The implicit mapping dynamics of a class of time-delay dynamical systems with impulses at fixed times

Han Xu; Xilin Fu; Han Xu; Xilin Fu

doi:10.3934/era.2026215

Electronic Research Archive

2026, Volume 34, Issue 7: 4868-4888. doi: 10.3934/era.2026215

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Research article

The implicit mapping dynamics of a class of time-delay dynamical systems with impulses at fixed times

Han Xu ^1,2,
Xilin Fu ^{1
,
,}

1.
School of Mathematics and Statistics, Shandong Normal University, Ji'nan 250014, Shandong, China
2.
School of Mathematics and Statistics, Linyi University, Linyi 276005, Shandong, China

Received: 06 March 2026 Revised: 08 May 2026 Accepted: 19 May 2026 Published: 08 June 2026

In this paper, we investigate the periodic flows of a class of time-delay dynamical systems with impulses at fixed times through an implicit mapping method. As the impulsive points coincide with the time-delay points, discrete implicit mappings at impulsive points are obtained based on the impulsive mappings and interpolation techniques according to the given accuracy. As the impulsive points coincide with the regular points, the integral limits of the impulsive points are constructed, and discrete implicit mappings at impulsive points and time-delay points are obtained respectively by using the impulsive functions and integral techniques. The approximate solutions of the periodic flows determined by regular and delay nodes in one period are presented. A second-order delay dynamical system with an impulse at a fixed time is presented as an example. The implicit mapping method provides a plan for the periodic flows of switching delay dynamical systems.
- time-delay systems,
- implicit mappings,
- mapping structures,
- periodic flows
Citation: Han Xu, Xilin Fu. The implicit mapping dynamics of a class of time-delay dynamical systems with impulses at fixed times[J]. Electronic Research Archive, 2026, 34(7): 4868-4888. doi: 10.3934/era.2026215

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Abstract

In this paper, we investigate the periodic flows of a class of time-delay dynamical systems with impulses at fixed times through an implicit mapping method. As the impulsive points coincide with the time-delay points, discrete implicit mappings at impulsive points are obtained based on the impulsive mappings and interpolation techniques according to the given accuracy. As the impulsive points coincide with the regular points, the integral limits of the impulsive points are constructed, and discrete implicit mappings at impulsive points and time-delay points are obtained respectively by using the impulsive functions and integral techniques. The approximate solutions of the periodic flows determined by regular and delay nodes in one period are presented. A second-order delay dynamical system with an impulse at a fixed time is presented as an example. The implicit mapping method provides a plan for the periodic flows of switching delay dynamical systems.

