The stability of anti-periodic solutions for fractional-order inertial BAM neural networks with time-delays

Yuehong Zhang; Zhiying Li; Wangdong Jiang; Wei Liu; Yuehong Zhang; Zhiying Li; Wangdong Jiang; Wei Liu

doi:10.3934/math.2023312

AIMS Mathematics

2023, Volume 8, Issue 3: 6176-6190. doi: 10.3934/math.2023312

Previous Article Next Article

Research article

The stability of anti-periodic solutions for fractional-order inertial BAM neural networks with time-delays

Shaoxing University Yuanpei College, Shaoxing 312000, China

Received: 11 October 2022 Revised: 05 December 2022 Accepted: 07 December 2022 Published: 03 January 2023
MSC : 92B20, 34K20

The dynamic signal transmission process can be regarded as an anti-periodic process, and fractional-order inertial neural networks are widely used in signal processing and other fields, so anti-periodicity is also regarded as an important dynamic feature of inertial neural networks. This paper mainly studies the existence and Mittag-Leffler stability of anti-periodic solutions for a class of fractional-order inertial BAM neural networks with time-delays. By introducing variable substitution, the model with two different fractional-order derivatives is transformed into a model with only one fractional-order derivative of the same order. Using the properties of fractional-order calculus, the relationship between the fractional-order integral of the state function with and without time-delays is given. Firstly, the sufficient conditions for the boundedness and the Mittag-Leffler stability of the solutions for the system are derived. Secondly, by constructing the sequence solution of the function for the system and applying Ascoli-Arzela theorem, the sufficient conditions for the existence and Mittag-Leffler stability of the anti-periodic solution are given. Finally, the correctness of the conclusion is verified by a numerical example.
- fractional-order,
- inertial BAM neural networks,
- Ascoli-Arzela theorem,
- anti-periodic solutions,
- Mittag-Leffler stability
Citation: Yuehong Zhang, Zhiying Li, Wangdong Jiang, Wei Liu. The stability of anti-periodic solutions for fractional-order inertial BAM neural networks with time-delays[J]. AIMS Mathematics, 2023, 8(3): 6176-6190. doi: 10.3934/math.2023312

Related Papers:

Abstract

The dynamic signal transmission process can be regarded as an anti-periodic process, and fractional-order inertial neural networks are widely used in signal processing and other fields, so anti-periodicity is also regarded as an important dynamic feature of inertial neural networks. This paper mainly studies the existence and Mittag-Leffler stability of anti-periodic solutions for a class of fractional-order inertial BAM neural networks with time-delays. By introducing variable substitution, the model with two different fractional-order derivatives is transformed into a model with only one fractional-order derivative of the same order. Using the properties of fractional-order calculus, the relationship between the fractional-order integral of the state function with and without time-delays is given. Firstly, the sufficient conditions for the boundedness and the Mittag-Leffler stability of the solutions for the system are derived. Secondly, by constructing the sequence solution of the function for the system and applying Ascoli-Arzela theorem, the sufficient conditions for the existence and Mittag-Leffler stability of the anti-periodic solution are given. Finally, the correctness of the conclusion is verified by a numerical example.

