Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter August 10, 2020

Global exponential stability analysis of anti-periodic solutions of discontinuous bidirectional associative memory (BAM) neural networks with time-varying delays

  • Xiangying Fu and Fanchao Kong EMAIL logo

Abstract

This paper is concerned with a class of bidirectional associative memory (BAM) neural networks with discontinuous activations and time-varying delays. Under the basic framework of differential inclusions theory, the existence result of solutions in sense of Filippov solution is firstly established by using the fundamental solution matrix of coefficients and inequality analysis technique. Also, the boundness of the solutions can be estimated. Secondly, based on the non-smooth Lyapunov-like approach and by construsting suitable Lyapunov–Krasovskii functionals, some new sufficient criteria are given to ascertain the globally exponential stability of the anti-periodic solutions for the proposed neural network system. Furthermore, we have collated our effort with some previous existing ones in the literatures and showed that it can take more advantages. Finally, two examples with numerical simulations are exploited to illustrate the correctness.


Corresponding author: Fanchao Kong, School of Mathematics and Statistics, Anhui Normal University, Wuhu, Anhui, 241000, PR China, E-mail:

Funding source: Anhui Normal University

Award Identifier / Grant number: 751965, China

Acknowledgments

The authors thank the anonymous reviewers for their insightful suggestions which improved this work significantly. The author expresses the sincere gratitude to Prof. Quanxin Zhu (Hunan Normal University) for the helpful discussion when this work was being carried out. This work is supported by Anhui Provincial Natural Science Foundation (No.2008085QA14) and Talent Foundation of Anhui Normal University (No.751965).

  1. Authors’ contributions: All authors read and approved the manuscript.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

References

[1] B. Kosko, “Adaptive bidirectional associative memories,” Appl. Opt., vol. 26, no. 23, pp. 4947–4960, 1987, https://doi.org/10.1364/AO.26.004947.Search in Google Scholar PubMed

[2] B. Kosko, “Bidirectional associative memories,” IEEE Trans. Syst. Man Cybern., vol. 18, no. 1, pp. 49–60, 1988, https://doi.org/10.1109/21.87054.Search in Google Scholar

[3] C. Maharajan, R. Raja, J. Cao, G. Rajchakit, Z. Tu, and A. Alsaedi, “LMI-based results on exponential stability of BAM-type neural networks with leakage and both time-varying delays: A non-fragile state estimation approach,” Appl. Math. Comput., vol. 326, pp. 33–55, 2018, https://doi.org/10.1016/j.amc.2018.01.001.Search in Google Scholar

[4] C. Maharajan, R. Raja, J. Cao, and G. Rajchakit, “Novel global robust exponential stability criterion for uncertain inertial-type BAM neural networks with discrete and distributed time-varying delays via Lagrange sense,” J. Franklin Inst., vol. 355, no. 11, pp. 4727–4754, 2018, https://doi.org/10.1016/j.jfranklin.2018.04.034.Search in Google Scholar

[5] C. Maharajan, R. Raja, J. Cao, G. Rajchakit, and A. Alsaedi, “Impulsive Cohen-Grossberg BAM neural networks with mixed time-delays: An exponential stability analysis issue,” Neurocomputing, vol. 275, pp. 2588–2602, 2018, https://doi.org/10.1016/j.neucom.2017.11.028.Search in Google Scholar

[6] C. Sowmiya, R. Raja, Q. X. Zhu, and G. Rajchakit, “Further mean-square asymptotic stability of impulsive discrete-time stochastic BAM neural networks with Markovian jumping and multiple time-varying delays,” J. Franklin Inst., vol. 356, no. 1, pp. 561–591, 2019, https://doi.org/10.1016/j.jfranklin.2018.09.037.Search in Google Scholar

[7] C. Sowmiya, R. Raja, J. Cao, and G. Rajchakit, “Impulsive discrete-time BAM neural networks with random parameter uncertainties and time-varying leakage delays: an asymptotic stability analysis,” Nonlinear Dynam., vol. 91, no. 4, pp. 2571–2592, 2018, https://doi.org/10.1007/s11071-017-4032-x.Search in Google Scholar

[8] Z. Wang and L. Huang, “Global stability analysis for delayed complex-valued BAM neural networks,” Neurocomputing, vol. 173, no. 4, pp. 2083–2089, 2016.10.1016/j.neucom.2015.09.086Search in Google Scholar

[9] Q. Zhu, C. Huang, and X. Yang, “Exponential stability for stochastic jumping BAM neural networks with time-varying and distributed delays,” Nonlinear Anal. Hybrid Syst., vol. 5, no. 1, pp. 52–77, 2011, https://doi.org/10.1016/j.nahs.2010.08.005.Search in Google Scholar

[10] Q. Zhu and J. Cao, “Stability analysis of Markovian jump stochastic BAM neural networks with impulse control and mixed time delays,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 3, pp. 467–479, 2012, https://doi.org/10.1109/TNNLS.2011.2182659.Search in Google Scholar PubMed

[11] Y. Chen, J. J. Nieto, and D. Oregan, “Anti-periodic solutions for fully nonlinear first-order differential equations,” Math. Comput. Model., vol. 46, no. 9–10, pp. 1183–1190, 2007, https://doi.org/10.1016/j.mcm.2006.12.006.Search in Google Scholar

