Weighted pseudo almost periodic solutions of octonion-valued neural networks with mixed time-varying delays and leakage delays

Jin Gao; Lihua Dai; Jin Gao; Lihua Dai

doi:10.3934/math.2023760

AIMS Mathematics

2023, Volume 8, Issue 6: 14867-14893. doi: 10.3934/math.2023760

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Research article

Weighted pseudo almost periodic solutions of octonion-valued neural networks with mixed time-varying delays and leakage delays

Jin Gao ^{1
,
,},
Lihua Dai ²

1.
School of Information, Yunnan Communications Vocational and Technical College, Kunming, Yunnan 650500, China
2.
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received: 22 February 2023 Revised: 27 March 2023 Accepted: 05 April 2023 Published: 21 April 2023
MSC : 34A34, 34C25, 34D23, 34K20

In this paper, we propose a class of octonion-valued neural networks with leakage delays and mixed delays. Considering that the multiplication of octonion algebras does not satisfy the associativity and commutativity, we can obtain the existence and global exponential stability of weighted pseudo almost periodic solutions for octonion-valued neural networks with leakage delays and mixed delays by using the Banach fixed point theorem, the proof by contradiction and the non-decomposition method. Finally, we will give one example to illustrate the feasibility and effectiveness of the main results.
- octonion algebra,
- neural networks,
- stability,
- weighted pseudo almost periodic solutions,
- delays
Citation: Jin Gao, Lihua Dai. Weighted pseudo almost periodic solutions of octonion-valued neural networks with mixed time-varying delays and leakage delays[J]. AIMS Mathematics, 2023, 8(6): 14867-14893. doi: 10.3934/math.2023760

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Abstract

In this paper, we propose a class of octonion-valued neural networks with leakage delays and mixed delays. Considering that the multiplication of octonion algebras does not satisfy the associativity and commutativity, we can obtain the existence and global exponential stability of weighted pseudo almost periodic solutions for octonion-valued neural networks with leakage delays and mixed delays by using the Banach fixed point theorem, the proof by contradiction and the non-decomposition method. Finally, we will give one example to illustrate the feasibility and effectiveness of the main results.

