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December  2016, 21(10): 3655-3667. doi: 10.3934/dcdsb.2016115

Finite-time synchronization of competitive neural networks with mixed delays

1. 

Department of Mathematics, Chongqing Normal University, Chongqing 401331, China, China

Received  September 2014 Revised  March 2016 Published  November 2016

In this paper, finite-time synchronization of competitive neural networks (CNNs) with bounded time-varying discrete and distributed delays (mixed delays) is investigated. A simple controller is added to response (slave) system such that it can be synchronized with the driving (master) CNN in a setting time. By introducing a suitable Lyapunov-Krasovskii's functional and utilizing some inequalities, several sufficient conditions are obtained to ensure the control object. Moreover, the setting time is explicitly given. Different from previous results, the setting is related to both the initial value of error system and the time delays. Finally, numerical examples are given to show the effectiveness of the theoretical results.
Citation: Tingting Su, Xinsong Yang. Finite-time synchronization of competitive neural networks with mixed delays. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3655-3667. doi: 10.3934/dcdsb.2016115
References:
[1]

M. P. Aghababa, S. Khanmohammadi and G. Alizadeh, Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique, Applied Mathematical Modelling, 35 (2011), 3080-3091. doi: 10.1016/j.apm.2010.12.020.

[2]

Y. Cheng, H. De, Y. He and R. Jia, Robust finite-time synchronization of coupled harmonic oscillations with external disturbance, Journal of the Franklin Institute, 352 (2015), 4366-4381. doi: 10.1016/j.jfranklin.2015.06.006.

[3]

D. Efimov, A. Polyakov, E. Fridman, W. Perruquetti and J.-P. Richard, Comments on finite-time stability of time-delay systems, Automatica, 50 (2014), 1944-1947. doi: 10.1016/j.automatica.2014.05.010.

[4]

Q. Gan, R. Hu and Y. Liang, Adaptive synchronization for stochastic competitive neural networks with mixed time-varying delays, Commun. Nonlinear Sci. Numer. Simul, 17 (2012), 3708-3718. doi: 10.1016/j.cnsns.2012.01.021.

[5]

H. Gu, H. Jiang and Z. Teng, Existence and global exponential stability of equlilbrium of competitive nearal networks with different time scales and multiple delays, Journal of the Franklin Institute, 347 (2010), 719-731. doi: 10.1016/j.jfranklin.2009.03.005.

[6]

W. He and J. Cao, Exponential synchronization of chaotic neural networks: A matrix measure approach, Nonlinear Dyn., 55 (2009), 55-65. doi: 10.1007/s11071-008-9344-4.

[7]

E. Moulay, M. Dambrine, N. Yeganefar and W. Perruquetti, Finite-time stability and stabilization of time-delay systems, Systems & Control Letters, 57 (2008), 561-566. doi: 10.1016/j.sysconle.2007.12.002.

[8]

X. Nie and J. Cao, Multistability of competive neural networks with time-varying and distributed delays, Nonlinear Analysis B: Real World Applications, 10 (2009), 928-942. doi: 10.1016/j.nonrwa.2007.11.014.

[9]

M. Pecora and L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824. doi: 10.1103/PhysRevLett.64.821.

[10]

Y. Shi and P. Zhu, Synchronization of stochastic competitive neural networks with different timescales and reaction-diffusion terms, Neural Comput., 26 (2014), 2005-2024. doi: 10.1162/NECO_a_00629.

[11]

N. Stepp, Anticipating in feedback-delayed manual tracking of a chaotic oscillation, Exp. Brain Res., (2009), 521-525.

[12]

Y. Tang, Terminal sliding mode control for rigid robots, Automatica, 34 (1998), 51-56. doi: 10.1016/S0005-1098(97)00174-X.

[13]

Y. Tang and J. Fang, Adaptive synchronization in an array of chaotic neural networks with mixed delays and jumping stochastically hybrid coupling, Commun. Nonlinear Sci. Numer. Simul, 14 (2009), 3615-3628. doi: 10.1016/j.cnsns.2009.02.006.

