Periodic solutions for stochastic Cohen–Grossberg neural networks with time-varying delays
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Wanqin Wu
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Yaping Ren
Abstract
This paper is concerned with the periodic solutions for a class of stochastic Cohen–Grossberg neural networks with time-varying delays. Since there is a non-linearity in the leakage terms of stochastic Cohen–Grossberg neural networks, some techniques are needed to overcome the difficulty in dealing with the nonlinearity. By applying fixed points principle and Gronwall–Bellman inequality, some sufficient conditions on the existence and exponential stability of periodic solution for the stochastic neural networks are established. Moreover, a numerical example is presented to validate the theoretical results. Our results are also applicable to the existence and exponential stability of periodic solution for the corresponding deterministic systems.
Funding source: Tian Yuan Fund of NSFC
Award Identifier / Grant number: 11526180
Funding source: Yunnan Province Education Department Scientific Research Fund Project
Award Identifier / Grant number: Nos.2018JS315, 2018JS309
Acknowledgments
All authors are grateful to the anonymous referees for their constructive comments and helpful suggestions.
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Author contribution: All authors contributed equally to the manuscript and typed, read and approved the final manuscript.
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Research funding: This work is supported by Tian Yuan Fund of NSFC (No. 11526180), Yunnan Province Education Department Scientific Research Fund Project (Nos. 2018JS315, 2018JS309), Special training program for outstanding young teachers of colleges and universities in Yunnan Province.
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Conflict of interest statement: All authors declare that they have no competing interests.
References
[1] M. A. Cohen, and S. Grossberg, “Absolute stability and global pattern formation and partial memory storage by compeitive neural networks,” IEEE Trans. Syst. Man Cybernet., vol. 13, pp. 815–826, 1983, https://doi.org/10.1109/tsmc.1983.6313075.Suche in Google Scholar
[2] J. R. Li, H. J. Jiang, C. Hu, and Z. Y. Yu, “Multiple types of synchronization analysis for discontinuous Cohen-Grossberg neural networks with time-varying delays,” Neural Netw., vol. 99, pp. 101–113, 2018, https://doi.org/10.1016/j.neunet.2017.12.011.Suche in Google Scholar PubMed
[3] D. S. Wang, and L. H. Huang, “Robust synchronization of discontinuous Cohen-Grossberg neural networks: Pinning control approach,” J. Franklin Inst., vol. 355, no. 13, pp. 5866–5892, 2018, https://doi.org/10.1016/j.jfranklin.2018.05.048.Suche in Google Scholar
[4] R. X. Li, J. D. Cao, A. Alsaedi, and F. Alsaadi, “Exponential and fixed-time synchronization of Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms,” Appl. Math. Comput., vol. 313, pp. 37–51, 2017, https://doi.org/10.1016/j.amc.2017.05.073.Suche in Google Scholar
[5] Z. Q. Zhang, and S. H. Yu, “Global asymptotic stability for a class of complex-valued Cohen-Grossberg neural networks with time delays,” Neurocomputing, vol. 171, pp. 1158–1166, 2016, https://doi.org/10.1016/j.neucom.2015.07.051.Suche in Google Scholar
[6] H. Y. Kang, X. C. Fu, and Z. L. Sun, “Global exponential stability of periodic solutions for impulsive Cohen-Grossberg neural networks with delays,” Appl. Math. Modell., vol. 39, no. 5-6, pp. 1526–1535, 2015, https://doi.org/10.1016/j.apm.2014.09.015.Suche in Google Scholar
[7] A. Abdurahman, and H. J. Jiang, “The existence and stability of the anti-periodic solution for delayed Cohen-Grossberg neural networks with impulsive effects,” Neurocomputing, vol. 149,Part A, pp. 22–28, 2015, https://doi.org/10.1016/j.neucom.2013.09.071.Suche in Google Scholar
