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Adaptive chaos synchronization in Chua’s systems with noisy parameters. (English) Zbl 1166.34029

The Chua’s system is modeled by the following differential equations

x ˙ 1 =px 2 -1 7(2x 1 3 -x 1 ),x ˙ 2 =x 1 -x 2 +x 3 ,x ˙ 3 =-qx 2 +rx 1 2 ,(1)

where p, q and r are real positive constants, and x i , i=1,,3, are state variables. The parameters p,q,r are unknown and in additional, due to system uncontrollable variations, their actual values deviate randomly around their mean values.

The synchronization problem of two chaotic Chua systems whose coefficients are unknown and stochastically time varying is studied. Stochastic behaviour of the parameters is modeled by white Gaussian noise generated by a Wiener process. Using the Lyapounov stability theory, a Markov adaptive control law is designed for synchronizing the stochastic chaotic behaviour of the two systems in the sense of mean value convergence. Simulation results indicate that the proposed adaptive controller has a high performance in synchronization of chaotic Chua circuits in noisy environment.

MSC:
34D05Asymptotic stability of ODE
34F05ODE with randomness
34C28Complex behavior, chaotic systems (ODE)
34C15Nonlinear oscillations, coupled oscillators (ODE)
34H05ODE in connection with control problems
93C40Adaptive control systems
References:
[1]Baker, G. L.; Blackburn, J. A.; Smith, H. J. T.: A stochastic model of synchronization for chaotic pendulums, Phys. lett. A 252, 191-197 (1999)
[2]Billings, L.; Bollt, E. M.; Schwartz, I. B.: Phase-space transport of stochastic chaos in population dynamics of virus spread, Phys. rev. Lett. 88, 234101 (2002)
[3]Botmart, T.; Niamsup, P.: Adaptive control and synchronization of the perturbed Chua’s system, Math. comput. Simul. 75, 37-55 (2007) · Zbl 1115.37072 · doi:10.1016/j.matcom.2006.08.008
[4]Carroll, T. L.; Pecora, L. M.: Synchronization in chaotic systems, Phys. rev. Lett. 64, 821-824 (1990)
[5]G. Chen, Control and Synchronization of Chaos, a Bibliography, Department of Electrical Engineering, University of Houston, TX, 1997.
[6]Chen, G.; Dong, X.: On feedback control of chaotic continuous time systems, IEEE trans. Circ. syst. 40, 591-601 (1993) · Zbl 0800.93758 · doi:10.1109/81.244908
[7]Chua, L.; Komuro, M.; Matsumoto, T.: The double scroll family, IEEE trans. Circ. syst. 33, 1073-1118 (1986) · Zbl 0634.58015 · doi:10.1109/TCS.1986.1085869
[8]Feng, J.; Chen, S.; Wang, C.: Adaptive synchronization of uncertain hyperchaotic systems based on parameter identification, Chaos solitons fractals 26, 1163-1169 (2005) · Zbl 1122.93401 · doi:10.1016/j.chaos.2005.02.027
[9]Freeman, W. J.: A proposed name for aperiodic brain activity: stochastic chaos, Neural networks 13, 11-13 (2000)
[10]Kakmeni, F. M. M.; Bowong, S.; Tchawoua, C.: Nonlinear adaptive synchronization of a class of chaotic systems, Phys. lett. A 355, 47-54 (2006)
[11]Kapitaniak, T.: Continuous control and synchronization in chaotic systems, Chaos solitons fractals 6, 237-244 (1995) · Zbl 0976.93504 · doi:10.1016/0960-0779(95)80030-K
[12]Mackevicius, V.: A note on synchronization of diffusion, Math. comput. Simul. 52, 491-495 (2000)
[13]Oksendal, B.: Stochastic differential equations, an introduction with applications, (1992) · Zbl 0747.60052
[14]Park, J. H.: Synchronization of Genesio chaotic system via backstepping approach, Chaos solitons fractals 27, 1369-1375 (2006) · Zbl 1091.93028 · doi:10.1016/j.chaos.2005.05.001
[15]Salarieh, H.; Shahrokhi, M.: Indirect adaptive control of discrete chaotic systems, Chaos solitons fractals 34, 1188-1201 (2007) · Zbl 1142.93359 · doi:10.1016/j.chaos.2006.03.115
[16]Salarieh, H.; Shahrokhi, M.: Adaptive synchronization of two chaotic systems with time varying unknown parameters, Chaos solitons fractals 37, 125-136 (2008) · Zbl 1147.93397 · doi:10.1016/j.chaos.2006.08.038
[17]Wu, C.; Fang, T.; Rong, H.: Chaos synchronization of two stochastic Duffing oscillators by feedback control, Chaos solitons fractals 32, 1201-1207 (2007) · Zbl 1129.37016 · doi:10.1016/j.chaos.2005.11.042
[18]Wu, C.; Lei, Y.; Fang, T.: Stochastic chaos in a Duffing oscillator and its control, Chaos solitons fractals 27, 459-469 (2005) · Zbl 1102.37312 · doi:10.1016/j.chaos.2005.04.035
[19]Yassen, M. T.: Adaptive control and synchronization of a modified Chua’s system, Appl. math. Comput. 135, 113-128 (2003) · Zbl 1038.34041 · doi:10.1016/S0096-3003(01)00318-6
[20]Yu, W.; Cao, J.: Synchronization control of stochastic delayed neural networks, Physica A 373, 252-260 (2006)
[21]Zhang, H.; Huang, W.; Wang, Z.; Chai, T.: Adaptive synchronization between two different chaotic systems with unknown parameters, Phys. lett. A 350, 363-366 (2006) · Zbl 1195.93121 · doi:10.1016/j.physleta.2005.10.033