[14]

M. Timme and F. Wolf, The simplest problem in the collective dynamics of neural networks: Is synchrony stable? Nonlinearity, 21 (2008), 1579-1599. doi: 10.1088/0951-7715/21/7/011.

[15]

H. U. Voss, Anticipating chaotic synchronization, Phys. Rev. E, (2001), 191-210.

[16]

X. Yang and J. Cao, Finite-time stochastic synchronization of complex networks, Applied Mathematical Modelling, 34 (2010), 3631-3641. doi: 10.1016/j.apm.2010.03.012.

[17]

X. Yang, J. Cao and J. Lu, Stochastic synchronization of complex networks with nonidentical nodes via hybrid adaptive and impulsive control, IEEE Trans. Circ. Syst. -I. Regular Paper, 59 (2012), 371-384. doi: 10.1109/TCSI.2011.2163969.

[18]

X. Yang, J. Cao and Z. Yang, Synchronization of coupled reaction-diffusion neural networks with time-varying delays via pinning-impulsive controller, SIAM J. Control Optim., 51 (2013), 3486-3510. doi: 10.1137/120897341.

[19]

X. Yang, D. W. C. Ho, J. Lu and Q. Song, Finite-time cluster synchronization of T-S fuzzy complex networks with discontinuous subsystems and random coupling delays, IEEE Transactions on Fuzzy Systems, 23 (2015), 2302-2316. doi: 10.1109/TFUZZ.2015.2417973.

[20]

X. Yang, C. Huang and Q. Zhu, Synchronization of switched neural networks with mixed delays via impulsive control, Chaos Solitons Fractals, 44 (2011), 817-826. doi: 10.1016/j.chaos.2011.06.006.

[21]

X. Yang, Z. Wu and J. Cao, Finite-time synchronization of complex networks with nonidentical discontinuous nodes, Nonlinear Dyn., 73 (2013), 2313-2327. doi: 10.1007/s11071-013-0942-4.

[22]

X. Yang, Z. Yang and X. Nie, Exponential synchronization of discontinuous chaotic systems via delayed impulsive control and its application to secure communication, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1529-1543. doi: 10.1016/j.cnsns.2013.09.012.

[23]

P. Zachary and C. Paul, Binocular rivalry in a competitive neural network with synaptic depression, SIAM Journal on Applied Dynamical Systems, 9 (2010), 1303-1347. doi: 10.1137/100788872.

show all references

References:
[1]

M. P. Aghababa, S. Khanmohammadi and G. Alizadeh, Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique, Applied Mathematical Modelling, 35 (2011), 3080-3091. doi: 10.1016/j.apm.2010.12.020.

[2]

Y. Cheng, H. De, Y. He and R. Jia, Robust finite-time synchronization of coupled harmonic oscillations with external disturbance, Journal of the Franklin Institute, 352 (2015), 4366-4381. doi: 10.1016/j.jfranklin.2015.06.006.

[3]

D. Efimov, A. Polyakov, E. Fridman, W. Perruquetti and J.-P. Richard, Comments on finite-time stability of time-delay systems, Automatica, 50 (2014), 1944-1947. doi: 10.1016/j.automatica.2014.05.010.

[4]

Q. Gan, R. Hu and Y. Liang, Adaptive synchronization for stochastic competitive neural networks with mixed time-varying delays, Commun. Nonlinear Sci. Numer. Simul, 17 (2012), 3708-3718. doi: 10.1016/j.cnsns.2012.01.021.

[5]

H. Gu, H. Jiang and Z. Teng, Existence and global exponential stability of equlilbrium of competitive nearal networks with different time scales and multiple delays, Journal of the Franklin Institute, 347 (2010), 719-731. doi: 10.1016/j.jfranklin.2009.03.005.

[6]

W. He and J. Cao, Exponential synchronization of chaotic neural networks: A matrix measure approach, Nonlinear Dyn., 55 (2009), 55-65. doi: 10.1007/s11071-008-9344-4.