References

[1]	C. Bai, Globally exponential stability and existence of periodic solution of Cohen-Grossberg type neural networks with delays and impulses, Nonlinear Anal. Real World Appl., 9 (2008), 747–761. https://doi.org/10.1016/j.nonrwa.2006.12.007 doi: 10.1016/j.nonrwa.2006.12.007
[2]	L. Zhang, J. Lu, F. Liu, J. Lou, Synchronization of time-delay coupled neural networks with st abilizing delayed impulsive control, IEEE Trans. Neural Netw. Learn. Syst., 35 (2024), 18899–18906. https://doi.org/10.1109/TNNLS.2023.3320651 doi: 10.1109/TNNLS.2023.3320651
[3]	S. L. Sun, L. S. Chen, Existence of positive periodic solution of an impulsive delay logistic model, Appl. Math. Comput., 184 (2007), 617–623. https://doi.org/10.1016/j.amc.2006.06.060 doi: 10.1016/j.amc.2006.06.060
[4]	F. Long, M. Q. Yang, Positive periodic solutions of delayed Nicholson's blowflies model with a linear harvesting term, Electron. J. Qual. Theory Differ. Equations, 41 (2011), 1–11.
[5]	R. Ming, Y. Xiao, Global dynamics of an impulsive vector-borne disease model with time delays, Math. Biosci. Eng., 20 (2024), 20939–20958. https://doi.org/10.3934/mbe.2023926 doi: 10.3934/mbe.2023926
[6]	W. Chen, W. Zheng, Input-to-state stabiliy and intergral input-to-state stability of nonlinear impulsive systems with delays, Automatica, 45 (2009), 1481–1488. https://doi.org/10.1016/j.automatica.2009.03.018 doi: 10.1016/j.automatica.2009.03.018
[7]	Q. Wan, Y. W. Wang, X. K. Liu, W. Yang, X. Chen, Impulsive load frequency control for multiarea interconnected power systems with network-induced delays, IEEE Trans. Ind. Electron., 73 (2026), 2916–2925. https://doi.org/10.1109/TIE.2025.3594429 doi: 10.1109/TIE.2025.3594429
[8]	P. S. Simeonov, D. D. Bainov, Stability with respect to part of the variables in systems with impulse effect, J. Math. Anal. Appl., 24 (1987), 547–560. https://doi.org/10.1016/0022-247X(86)90259-3 doi: 10.1016/0022-247X(86)90259-3
[9]	H. B. Bao, J. D. Cao, Exponential stability for stochastic BAM networks with descrete and distributed delays, Appl. Math. Comput., 218 (2011), 6188–6199. https://doi.org/10.1016/j.amc.2011.11.035 doi: 10.1016/j.amc.2011.11.035
[10]	X. Fu, X. Liu, S. Sivaloganathan, Oscillation criteria for impulsive parabolic differential equations with delay, J. Math. Anal. Appl., 268 (2002), 647–664. https://doi.org/10.1006/jmaa.2001.7786 doi: 10.1006/jmaa.2001.7786
[11]	X. Fu, B. Yan, Y. Liu, Nonlinear Impulsive Differential Systems, Science press, Beijing, 2008.
[12]	V. Lakshmikantham, D. Bainov, P. Simeonov, Theory of Impulsive Differential Equations, World scientific, Singapore, 1989. https://doi.org/10.1142/0906
[13]	V. Lakshmikantham, X. Lin, On quasi stability for impulsive differential systems, Nonlinear Anal. Theory Methods Appl., 13 (1989), 819–828. https://doi.org/10.1016/0362-546X(89)90074-6 doi: 10.1016/0362-546X(89)90074-6
[14]	X. Y. Liu, J. D. Cao, On periodic solutions of neural networks via differential inclusions, Neural Networks, 22 (2009), 329–334. https://doi.org/10.1016/j.neunet.2008.11.003 doi: 10.1016/j.neunet.2008.11.003
[15]	H. J. Xiang, J. D. Cao, Almost periodic solutions of recurrent neural networks with continuously distributed delays, Nonlinear Anal. Theory Methods Appl., 71 (2009), 6018–6097. https://doi.org/10.1016/j.na.2009.05.079 doi: 10.1016/j.na.2009.05.079
[16]	H. F. Huo, Existence of positive periodic solutions of a neutral delay lotka-volterra system with impulses, Comput. Math. Appl., 48 (2004), 1833–1846. https://doi.org/10.1016/j.camwa.2004.07.009 doi: 10.1016/j.camwa.2004.07.009
[17]	D. Chen, B. Dai, Periodic solutions of second-order impulsive delay differential system via generalized mountain pass theorem, Boundary Value Probl., 2014 (2014), 1–15. https://doi.org/10.1186/s13661-014-0234-z doi: 10.1186/s13661-014-0234-z