[12] Y. Li, L. Yang, and W. Wu, “Anti-periodic solution for impulsive BAM neural networks with time-varying leakage delays on time scales,” Neurocomputing, vol. 149, pp. 536–545, 2015, https://doi.org/10.1016/j.neucom.2014.08.020.Search in Google Scholar

[13] X. Wei, Z. Qiu, “Anti-periodic solutions for BAM neural networks with time delays,” Appl. Math. Comput., vol. 221, pp. 221–229, 2013, https://doi.org/10.1016/j.amc.2013.06.063.Search in Google Scholar

[14] C. Xu and Q. Zhang, “Existence and global exponential stability of anti-periodic solutions for BAM neural networks with inertial term and delay,” Neurocomputing, vol. 153, pp. 108–116, 2015, https://doi.org/10.1016/j.neucom.2014.11.047.Search in Google Scholar

[15] M. Forti, P. Nistri, and D. Papini, “Global convergence of neural networks with discontinuous neuron activations,” IEEE Trans. Circ. Syst. Fund. Theor. Appl., vol. 50, no. 11, pp. 1421–1435, 2003, https://doi.org/10.1109/TCSI.2003.818614.Search in Google Scholar

[16] Z. W. Cai, L. H. Huang, Z. Y. Guo, and X. Y. Chen, “On the periodic dynamics of a class of time-varying delayed neural networks via differential inclusions,” Neural Network., vol. 33, pp. 97–113, 2012, https://doi.org/10.1016/j.neunet.2012.04.009.Search in Google Scholar PubMed

[17] L. H. Huang, Z. Y. Guo, and J. F. Wang, Theory and Applications of Differential Equations with Discontinuous Right-Hand Sides, Beijing, Science Press, 2011.(in Chinese).Search in Google Scholar

[18] C. X. Huang, Z. C. Yang, T. S. Yi, and X. F. Zou, “On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities,” J. Differ. Equ., vol. 256, no. 7, pp. 2101–2114, 2014, https://doi.org/10.1016/j.jde.2013.12.015.Search in Google Scholar

[19] C. X. Huang, H. Zhang, J. D. Cao, and H. J. Hu, “Stability and Hopf bifurcation of a delayed prey-predator model with disease in the predator,” Int. J. Bifurcat. Chaos, vol. 29, no. 07, pp. 1950091, 2019, https://doi.org/10.1142/S0218127419500913.Search in Google Scholar

[20] C. X. Huang, H. Zhang, and L. H. Huang, “Almost periodicity analysis for a delayed Nicholson’s blowflies model with nonlinear density-dependent mortality term,” Commun. Pure Appl. Anal., vol. 18, no. 6, pp. 3337–3349, 2019, https://doi.org/10.3934/cpaa.2019150.Search in Google Scholar

[21] F. C. Kong, Q. X. Zhu, and J. J. Nieto, “Robust fixed-time synchronization of discontinuous Cohen-Grossberg neural networks with mixed time delays,” Nonlinear Anal. Model. Contr., vol. 24, no. 4, pp. 603–625, 2019, https://doi.org/10.15388/NA.2019.4.7.Search in Google Scholar

[22] F. C. Kong and J. J. Nieto, “Almost periodic dynamical behaviors of the hematopoiesis model with mixed discontinuous harvesting terms,” Discrete Contin. Dyn. Syst. Ser. B., vol. 42, pp. 233–239, 2019.10.3934/dcdsb.2019107Search in Google Scholar

[23] F. C. Kong, Q. X. Zhu, K. Wang, and J. J. Nieto, “Stability analysis of almost periodic solutions of discontinuous BAM neural networks with hybrid time-varying delays and D operator,” J. Franklin Inst., vol. 356, no. 18, pp. 11605–11637, 2019, https://doi.org/10.1016/j.jfranklin.2019.09.030.Search in Google Scholar

[24] R. Tang, X. Yang, X. Wan, Y. Zou, Z. Cheng, and H. M. Fardoun, “Finite-time synchronization of nonidentical BAM discontinuous fuzzy neural networks with delays and impulsive effects via non-chattering quantized control,” Commun. Nonlinear Sci. Numer. Simul., vol. 78, pp. 104893, 2019, https://doi.org/10.1016/j.cnsns.2019.104893.Search in Google Scholar

[25] H. Wu and Y. Li, “Existence and stability of periodic solution for BAM neural networks with discontinuous neuron activations,” Comput. Math. Appl., vol. 56, no. 8, pp. 1981–1993, 2008, https://doi.org/10.1016/j.camwa.2008.04.027.Search in Google Scholar

[26] X. Yang, Q. Song, J. Liang, and B. He, “Finite-time synchronization of coupled discontinuous neural networks with mixed delays and nonidentical perturbations,” J. Franklin Inst., vol. 352, no. 10, pp. 4382–4406, 2015, https://doi.org/10.1016/j.jfranklin.2015.07.001.Search in Google Scholar

[27] J. Aubin and A. Cellina, Differential Inclusions, Berlin, Springer-Verlag, 1984.10.1007/978-3-642-69512-4Search in Google Scholar

Received: 2019-09-06
Accepted: 2020-04-08
Published Online: 2020-08-10
Published in Print: 2020-11-18

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 29.3.2024 from https://www.degruyter.com/document/doi/10.1515/ijnsns-2019-0220/html
Scroll to top button