[7]

E. Moulay, M. Dambrine, N. Yeganefar and W. Perruquetti, Finite-time stability and stabilization of time-delay systems, Systems & Control Letters, 57 (2008), 561-566. doi: 10.1016/j.sysconle.2007.12.002.

[8]

X. Nie and J. Cao, Multistability of competive neural networks with time-varying and distributed delays, Nonlinear Analysis B: Real World Applications, 10 (2009), 928-942. doi: 10.1016/j.nonrwa.2007.11.014.

[9]

M. Pecora and L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824. doi: 10.1103/PhysRevLett.64.821.

[10]

Y. Shi and P. Zhu, Synchronization of stochastic competitive neural networks with different timescales and reaction-diffusion terms, Neural Comput., 26 (2014), 2005-2024. doi: 10.1162/NECO_a_00629.

[11]

N. Stepp, Anticipating in feedback-delayed manual tracking of a chaotic oscillation, Exp. Brain Res., (2009), 521-525.

[12]

Y. Tang, Terminal sliding mode control for rigid robots, Automatica, 34 (1998), 51-56. doi: 10.1016/S0005-1098(97)00174-X.

[13]

Y. Tang and J. Fang, Adaptive synchronization in an array of chaotic neural networks with mixed delays and jumping stochastically hybrid coupling, Commun. Nonlinear Sci. Numer. Simul, 14 (2009), 3615-3628. doi: 10.1016/j.cnsns.2009.02.006.

[14]

M. Timme and F. Wolf, The simplest problem in the collective dynamics of neural networks: Is synchrony stable? Nonlinearity, 21 (2008), 1579-1599. doi: 10.1088/0951-7715/21/7/011.

[15]

H. U. Voss, Anticipating chaotic synchronization, Phys. Rev. E, (2001), 191-210.

[16]

X. Yang and J. Cao, Finite-time stochastic synchronization of complex networks, Applied Mathematical Modelling, 34 (2010), 3631-3641. doi: 10.1016/j.apm.2010.03.012.

[17]

X. Yang, J. Cao and J. Lu, Stochastic synchronization of complex networks with nonidentical nodes via hybrid adaptive and impulsive control, IEEE Trans. Circ. Syst. -I. Regular Paper, 59 (2012), 371-384. doi: 10.1109/TCSI.2011.2163969.

[18]

X. Yang, J. Cao and Z. Yang, Synchronization of coupled reaction-diffusion neural networks with time-varying delays via pinning-impulsive controller, SIAM J. Control Optim., 51 (2013), 3486-3510. doi: 10.1137/120897341.

[19]

X. Yang, D. W. C. Ho, J. Lu and Q. Song, Finite-time cluster synchronization of T-S fuzzy complex networks with discontinuous subsystems and random coupling delays, IEEE Transactions on Fuzzy Systems, 23 (2015), 2302-2316. doi: 10.1109/TFUZZ.2015.2417973.

[20]

X. Yang, C. Huang and Q. Zhu, Synchronization of switched neural networks with mixed delays via impulsive control, Chaos Solitons Fractals, 44 (2011), 817-826. doi: 10.1016/j.chaos.2011.06.006.

[21]

X. Yang, Z. Wu and J. Cao, Finite-time synchronization of complex networks with nonidentical discontinuous nodes, Nonlinear Dyn., 73 (2013), 2313-2327. doi: 10.1007/s11071-013-0942-4.

[22]

X. Yang, Z. Yang and X. Nie, Exponential synchronization of discontinuous chaotic systems via delayed impulsive control and its application to secure communication, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1529-1543. doi: 10.1016/j.cnsns.2013.09.012.

[23]

P. Zachary and C. Paul, Binocular rivalry in a competitive neural network with synaptic depression, SIAM Journal on Applied Dynamical Systems, 9 (2010), 1303-1347. doi: 10.1137/100788872.

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