References

[1]	A. R. Aftabizadeh, S. Aizicovici, N. H. Pavel, On a class of second-order anti-periodic boundary value problems, J. Math. Anal. Appl., 171 (1992), 301–320. https://doi.org/10.1016/0022-247X(92)90345-E doi: 10.1016/0022-247X(92)90345-E
[2]	S. Aizicovici, M. Mckibben, S. Reich, Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities, Nonlinear Anal., 43 (2001), 233–251. https://doi.org/10.1016/S0362-546X(99)00192-3 doi: 10.1016/S0362-546X(99)00192-3
[3]	Y. Q. Chen, J. J. Nieto, D. O'Regan, Anti-periodic solutions for fully nonlinear first-order differential equations, Math. Comput. Model., 46 (2007), 1183–1190. https://doi.org/10.1016/j.mcm.2006.12.006 doi: 10.1016/j.mcm.2006.12.006
[4]	H. L. Chen, Antiperiodic wavelets, J. Comput. Math., 14 (1996), 32–39.
[5]	Y. Q. Li, L. H. Huang, Anti-periodic solutions for a class of Liénard-type systems with continuously distributed delays, Nonlinear Anal. Real World Appl., 10 (2009), 2127–2132. https://doi.org/10.1016/j.nonrwa.2008.03.020 doi: 10.1016/j.nonrwa.2008.03.020
[6]	Z. Q. Zhang, Z. Y. Quan, Global exponential stability via inequality technique for inertial BAM neural networks with time delays, Neurocomputing, 151 (2015), 1316–1326. https://doi.org/10.1016/j.neucom.2014.10.072 doi: 10.1016/j.neucom.2014.10.072
[7]	Z. Q. Zhang, W. B. Liu, D. M. Zhou, Global asymptotic stability to a generalized Cohen-Grossberg BAM neural networks of neutral type delays, Neural Networks, 25 (2012), 94–105. https://doi.org/10.1016/j.neunet.2011.07.006 doi: 10.1016/j.neunet.2011.07.006
[8]	Z. Q. Zhang, J. D. Cao, D. M. Zhou, Novel LMI-based condition on global asymptotic stability for a class of Cohen-Grossberg BAM networks with extended activation functions, IEEE Trans. Neural Net. Learn. Syst., 25 (2014), 1161–1172. https://doi.org/10.1109/TNNLS.2013.2289855 doi: 10.1109/TNNLS.2013.2289855
[9]	X. Y. Fu, F. C. Kong, Global exponential stability analysis of anti-periodic solutions of discontinuous bidirectional associative memory (BAM) neural networks with time-varying delays, Int. J. Nonlinear Sci. Numer. Simul., 21 (2020), 807–820. https://doi.org/10.1515/ijnsns-2019-0220 doi: 10.1515/ijnsns-2019-0220
[10]	N. Radhakrishnan, R. Kodeeswaran, R. Raja, C. Maharajan, A. Stephen, Global exponential stability analysis of anti-periodic of discontinuous BAM neural networks with time-varying delays, J. Phys. Conf. Ser., 1850 (2021), 1–20. https://doi.org/10.1088/1742-6596/1850/1/012098 doi: 10.1088/1742-6596/1850/1/012098
[11]	C. F. Xu, F. C. Kong, Global exponential stability of anti-periodic solutions for discontinuous Cohen-Grossberg neural networks with time-varying delays, J. Exp. Theor. Artif. Intell., 33 (2021), 263–281. https://doi.org/10.1080/0952813X.2020.1737244 doi: 10.1080/0952813X.2020.1737244
[12]	P. Cui, Z. B. Li, Anti-periodic solutions for BAM-type Cohen-Grossberg neural networks with time delays, J. Nonlinear Sci. Appl., 10 (2017), 2171–2180. https://doi.org/10.22436/jnsa.010.04.69 doi: 10.22436/jnsa.010.04.69
[13]	Y. Q. Ke, C. F. Miao, Anti-periodic solutions of inertial neural networks with time delays, Neural Process. Lett., 45 (2017), 523–538. https://doi.org/10.1007/s11063-016-9540-z doi: 10.1007/s11063-016-9540-z
[14]	A. Arbi, N. Tahri, C. Jammazi, C. X. Huang, J. D. Cao, Almost anti-periodic solution of inertial neural networks with leakage and time-varying delays on timescales, Circuits Syst. Signal Process., 41 (2022), 1940–1956. https://doi.org/10.1007/s00034-021-01894-4 doi: 10.1007/s00034-021-01894-4
