Stationary oscillation of delayed complex dynamical networks under impulsive control involving delays

Guixia Sui; Jingting Hu; Guixia Sui; Jingting Hu

doi:10.3934/math.2026828

AIMS Mathematics

2026, Volume 11, Issue 7: 20389-20406. doi: 10.3934/math.2026828

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Stationary oscillation of delayed complex dynamical networks under impulsive control involving delays

Guixia Sui ^{1
,
,},
Jingting Hu ²

1.
Jinan Preschool Education College, Jinan 250307, Shandong, China
2.
School of Mathematical Sciences, University of Jinan, Jinan 250022, Shandong, China

Received: 08 April 2026 Revised: 23 June 2026 Accepted: 30 June 2026 Published: 10 July 2026

This paper focused on the stationary oscillation problems for delayed complex dynamical networks, with the full consideration of time delays in impulsive actions. With the help of inequality-based techniques, several novel sufficient conditions that guarantee the existence of stationary oscillations for error system between complex dynamical network and its isolated node were derived. Our presented method, unlike the existing results, demonstrated that such an error system can exhibit stationary oscillations under impulsive control strategies involving time delays, even when the system was initially divergent or unstable. Finally, two illustrative examples along with their simulation results were put forward for demonstrating the efficacy and effectiveness of the developed control method.
- impulsive control,
- complex dynamical networks,
- time delays,
- stationary oscillation,
- inequality technique
Citation: Guixia Sui, Jingting Hu. Stationary oscillation of delayed complex dynamical networks under impulsive control involving delays[J]. AIMS Mathematics, 2026, 11(7): 20389-20406. doi: 10.3934/math.2026828

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Abstract

This paper focused on the stationary oscillation problems for delayed complex dynamical networks, with the full consideration of time delays in impulsive actions. With the help of inequality-based techniques, several novel sufficient conditions that guarantee the existence of stationary oscillations for error system between complex dynamical network and its isolated node were derived. Our presented method, unlike the existing results, demonstrated that such an error system can exhibit stationary oscillations under impulsive control strategies involving time delays, even when the system was initially divergent or unstable. Finally, two illustrative examples along with their simulation results were put forward for demonstrating the efficacy and effectiveness of the developed control method.

