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2015, 5(4): 827-844. doi: 10.3934/mcrf.2015.5.827

Adaptive projective synchronization of memristive neural networks with time-varying delays and stochastic perturbation

1. 

Department of Mathematics and Research Center for Complex Systems and Network Sciences, Southeast University, 210096, Nanjing, China

2. 

Department of Applied Mathematics, Yanshan University, 066001, Qinhuangdao, China, China, China

Received  May 2014 Revised  January 2015 Published  October 2015

This paper is concerned with the projective synchronization issue for memristive neural networks with time-varying delays and stochastic perturbations. Based on LaSalle-type invariance principle of stochastic functional-differential equations, by applying Lyapunov functional approach, several sufficient conditions are developed to achieve the projective synchronization between the master-slave systems with time-varying delays under stochastic perturbation and adaptive controller. A numerical example and its simulation is given to show the effectiveness of the theoretical results in this paper.
Citation: Ruoxia Li, Huaiqin Wu, Xiaowei Zhang, Rong Yao. Adaptive projective synchronization of memristive neural networks with time-varying delays and stochastic perturbation. Mathematical Control & Related Fields, 2015, 5 (4) : 827-844. doi: 10.3934/mcrf.2015.5.827
References:
[1]

J. Aubin and H. Frankowska, Set-valued Analysis,, Birkhäuser, (1990).

[2]

S. Bowong, F. M. Moukam Kakmeni and H. Fotsin, A new adaptive observer-based synchronization scheme for private communication,, Physics Letters A, 355 (2006), 193. doi: 10.1016/j.physleta.2006.02.035.

[3]

L. Chua, Memristor-the missing circut element,, IEEE Trans, 18 (1971), 507.

[4]

T. Driscoll, J. Quinn, S. Klein, H. T. Kim, B. J. Kim, Y. V. Pershin, M. DiVentra and D. N. Basov, Memristive adaptive filters,, Appl. Phys. Lett., 97 (2010). doi: 10.1063/1.3485060.

[5]

A. Friedman, Stochastic Differential Equations and Applications,, Academic Press, (1976).

[6]

L. Hu, H. J. Gao and W. X. Zheng, Novel stability of cellular neural networks with interval time-varying delay,, Neural Networks, 21 (2008), 1458. doi: 10.1016/j.neunet.2008.09.002.

[7]

X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations,, Journal of Mathematical Analysis and Applications, 268 (2002), 125. doi: 10.1006/jmaa.2001.7803.

[8]

L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems,, Physical Review Letters, 64 (1990), 821. doi: 10.1103/PhysRevLett.64.821.

[9]

Y. V. Pershin, S. La Fontaine and M. Di Ventra, Memristive model of amoeba learning, preprint,, 2009., ().

[10]

Y. V. Pershin and M. Di Ventra, Experimental demonstration of associative memory with memristive neural networks,, Neural Networks, 23 (2010), 881. doi: 10.1016/j.neunet.2010.05.001.

[11]

Y. Shen and J. Wang, An improved algebraic criterion for global exponential stability of recurrent neural networks with time-varying delays,, Neural Networks, 19 (2008), 528.

[12]

D. B. Strukov, G. S. Snider, D. R. Stewart and R. S. Williams, The missing memristor found,, Nature, 453 (2008), 80.

[13]

S. Sundar and A. Minai, Synchronization of randomly multiplexed chaotic systems with application to communication,, Physical Review Letters, 85 (2000), 5456. doi: 10.1103/PhysRevLett.85.5456.

[14]

J. M. Tour and T. He. Electronics, Electronics: The fourth element,, Nature, 453 (2008), 42. doi: 10.1038/453042a.

[15]

A. H. Wan, J. G. Peng and M. S. Wang, Exponential stability of a class of generalized neural networks with time-varying delays,, Neurocomputing, 69 (2006), 959. doi: 10.1016/j.neucom.2005.06.012.

[16]

R. L. Wang, Z. Tang and Q. P. Cao, A learning method in Hopfield neural network for combinatorial optimization problem,, Neurocomputing, 48 (2002), 1021. doi: 10.1016/S0925-2312(02)00596-9.

[17]

S. P. Wen, Z. G. Zeng and T. W. Huang, Adaptive synchronization of memristor-based Chua's circuits,, Physics Letters A, 376 (2012), 2775.