[15]	Q. Cao, X. J. Guo, Anti-periodic dynamics on high-order inertial Hopfield neural networks involving time-varying delays, AIMS Math., 5 (2020), 5402–5421. https://doi.org/10.3934/math.2020347 doi: 10.3934/math.2020347
[16]	L. G. Yao, Q. Cao, Anti-periodicity on high-order inertial Hopfield neural networks involving mixed delays, J. Inequal. Appl., 2020 (2020), 182. https://doi.org/10.1186/s13660-020-02444-3 doi: 10.1186/s13660-020-02444-3
[17]	B. S. Chen, J. J. Chen, Global asymptotical $\omega$-periodicity of a fractional-order non-autonomous neural networks, Neural Networks, 68 (2015), 78–88. https://doi.org/10.1016/j.neunet.2015.04.006 doi: 10.1016/j.neunet.2015.04.006
[18]	Y. Y. Hou, L. H. Dai, S-asymptotically $\omega$-periodic solutions of fractional-order complex-valued recurrent neural networks with delays, IEEE Access, 9 (2021), 37883–37893. https://doi.org/10.1109/ACCESS.2021.3063746 doi: 10.1109/ACCESS.2021.3063746
[19]	A. L. Wu, Z. G. Zeng, Boundedness, Mittag-Leffler stability and asymptotical $\omega$-periodicity of fractional-order fuzzy neural networks, Neural Networks, 74 (2016), 73–84. https://doi.org/10.1016/j.neunet.2015.11.003 doi: 10.1016/j.neunet.2015.11.003
[20]	Y. K. Li, M. Huang, B. Li, Besicovitch almost periodic solutions for fractional-order quaternion-valued neural networks with discrete and distributed delays, Math. Methods Appl. Sci., 45 (2022), 4791–4808. https://doi.org/10.1002/mma.8070 doi: 10.1002/mma.8070
[21]	P. Wan, D. H. Sun, M. Zhao, H. Zhao, Monostability and multistability for almost-periodic solutions of fractional-order neural networks with unsaturating piecewise linear activation functions, IEEE Trans. Neural Net. Learn. Syst., 31 (2020), 5138–5152. https://doi.org/10.1109/TNNLS.2020.2964030 doi: 10.1109/TNNLS.2020.2964030
[22]	C. J. Xu, P. L. Li, On anti-periodic solutions for neutral shunting inhibitory cellular neural networks with time-varying delays and D operator, Neurocomputing, 275 (2018), 377–382. https://doi.org/10.1016/j.neucom.2017.08.030 doi: 10.1016/j.neucom.2017.08.030
[23]	C. X. Huang, S. G. Wen, L. H. Huang, Dynamics of anti-periodic solutions on shunting inhibitory cellular neural networks with multi-proportional delays, Neurocomputing, 357 (2019), 47–52. https://doi.org/10.1016/j.neucom.2019.05.022 doi: 10.1016/j.neucom.2019.05.022
[24]	Y. K. Li, J. L. Xiang, Existence and global exponential stability of anti-periodic solution for Clifford-valued inertial Cohen-Grossberg neural networks with delays, Neurocomputing, 332 (2019), 259–269. https://doi.org/10.1016/j.neucom.2018.12.064 doi: 10.1016/j.neucom.2018.12.064
[25]	I. Podlubny, Fractional differential equations, New York: Academic Press, 1998.
[26]	L. Ke, Mittag-Leffler stability and asymptotic $\omega$-periodicity of fractional-order inertial neural networks with time-delays, Neurocomputing, 465 (2021), 53–62. https://doi.org/10.1016/j.neucom.2021.08.121 doi: 10.1016/j.neucom.2021.08.121
[27]	Y. J. Gu, H. Wang, Y. G. Yu, Stability and synchronization for Riemann-Liouville fractional-order time-delayed inertial neural networks, Neurocomputing, 340 (2019), 270–280. https://doi.org/10.1016/j.neucom.2019.03.005 doi: 10.1016/j.neucom.2019.03.005

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)