[8] S. T. Qin, J. X. Xu, and X. Shi, “Convergence analysis for second-order interval Cohen-Grossberg neural networks,” Commun. Nonlinear Sci. Numerical Simul., vol. 19, no. 8, pp. 2747–2757, 2014, https://doi.org/10.1016/j.cnsns.2014.01.008.Suche in Google Scholar
[9] Z. Q. Zhang, W. B Liu, and D. M. Zhou, “Global asymptotic stability to a generalized Cohen-Grossberg BAM neural networks of neutral type delays,” Neural Netw., vol. 25, pp. 94–105, 2012, https://doi.org/10.1016/j.neunet.2011.07.006.Suche in Google Scholar PubMed
[10] Z. Q. Zhang, and T. Zheng, “Global asymptotic stability of periodic solutions for delayed complex-value d Cohen-Grossberg neural networks by combining coincidence degree theory with LMI method,” Neurocomputing, vol. 289, pp. 220–230, 2018, https://doi.org/10.1016/j.neucom.2018.02.033.Suche in Google Scholar
[11] S. Blythe, X. Mao, and X. Liao, “Stability of stochastic delay neural networks,” J. Franklin Inst., vol. 338, no. 5, pp. 481–495, 2001, https://doi.org/10.1016/s0016-0032(01)00016-3.Suche in Google Scholar
[12] Y. Sheng, and Z. G. Zeng, “Impulsive synchronization of stochastic reaction-diffusion neural networks with mixed time delays,” Neural Netw., vol. 103, pp. 83–93, 2018, https://doi.org/10.1016/j.neunet.2018.03.010.Suche in Google Scholar PubMed
[13] Y. Ren, Q. He, Y. F. Gu, and R. Sakthivel, “Mean-square stability of delayed stochastic neural networks with impulsive effects driven by G-Brownian motion,” Stat. Prob. Lett., vol. 143, pp. 56–66, 2018, https://doi.org/10.1016/j.spl.2018.07.024.Suche in Google Scholar
[14] G. L. Chen, D. S. Li, L. Shi, O. V. Gaans, and S. V. Lunel, “Stability results for stochastic delayed recurrent neural networks with discrete and distributed delays,” J. Different. Equat., vol. 264, no. 6, pp. 3864–3898, 2018, https://doi.org/10.1016/j.jde.2017.11.032.Suche in Google Scholar
[15] L. N. Liu, and F. D. Deng, “Stability analysis of time varying delayed stochastic Hopfield neural networks in numerical simulation,” Neurocomputing, vol. 316, pp. 294–305, 2018, https://doi.org/10.1016/j.neucom.2018.08.004.Suche in Google Scholar
[16] X. M. Jiang, S. Mahadevan, and Y. Yuan, “Fuzzy stochastic neural network model for structural system identification,” Mech. Syst. Signal Process., vol. 82, pp. 394–411, 2017, https://doi.org/10.1016/j.ymssp.2016.05.030.Suche in Google Scholar
[17] X. F. Li, and D. Ding, “Mean square exponential stability of stochastic Hopfield neural networks with mixed delays,” Stat. Prob. Lett., vol. 126, pp. 88–96, 2017, https://doi.org/10.1016/j.spl.2017.02.029.Suche in Google Scholar
[18] W. J. Xie, Q. X. Zhu, and F. Jiang, “Exponential stability of stochastic neural networks with leakage delays and expectations in the coefficients,” Neurocomputing, vol. 173, Part 3, pp. 1268–1275, 2016, https://doi.org/10.1016/j.neucom.2015.08.086.Suche in Google Scholar
[19] S. Senthilraj, R. Raja, Q. X. Zhu, R. Samidurai, and Z. S. Yao, “Delay-interval-dependent passivity analysis of stochastic neural networks with Markovian jumping parameters and time delay in the leakage term,” Nonlinear Anal. Hybrid Syst., vol. 22, pp. 262–275, 2016, https://doi.org/10.1016/j.nahs.2016.05.002.Suche in Google Scholar
[20] L. Yang, and Y. K. Li, “Existence and exponential stability of periodic solution for stochastic Hopfield neural networks on time scales,” Neurocomputing, vol. 167, pp. 543–550, 2015, https://doi.org/10.1016/j.neucom.2015.04.038.Suche in Google Scholar
[21] S. Zhu, and Y. Shen, “Robustness analysis for connection weight matrices of global exponential stability of stochastic recurrent neural networks,” Neural Netw., vol. 38, pp. 17–22, 2013, https://doi.org/10.1016/j.neunet.2012.10.004.Suche in Google Scholar PubMed