References

[1]	C. Huang, X. Long, J. Cao, Stability of antiperiodic recurrent neural networks with multiproportional delays, Math. Methods Appl. Sci., 43 (2020), 6093–6102. https://doi.org/10.1002/mma.6350 doi: 10.1002/mma.6350
[2]	X. Fu, F. 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
[3]	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), 012098. https://doi.org/10.1088/1742-6596/1850/1/012098 doi: 10.1088/1742-6596/1850/1/012098
[4]	M. Khuddush, K. R. Prasad, Global exponential stability of almost periodic solutions for quaternion-valued RNNs with mixed delays on time scales, Bol. Soc. Mat. Mex., 28 (2022), 75. https://doi.org/10.1007/s40590-022-00467-y doi: 10.1007/s40590-022-00467-y
[5]	L. T. H. Dzung, L. V. Hien, Positive solutions and exponential stability of nonlinear time-delay systems in the model of BAM-Cohen-Grossberg neural networks, Differ. Equ. Dyn. Syst., 159 (2022). https://doi.org/10.1007/s12591-022-00605-y doi: 10.1007/s12591-022-00605-y
[6]	L. Li, D. W. C. Ho, J. Cao, J. Lu, Pinning cluster synchronization in an array of coupled neural networks under event-based mechanism, Neural Networks, 76 (2016), 1–12. https://doi.org/10.1016/j.neunet.2015.12.008 doi: 10.1016/j.neunet.2015.12.008
[7]	R. Li, X. Gao, J. Cao, Exponential synchronization of stochastic memristive neural networks with time-varying delays, Neural Process. Lett., 50 (2019), 459–475. https://doi.org/10.1007/s11063-019-09989-5 doi: 10.1007/s11063-019-09989-5
[8]	Y. Sun, L. Li, X. Liu, Exponential synchronization of neural networks with time-varying delays and stochastic impulses, Neural Networks, 132 (2020), 342–352. https://doi.org/10.1016/j.neunet.2020.09.014 doi: 10.1016/j.neunet.2020.09.014
[9]	Q. Xiao, T. Huang, Z. Zeng, Synchronization of timescale-type nonautonomous neural networks with proportional delays, IEEE Trans. Syst., Man, Cybern.: Syst., 52 (2021), 2167–2173. https://doi.org/10.1109/tsmc.2021.3049363 doi: 10.1109/tsmc.2021.3049363
[10]	J. Gao, L. Dai, Anti-periodic synchronization of quaternion-valued high-order Hopfield neural networks with delays, AIMS Math., 7 (2022), 14051–14075. https://doi.org/10.3934/math.2022775 doi: 10.3934/math.2022775
[11]	B. Liu, S. Gong, Periodic solution for impulsive cellar neural networks with time-varying delays in the leakage terms, Abstr. Appl. Anal., 2013 (2013), 1–10. https://doi.org/10.1155/2013/701087 doi: 10.1155/2013/701087
[12]	L. Peng, W. Wang, Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying delays in leakage terms, Neurocomputing, 111 (2013), 27–33. https://doi.org/10.1016/j.neucom.2012.11.031 doi: 10.1016/j.neucom.2012.11.031
[13]	Y. Xu, Periodic solutions of BAM neural networks with continuously distributed delays in the leakage terms, Neural Process. Lett., 41 (2015), 293–307. https://doi.org/10.1007/s11063-014-9346-9 doi: 10.1007/s11063-014-9346-9
[14]	H. Zhang, J. Shao, Almost periodic solutions for cellular neural networks with time-varying delays in leakage terms, Appl. Math. Comput., 219 (2013), 11471–11482. https://doi.org/10.1016/j.amc.2013.05.046 doi: 10.1016/j.amc.2013.05.046
[15]	P. Jiang, Z. Zeng, J. Chen, Almost periodic solutions for a memristor-based neural networks with leakage, time-varying and distributed delays, Neural Networks, 68 (2015), 34–45. https://doi.org/10.1016/j.neunet.2015.04.005 doi: 10.1016/j.neunet.2015.04.005
[16]	H. Zhou, Z. Zhou, W. Jiang, Almost periodic solutions for neutral type BAM neural networks with distributed leakage delays on time scales, Neurocomputing, 157 (2015), 223–230. https://doi.org/10.1016/j.neucom.2015.01.013 doi: 10.1016/j.neucom.2015.01.013
[17]	M. Song, Q. Zhu, H. Zhou, Almost sure stability of stochastic neural networks with time delays in the leakage terms, Discrete Dyn. Nat. Soc., 2016 (2016), 1–10. https://doi.org/10.1155/2016/2487957 doi: 10.1155/2016/2487957
[18]	H. Li, H. Jiang, J. Cao, Global synchronization of fractional-order quaternion-valued neural networks with leakage and discrete delays, Neurocomputing, 383 (2020), 211–219. https://doi.org/10.1016/j.neucom.2019.12.018 doi: 10.1016/j.neucom.2019.12.018
[19]	W. Zhang, H. Zhang, J. Cao, H. Zhang, D. Chen, Synchronization of delayed fractional-order complex-valued neural networks with leakage delay, Phys. A: Stat. Mech. Appl., 556 (2020), 124710. https://doi.org/10.1016/j.physa.2020.124710 doi: 10.1016/j.physa.2020.124710