[18]

S. P. Wen, Z. G. Zeng and T. W. Huang, Dynamic behaviors of memristor-based delayed recurrent networks,, Neural Computing and Applications, 23 (2013), 815. doi: 10.1007/s00521-012-0998-y.

[19]

A. L. Wu, S. P. Wen and Z. G. Zeng, Exponential synchronization of memristor-based recurrent neural networks with time delays,, Neurocomputing, 74 (2011), 3043. doi: 10.1016/j.neucom.2011.04.016.

[20]

A. L. Wu, S. P. Wen and Z. G. Zeng, Synchronization control of a class of memristor-based recurrent neural networks,, Information Sciences, 183 (2012), 106. doi: 10.1016/j.ins.2011.07.044.

[21]

A. L. Wu and Z. G. Zeng, Anti-synchronization control of a class of memristive recurrent neural networks,, Communications in Nonlinear Science and Numerical, 18 (2013), 373. doi: 10.1016/j.cnsns.2012.07.005.

[22]

S. Y. Xu, Y. M. Chu and J. W. Lu, New results on global exponential stability of recurrent neural networks with time-varying delays,, Physics. Letters. A, 352 (2006), 371. doi: 10.1016/j.physleta.2005.12.031.

[23]

G. D. Zhang and Y. Shen, Global exponential periodicity and stability of a class of memristor-based recurrent neural networks with multiple delays,, Information Sciences, 232 (2013), 386. doi: 10.1016/j.ins.2012.11.023.

[24]

H. G. Zhang and G. Wang, New criteria of global exponential stability for a class of generalized neural networks with time-varying delays,, Neurocomputing, 70 (2007), 2486. doi: 10.1016/j.neucom.2006.08.002.

[25]

L. Zhang, Y. Zhang and J. L. Yu, Multiperiodicity and attractivity of delayed recurrent neural networks with unsaturating piecewise linear transfer functions,, Neural Networks, 19 (2008), 158. doi: 10.1109/TNN.2007.904015.

[26]

L. Zhang and Y. Zhang, Selectable and unselectable sets of neurons in recurrent neural networks with saturated piecewise linear transfer function,, Neural Networks, 22 (2011), 1021. doi: 10.1109/TNN.2011.2132762.

[27]

L. Zhang, Y. Zhang, S. L. Zhang and P. A. Heng, Activity invariant sets and exponentially stable attractors of linear threshold discrete-time recurrent neural networks,, Automatic Control, 54 (2009), 1341. doi: 10.1109/TAC.2009.2015552.

[28]

D. M. Zhou and J. D. Cao, Globally exponential stability conditions for cellular neural networks with time-varying delays,, Applied Mathematics and Computation, 131 (2002), 487. doi: 10.1016/S0096-3003(01)00162-X.

show all references

References:
[1]

J. Aubin and H. Frankowska, Set-valued Analysis,, Birkhäuser, (1990).

[2]

S. Bowong, F. M. Moukam Kakmeni and H. Fotsin, A new adaptive observer-based synchronization scheme for private communication,, Physics Letters A, 355 (2006), 193. doi: 10.1016/j.physleta.2006.02.035.

[3]

L. Chua, Memristor-the missing circut element,, IEEE Trans, 18 (1971), 507.

[4]

T. Driscoll, J. Quinn, S. Klein, H. T. Kim, B. J. Kim, Y. V. Pershin, M. DiVentra and D. N. Basov, Memristive adaptive filters,, Appl. Phys. Lett., 97 (2010). doi: 10.1063/1.3485060.

[5]

A. Friedman, Stochastic Differential Equations and Applications,, Academic Press, (1976).

[6]

L. Hu, H. J. Gao and W. X. Zheng, Novel stability of cellular neural networks with interval time-varying delay,, Neural Networks, 21 (2008), 1458. doi: 10.1016/j.neunet.2008.09.002.

[7]

X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations,, Journal of Mathematical Analysis and Applications, 268 (2002), 125. doi: 10.1006/jmaa.2001.7803.

[8]

L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems,, Physical Review Letters, 64 (1990), 821. doi: 10.1103/PhysRevLett.64.821.