References

[1]	O.-M. Kwon, S.-H. Lee, M.-J. Park, Some novel results on stability analysis of generalized neural networks with time-varying delays via augmented approach, IEEE Trans. Cybernetics, 52 (2022), 2238–2248. https://doi.org/10.1109/tcyb.2020.3001341 doi: 10.1109/tcyb.2020.3001341
[2]	S. H. Strogatz, Exploring complex networks, Nature, 410 (2001), 268–276. https://doi.org/10.1038/35065725 doi: 10.1038/35065725
[3]	L. Ding, Q.-L. Han, X.-M. Zhang, Distributed secondary control for active power sharing and frequency regulation in islanded microgrids using an event-triggered communication mechanism, IEEE Trans. Ind. Inform., 15 (2019), 3910–3922. https://doi.org/10.1109/tii.2018.2884494 doi: 10.1109/tii.2018.2884494
[4]	H. Zhang, P. Q. Gao, R. Y. Ye, I. Stamova, J. D. Cao, Fixed/Predefined time synchronization of fractional quaternion delayed neural networks with disturbances, Math. Comput. Simulat., 232 (2025), 276–294. https://doi.org/10.1016/j.matcom.2025.01.010 doi: 10.1016/j.matcom.2025.01.010
[5]	H.-L. Li, J. D. Cao, C. Hu, H. J. Jiang, F. E. Alsaadi, Synchronization analysis of discrete-time fractional-order quaternion-valued uncertain neural networks, IEEE Trans. Neur. Netw. Learn., 35 (2023), 14178–14189. https://doi.org/10.1109/tnnls.2023.3274959 doi: 10.1109/tnnls.2023.3274959
[6]	H. Long, J. X. Ci, Z. Y. Guo, S. P. Wen, T. W. Huang, Synchronization of coupled switched neural networks subject to hybrid stochastic disturbances, Neural Networks, 166 (2023), 459–470. https://doi.org/10.1016/j.neunet.2023.07.045 doi: 10.1016/j.neunet.2023.07.045
[7]	Y. He, Q.-G. Wang, M. Wu, C. Lin, Delay-dependent state estimation for delayed neural networks, IEEE Transactions on Neural Networks, 17 (2006), 1077–1081. https://doi.org/10.1109/TNN.2006.875969 doi: 10.1109/TNN.2006.875969
[8]	Z. L. Yan, X. Huang, J. D. Cao, Variable-sampling-period dependent global stabilization of delayed memristive neural networks based on refined switching event-triggered control, Sci. China Inform. Sci., 63 (2020), 212201. https://doi.org/10.1007/s11432-019-2664-7 doi: 10.1007/s11432-019-2664-7
[9]	K. B. Shi, J. Wang, S. M. Zhong, Y. Y. Tang, J. Cheng, Non-fragile memory filtering of TS fuzzy delayed neural networks based on switched fuzzy sampled-data control, Fuzzy Set. Syst., 394 (2020), 40–64. https://doi.org/10.1016/j.fss.2019.09.001 doi: 10.1016/j.fss.2019.09.001
[10]	K. Godfrey, Compartmental models and their application, London: Academic Press, 1983.
[11]	S.-I. Niculescu, Delay effects on stability: a robust control approach, London: Springer, 2001. https://doi.org/10.1007/1-84628-553-4
[12]	K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, Dordrecht: Springer, 1992. https://doi.org/10.1007/978-94-015-7920-9
[13]	H.-C. Lin, H.-B. Zeng, X.-M. Zhang, W. Wang, Stability analysis for delayed neural networks via a generalized reciprocally convex inequality, IEEE Trans. Neur. Netw. Learn., 34 (2023), 7491–7499. https://doi.org/10.1109/TNNLS.2022.3144032 doi: 10.1109/TNNLS.2022.3144032
[14]	L. Li, L. F. Yan, C. X. Huang, J. D. Cao, X. D. Ding, Linear formation of Cucker-Smale model with distributed time delays, Math. Comput. Simulat., 222 (2024), 296–310. https://doi.org/10.1016/j.matcom.2023.08.034 doi: 10.1016/j.matcom.2023.08.034
[15]	B. Yan, X. J. Wang, H. Y. Ma, W. Q. Lu, Q. C. Li, Hybrid time-delayed feedforward and feedback control of lever-type quasi-zero-stiffness vibration isolators, IEEE Trans. Ind. Electron., 71 (2024), 2810–2819. https://doi.org/10.1109/TIE.2023.3269481 doi: 10.1109/TIE.2023.3269481
[16]	Y. H. Xia, Impulsive effect on the delayed Cohen–Grossberg-type BAM neural networks, Neurocomputing, 73 (2010), 2754–2764. https://doi.org/10.1016/j.neucom.2010.04.011 doi: 10.1016/j.neucom.2010.04.011
[17]	R. V. Aravind, P. Balasubramaniam, Global asymptotic stability of delayed fractional-order complex-valued fuzzy cellular neural networks with impulsive disturbances, J. Appl. Math. Comput., 68 (2022), 4713–4731. https://doi.org/10.1007/s12190-022-01726-x doi: 10.1007/s12190-022-01726-x
[18]	X. Z. Liu, Stability results for impulsive differential systems with applications to population growth models, Dynamics and Stability of Systems, 9 (1994), 163–174. https://doi.org/10.1080/02681119408806175 doi: 10.1080/02681119408806175
