Zbl 1192.93063 · doi:10.1016/j.amc.2010.02.026 |
| [21] | Zhang, Q., Chen, S., Hu, Y.: Synchronizing the noise-perturbed unified chaotic system by sliding mode control. Physica A 371(2), 317–324 (2006) · doi:10.1016/j.physa.2006.03.057 |
| [22] | Haeri, M., Tavazoei, M., Naseh, M.: Synchronization of uncertain chaotic systems using active sliding mode control. Chaos Solitons Fractals 33(4), 1230–1239 (2007) · Zbl 1138.93045 · doi:10.1016/j.chaos.2006.01.076 |
| [23] | Cao, J., Wang, Z., Sun, Y.: Synchronization in an array of linearly stochastically coupled networks with time delays. Physica A 385(2), 718–728 (2007) · doi:10.1016/j.physa.2007.06.043 |
| [24] | Mao, X.: Stochastic Differential Equations and Applications. Horwood, Chichester (1997) · Zbl 0892.60057 |
| [25] | Lü, J., Chen, G.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 12, 659–661 (2002) · Zbl 1063.34510 · doi:10.1142/S0218127402004620 |
| [26] | Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9, 1465–1466 (1999) · Zbl 0962.37013 · doi:10.1142/S0218127499001024 |
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