[9]

Y. V. Pershin, S. La Fontaine and M. Di Ventra, Memristive model of amoeba learning, preprint,, 2009., ().

[10]

Y. V. Pershin and M. Di Ventra, Experimental demonstration of associative memory with memristive neural networks,, Neural Networks, 23 (2010), 881. doi: 10.1016/j.neunet.2010.05.001.

[11]

Y. Shen and J. Wang, An improved algebraic criterion for global exponential stability of recurrent neural networks with time-varying delays,, Neural Networks, 19 (2008), 528.

[12]

D. B. Strukov, G. S. Snider, D. R. Stewart and R. S. Williams, The missing memristor found,, Nature, 453 (2008), 80.

[13]

S. Sundar and A. Minai, Synchronization of randomly multiplexed chaotic systems with application to communication,, Physical Review Letters, 85 (2000), 5456. doi: 10.1103/PhysRevLett.85.5456.

[14]

J. M. Tour and T. He. Electronics, Electronics: The fourth element,, Nature, 453 (2008), 42. doi: 10.1038/453042a.

[15]

A. H. Wan, J. G. Peng and M. S. Wang, Exponential stability of a class of generalized neural networks with time-varying delays,, Neurocomputing, 69 (2006), 959. doi: 10.1016/j.neucom.2005.06.012.

[16]

R. L. Wang, Z. Tang and Q. P. Cao, A learning method in Hopfield neural network for combinatorial optimization problem,, Neurocomputing, 48 (2002), 1021. doi: 10.1016/S0925-2312(02)00596-9.

[17]

S. P. Wen, Z. G. Zeng and T. W. Huang, Adaptive synchronization of memristor-based Chua's circuits,, Physics Letters A, 376 (2012), 2775.

[18]

S. P. Wen, Z. G. Zeng and T. W. Huang, Dynamic behaviors of memristor-based delayed recurrent networks,, Neural Computing and Applications, 23 (2013), 815. doi: 10.1007/s00521-012-0998-y.

[19]

A. L. Wu, S. P. Wen and Z. G. Zeng, Exponential synchronization of memristor-based recurrent neural networks with time delays,, Neurocomputing, 74 (2011), 3043. doi: 10.1016/j.neucom.2011.04.016.

[20]

A. L. Wu, S. P. Wen and Z. G. Zeng, Synchronization control of a class of memristor-based recurrent neural networks,, Information Sciences, 183 (2012), 106. doi: 10.1016/j.ins.2011.07.044.

[21]

A. L. Wu and Z. G. Zeng, Anti-synchronization control of a class of memristive recurrent neural networks,, Communications in Nonlinear Science and Numerical, 18 (2013), 373. doi: 10.1016/j.cnsns.2012.07.005.

[22]

S. Y. Xu, Y. M. Chu and J. W. Lu, New results on global exponential stability of recurrent neural networks with time-varying delays,, Physics. Letters. A, 352 (2006), 371. doi: 10.1016/j.physleta.2005.12.031.

[23]

G. D. Zhang and Y. Shen, Global exponential periodicity and stability of a class of memristor-based recurrent neural networks with multiple delays,, Information Sciences, 232 (2013), 386. doi: 10.1016/j.ins.2012.11.023.

[24]

H. G. Zhang and G. Wang, New criteria of global exponential stability for a class of generalized neural networks with time-varying delays,, Neurocomputing, 70 (2007), 2486. doi: 10.1016/j.neucom.2006.08.002.

[25]

L. Zhang, Y. Zhang and J. L. Yu, Multiperiodicity and attractivity of delayed recurrent neural networks with unsaturating piecewise linear transfer functions,, Neural Networks, 19 (2008), 158. doi: 10.1109/TNN.2007.904015.

[26]

L. Zhang and Y. Zhang, Selectable and unselectable sets of neurons in recurrent neural networks with saturated piecewise linear transfer function,, Neural Networks, 22 (2011), 1021. doi: 10.1109/TNN.2011.2132762.

[27]

L. Zhang, Y. Zhang, S. L. Zhang and P. A. Heng, Activity invariant sets and exponentially stable attractors of linear threshold discrete-time recurrent neural networks,, Automatic Control, 54 (2009), 1341. doi: 10.1109/TAC.2009.2015552.

[28]

D. M. Zhou and J. D. Cao, Globally exponential stability conditions for cellular neural networks with time-varying delays,, Applied Mathematics and Computation, 131 (2002), 487. doi: 10.1016/S0096-3003(01)00162-X.

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