[18]	B. Dai, L. Bao, Positive periodic solutions generated by impulses for the delay Nicholson's blowflies model, Electron. J. Qual. Theory Differ. Equations, 4 (2016), 1–11. https://doi.org/10.14232/ejqtde.2016.1.4 doi: 10.14232/ejqtde.2016.1.4
[19]	T. Li, W. Zhao, Periodic solution of a neutral delay Leslie predator-prey model and the effect of random perturbation on the Smith growth model, Complexity, 2020 (2020), 8428269. https://doi.org/10.1155/2020/8428269 doi: 10.1155/2020/8428269
[20]	S. Chen, K. Wang, J. Liu, X. Lin, Periodic solutions of Cohen-grossberg-type bi-directional associative memory neural networks with neutral delays and impulse, AIMS Math., 6 (2020), 2539–2558. https://doi.org/10.3934/math.2021154 doi: 10.3934/math.2021154
[21]	Y. Yan, Positive periodic solutions for discrete time-delay hematopoiesis model with impulses, Open Math., 21 (2023), 20230613. https://doi.org/10.1515/math-2023-0613 doi: 10.1515/math-2023-0613
[22]	H. Qiu, Y. Zhang, F. Kong, Stability analysis of periodic solutions of the neutral-type neural networks with impulses and time-varying delays, Filomat, 37 (2023), 6021–6040. https://doi.org/10.2298/FIL2318021Q doi: 10.2298/FIL2318021Q
[23]	M. Bachar, Linearized stability analysis of nonlinear delay differential equations with impulses, Axioms, 13 (2024), 1–13. https://doi.org/10.3390/axioms13080524 doi: 10.3390/axioms13080524
[24]	M. Benhadri, T. Caraballo, Positivity of periodic solutions for neutral-type impulsive neural networks with distributed delays and two parameters, Math. Methods Appl. Sci., 48 (2025), 3759–3781. https://doi.org/10.1002/mma.10512 doi: 10.1002/mma.10512
[25]	A. C. J. Luo, Periodic flows in nonlinear dynamical systems based on discrete implicit maps, Int. J. Bifurcation Chaos, 25 (2015), 1550044.
[26]	A. C. J. Luo, S. Xing, Multiple bifurcation trees of period-1 motions to chaos in a periodically forced, time-delayed, hardening Duffing oscillator, Chaos, Solitons Fractals, 89 (2016), 405–434. https://doi.org/10.1016/j.chaos.2016.02.005 doi: 10.1016/j.chaos.2016.02.005
[27]	A. C. J. Luo, S. Xing, Bifurcation trees of period-3 motions to chaos in a time-delayed Duffing oscillatoy, Nonlinear Dyn., 88 (2017), 2831–2862. https://doi.org/10.1007/s11071-017-3415-3 doi: 10.1007/s11071-017-3415-3
[28]	S. Xing, A. C. J. Luo, Period motions to chaos in 1-dimensional time-delayed, nonlinear system, Eur. Phys. J. Spec. Top., 228 (2019), 1747–1765. https://doi.org/10.1140/epjst/e2019-800243-y doi: 10.1140/epjst/e2019-800243-y
[29]	C. Guo, A. C. J. Luo, Symmetric and asymmetric periodic motions of a nonlinear oscillator with a tuned mass damper inerter, Eur. Phys. J. Spec. Top., 230 (2021), 3533–3549. https://doi.org/10.1140/epjs/s11734-021-00235-5 doi: 10.1140/epjs/s11734-021-00235-5
[30]	S. Guo, A. C. J. Luo, Period-1 to period-4 motions in a 5D Lorenz system, Int. J. Bifurcation Chaos, 34 (2024), 2450065. https://doi.org/10.1142/S0218127424500652 doi: 10.1142/S0218127424500652
[31]	A. C. J. Luo, Y. Zhu, Periodic motions with impact chatters in an impact Duffing oscillator, Chaos, 34 (2024), 053124. https://doi.org/10.1063/5.0210693 doi: 10.1063/5.0210693
[32]	H. Xu, X. L. Fu, Periodic flows in switching dynamical systems through discrete implicit mappinngs, J. Appl. Anal. Comput., 16 (2026), 229–245. https://doi.org/10.11948/20240545 doi: 10.11948/20240545
[33]	L. Li, A. C. J. Luo, On periodic solutions of a second-order, time-delayed, discontinuous dynamical system, Chaos, Solitons Fractals, 114 (2018), 216–229. https://doi.org/10.1016/j.chaos.2018.07.005 doi: 10.1016/j.chaos.2018.07.005
[34]	A. C. J. Luo, Discretization and Implicit Mapping Dynamics, Higher Education Press, Beijing, 2015. https://doi.org/10.1007/978-3-662-47275-0

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