[19]	B. W. Li, Y. Cai, Self-triggered delay impulsive control for nonlinear systems with exogenous disturbances, AIMS Math., 11 (2026), 8014–8030. https://doi.org/10.3934/math.2026330 doi: 10.3934/math.2026330
[20]	X. F. Xing, H. Q. Wu, J. D. Cao, Finite-time synchronization of impulsive stochastic systems with DoS attacks via dynamic event-triggered control, Math. Comput. Simulat., 219 (2024), 573–593. https://doi.org/10.1016/j.matcom.2023.12.041 doi: 10.1016/j.matcom.2023.12.041
[21]	Y. Chu, X. P. Han, R. Rakkiyappan, Finite-time lag synchronization for two-layer complex networks with impulsive effects, Math. Model. Control, 4 (2024), 71–85. https://doi.org/10.3934/mmc.2024007 doi: 10.3934/mmc.2024007
[22]	A. Khadra, X. Z. Liu, X. M. Shen, Application of impulsive synchronization to communication security, IEEE Trans. Circuits Syst. I, 50 (2003), 341–351. https://doi.org/10.1109/TCSI.2003.808839 doi: 10.1109/TCSI.2003.808839
[23]	H. T. Zhang, X. Z. Liu, X. M. Shen, J. Liu, Intermittent impulsive synchronization of hyperchaos with application to secure communication, Asian J. Control, 15 (2013), 1686–1699. https://doi.org/10.1002/asjc.728 doi: 10.1002/asjc.728
[24]	F. Cordova-Lepe, G. Robledo, J. Cabrera-Villegas, Population growth modeling with boom and bust patterns: the impulsive differential equation formalism, J. Biol. Syst., 23 (2015), S135–S149. https://doi.org/10.1142/S0218339015400112 doi: 10.1142/S0218339015400112
[25]	Z. Tang, C. H. Jiang, Y. Wang, J. W. Feng, J. H. Park, Average impulsive weight based event-triggered impulsive synchronization on coupled neural networks, IEEE Trans. Netw. Sci. Eng., 10 (2023), 2180–2189. https://doi.org/10.1109/tnse.2023.3243248 doi: 10.1109/tnse.2023.3243248
[26]	D. Ding, Z. Tang, J. H. Park, Y. Wang, Z. C. Ji, Dynamic self-triggered impulsive synchronization of complex networks with mismatched parameters and distributed delay, IEEE Trans. Cybernetics, 53 (2023), 887–899. https://doi.org/10.1109/tcyb.2022.3168854 doi: 10.1109/tcyb.2022.3168854
[27]	X. Z. Liu, Impulsive stabilization and applications to population growth models, Rocky Mountain J. Math., 25 (1995), 381–395. https://doi.org/10.1216/rmjm/1181072290 doi: 10.1216/rmjm/1181072290
[28]	T. Yang, Impulsive systems and control: theory and applications, New York: Nova Science Publishers, 2001.
[29]	X. D. Li, M. Bohner, C.-K. Wang, Impulsive differential equations: periodic solutions and applications, Automatica, 52 (2015), 173–178. https://doi.org/10.1016/j.automatica.2014.11.009 doi: 10.1016/j.automatica.2014.11.009
[30]	M. Al Nuwairan, A. G. Ibrahim, Solutions and anti-periodic solutions for impulsive differential equations and inclusions containing Atangana-Baleanu fractional derivative of order $\zeta\in(1, 2)$ in infinite dimensional Banach spaces, AIMS Math., 9 (2024), 10386–10415. https://doi.org/10.3934/math.2024508 doi: 10.3934/math.2024508
[31]	S. P. Li, Impulsive control for stationary oscillation of nonlinear delay systems and applications, Math. Model. Control, 3 (2023), 267–277. https://doi.org/10.3934/mmc.2023023 doi: 10.3934/mmc.2023023
[32]	M. Akhmet, E. Yilmaz, Neural networks with discontinuous/impact activations, New York: Springer, 2014. https://doi.org/10.1007/978-1-4614-8566-7
[33]	X. D. Li, S. J. Song, Stabilization of delay systems: delay-dependent impulsive control, IEEE Trans. Automat. Contr., 62 (2017), 406–411. https://doi.org/10.1109/tac.2016.2530041 doi: 10.1109/tac.2016.2530041
[34]	M. Z. Wang, X. D. Li, S. J. Song, Local synchronization for delayed complex dynamical networks via self-triggered impulsive control involving delays, IEEE Trans. Neur. Netw. Learn., 36 (2025), 9663–9669. https://doi.org/10.1109/TNNLS.2024.3414126 doi: 10.1109/TNNLS.2024.3414126
[35]	H. Shen, C. J. Peng, H. C. Yan, S. Y. Xu, Data-driven near optimization for fast sampling singularly perturbed systems, IEEE Trans. Automat. Contr., 69 (2024), 4689–4694. https://doi.org/10.1109/TAC.2024.3352703 doi: 10.1109/TAC.2024.3352703
[36]	H. Shen, J. C. Wu, J. Wang, Z. G. Wu, Adversarial dynamic games for Markov jump systems: A policy iteration Q-learning method, Automatica, 183 (2026), 112591. https://doi.org/10.1016/j.automatica.2025.112591 doi: 10.1016/j.automatica.2025.112591

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