[22] Q. X. Zhu, and X. D. Li, “Exponential and almost sure exponential stability of stochastic fuzzy delayed Cohen-Grossberg neural networks,” Fuzzy Sets Syst., vol. 203, pp. 74–94, 2012, https://doi.org/10.1016/j.fss.2012.01.005.Suche in Google Scholar
[23] Z. T. Huang, and Q. G. Yang, “Existence and exponential stability of almost periodic solution for stochastic cellular neural networks with delay,” Chaos Sol. Fractals., vol. 42, pp. 773–780, 2009, https://doi.org/10.1016/j.chaos.2009.02.008.Suche in Google Scholar
[24] H. Zhou, Z. F. Zhou, and Z. M. Qiao, “Mean-square almost periodic solution for impulsive stochastic Nicholson’s blowflies model with delays,” Appl. Math. Comput., vol. ,219, pp. 5943–5948, 2013, https://doi.org/10.1016/j.amc.2012.12.005.Suche in Google Scholar
[25] R. Has’minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Maryland, 1980.10.1007/978-94-009-9121-7Suche in Google Scholar
[26] Bernt∅$\varnothing $ksendal, Stochastic Differential Equations: An Introduction With Applications, 4.ed, Springer-Verlag, 1995.10.1007/978-3-662-03185-8Suche in Google Scholar
[27] F. Wu, S. Hu, and Y. Liu, “Positive solution and its asymptotic behavior of stochastic functional Kolmogorov-type system,” J. Math. Anal. Applic., vol. 364, pp. 104–118, 2010, https://doi.org/10.1016/j.jmaa.2009.10.072.Suche in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Original Research Articles
- Stability control of a novel multidimensional fractional-order financial system with time‐delay via impulse control
- Periodic solutions for stochastic Cohen–Grossberg neural networks with time-varying delays
- The monotone iterative method for the integral boundary value problems of fractional p-Laplacian equations with delay
- Solvability of fractional differential inclusions with nonlocal initial conditions via resolvent family of operators
- Synchronization of Cohen-Grossberg fuzzy cellular neural networks with time-varying delays
- Application of Hermite–Padé approximation for detecting singularities of some boundary value problems
- The extended auxiliary equation mapping method to determine novel exact solitary wave solutions of the nonlinear fractional PDEs
- A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument
- Using the generalized Adams-Bashforth-Moulton method for obtaining the numerical solution of some variable-order fractional dynamical models
- Synchronization stability on the BAM neural networks with mixed time delays
- Nonlinear solution of the reaction–diffusion equation using a two-step third–fourth-derivative block method
Artikel in diesem Heft
- Frontmatter
- Original Research Articles
- Stability control of a novel multidimensional fractional-order financial system with time‐delay via impulse control
- Periodic solutions for stochastic Cohen–Grossberg neural networks with time-varying delays
- The monotone iterative method for the integral boundary value problems of fractional p-Laplacian equations with delay
- Solvability of fractional differential inclusions with nonlocal initial conditions via resolvent family of operators
- Synchronization of Cohen-Grossberg fuzzy cellular neural networks with time-varying delays
- Application of Hermite–Padé approximation for detecting singularities of some boundary value problems
- The extended auxiliary equation mapping method to determine novel exact solitary wave solutions of the nonlinear fractional PDEs
- A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument
- Using the generalized Adams-Bashforth-Moulton method for obtaining the numerical solution of some variable-order fractional dynamical models
- Synchronization stability on the BAM neural networks with mixed time delays
- Nonlinear solution of the reaction–diffusion equation using a two-step third–fourth-derivative block method