[20]	A. Singh, J. N. Rai, Stability analysis of fractional order fuzzy cellular neural networks with leakage delay and time varying delays, Chinese J. Phys., 73 (2021), 589–599. https://doi.org/10.1016/j.cjph.2021.07.029 doi: 10.1016/j.cjph.2021.07.029
[21]	C. Xu, M. Liao, P. Li, S. Yuan, Impact of leakage delay on bifurcation in fractional-order complex-valued neural networks, Chaos, Solitons Fract., 142 (2021), 110535. https://doi.org/10.1016/j.chaos.2020.110535 doi: 10.1016/j.chaos.2020.110535
[22]	C. Xu, Z. Liu, C. Aouiti, P. Li, L. Yao, J. Yan, New exploration on bifurcation for fractional-order quaternion-valued neural networks involving leakage delays, Cogn. Neurodyn., 16 (2022), 1233–1248. https://doi.org/10.1007/s11571-021-09763-1 doi: 10.1007/s11571-021-09763-1
[23]	C. A. Popa, Octonion-valued neural networks, In: A. Villa, P. Masulli, A. Pons Rivero, Artificial Neural Networks and Machine Learning–ICANN 2016, Lecture Notes in Computer Science, Cham: Springer, 2016,435–443. https://doi.org/10.1007/978-3-319-44778-0_51
[24]	J. C. Baez, The octonions, Bull. Am. Math. Soc., 39 (2002), 145–205.
[25]	A. K. Kwaśniewski, Glimpses of the octonions and quaternions history and today's applications in quantum physics, Adv. Appl. Clifford Algebras, 22 (2012), 87–105. https://doi.org/10.1007/s00006-011-0299-z doi: 10.1007/s00006-011-0299-z
[26]	C. A. Popa, Global asymptotic stability for octonion-valued neural networks with delay, In: F. Cong, A. Leung, Q. Wei, Advances in Neural Networks–ISNN 2017, Lecture Notes in Computer Science, Cham: Springer, 2017,439–448. https://doi.org/10.1007/978-3-319-59072-1_52
[27]	C. A. Popa, Exponential stability for delayed octonion-valued recurrent neural networks, In: I. Rojas, G. Joya, A. Catala, Advances in Computational Intelligence–IWANN 2017, Lecture Notes in Computer Science, Cham: Springer, 2017,375–385. https://doi.org/10.1007/978-3-319-59153-7_33
[28]	C. A. Popa, Global exponential stability of octonion-valued neural networks with leakage delay and mixed delays, Neural Networks, 105 (2018), 277–293. https://doi.org/10.1016/j.neunet.2018.05.006 doi: 10.1016/j.neunet.2018.05.006
[29]	C. A. Popa, Global exponential stability of neutral-type octonion-valued neural networks with time-varying delays, Neurocomputing, 309 (2018), 117–133. https://doi.org/10.1016/j.neucom.2018.05.004 doi: 10.1016/j.neucom.2018.05.004
[30]	J. Wang, X. Liu, Global $\mu$-stability and finite-time control of octonion-valued neural networks with unbounded delays, arXiv Preprint, 2020. https://doi.org/10.48550/arXiv.2003.11330
[31]	M. S. M'hamdi, C. Aouiti, A. Touati, A. M. Alimi, V. Snasel, Weighted pseudo almost-periodic solutions of shunting inhibitory cellular neural networks with mixed delays, Acta Math. Sci., 36 (2016), 1662–1682. https://doi.org/10.1016/s0252-9602(16)30098-4 doi: 10.1016/s0252-9602(16)30098-4
[32]	G. Yang, W. Wan, Weighted pseudo almost periodic solutions for cellular neural networks with multi-proportional delays, Neural Process. Lett., 49 (2019), 1125–1138. https://doi.org/10.1007/s11063-018-9851-3 doi: 10.1007/s11063-018-9851-3
[33]	X. Yu, Q. Wang, Weighted pseudo-almost periodic solutions for shunting inhibitory cellular neural networks on time scales, Bull. Malay. Math. Sci. Soc., 42 (2019), 2055–2074. https://doi.org/10.1007/s40840-017-0595-4 doi: 10.1007/s40840-017-0595-4
[34]	C. Huang, H. Yang, J. Cao, Weighted pseudo almost periodicity of multi-proportional delayed shunting inhibitory cellular neural networks with D operator, Discrete Cont. Dyn. Syst.-Ser. S, 14 (2020), 1259–1272. https://doi.org/10.3934/dcdss.2020372 doi: 10.3934/dcdss.2020372
[35]	M. Ayachi, Existence and exponential stability of weighted pseudo-almost periodic solutions for genetic regulatory networks with time-varying delays, Int. J. Biomath., 14 (2021), 2150006. https://doi.org/10.1142/s1793524521500066 doi: 10.1142/s1793524521500066
[36]	M. M'hamdi, On the weighted pseudo almost-periodic solutions of static DMAM neural network, Neural Process. Lett., 54 (2022), 4443–4464. https://doi.org/10.1007/s11063-022-10817-6 doi: 10.1007/s11063-022-10817-6
[37]	A. Fink, Almost periodic differential equations, Berlin: Springer, 1974. https://doi.org/10.1007/BFb0070324

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