Geometric trajectory stabilization and parabolic modal rate matching in impulsive delayed Cohen–Grossberg neural networks

Gülden Altay Suroğlu; Münevver Tuz; Gülden Altay Suroğlu; Münevver Tuz

doi:10.3934/math.20261062

AIMS Mathematics

2026, Volume 11, Issue 8: 26470-26502. doi: 10.3934/math.20261062

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Research article

Geometric trajectory stabilization and parabolic modal rate matching in impulsive delayed Cohen–Grossberg neural networks

Gülden Altay Suroğlu ,
Münevver Tuz ^,

Department of Mathematics, Faculty of Science, Fırat University, Elazığ 23119, Türkiye; galtay@firat.edu.tr

Received: 15 May 2026 Revised: 29 July 2026 Accepted: 10 August 2026 Published: 25 August 2026
MSC : 34A37, 34D23, 53A04, 34C27, 92B20

This study investigated how exponential stability in impulsive delayed Cohen–Grossberg neural networks is reflected in the geometry of the error trajectory and in an auxiliary reaction–diffusion description. The analysis distinguished rigorous consequences of exponential stability from regular-point and finite-horizon geometric diagnostics. It was shown that exponential convergence of the error norm does not, in general, imply decay of the classical curvature. Instead, a speed-weighted curvature, representing the normal component of the error acceleration, satisfies an exponential upper bound under the stated higher-order regularity assumptions. An auxiliary reaction–diffusion model with multiplicative contractive impulses was then analysed by semigroup and Duhamel estimates, and a periodic Fourier-mode formulation was used to characterize transient rate matching between modal and forcing decay rates. Spectral graph estimates further provided a rigorous exponential envelope for the network disagreement energy. Numerical experiments supported these distinctions: The weighted-curvature diagnostic exhibited a fitted decay rate of $ 1.0478 $ with $ R^2 = 0.9914 $, while $ 456 $ of $ 500 $ Monte Carlo realizations satisfied the convergence criterion. The near-matching modal response had a peak ratio of $ 1.2553 $ relative to the selected detuned case, and the topology experiment detected statistically significant differences in both classical-curvature behavior and finite-horizon accumulated curvature. Overall, the results showed that norm stability rigorously controls weighted geometric and graph-energy quantities, whereas classical curvature and topology-dependent geometric effects should be interpreted as empirical finite-horizon diagnostics.
- Cohen–Grossberg neural networks,
- impulsive systems,
- trajectory geometry,
- parabolic modal rate matching,
- spectral graph analysis
Citation: Gülden Altay Suroğlu, Münevver Tuz. Geometric trajectory stabilization and parabolic modal rate matching in impulsive delayed Cohen–Grossberg neural networks[J]. AIMS Mathematics, 2026, 11(8): 26470-26502. doi: 10.3934/math.20261062

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Abstract

This study investigated how exponential stability in impulsive delayed Cohen–Grossberg neural networks is reflected in the geometry of the error trajectory and in an auxiliary reaction–diffusion description. The analysis distinguished rigorous consequences of exponential stability from regular-point and finite-horizon geometric diagnostics. It was shown that exponential convergence of the error norm does not, in general, imply decay of the classical curvature. Instead, a speed-weighted curvature, representing the normal component of the error acceleration, satisfies an exponential upper bound under the stated higher-order regularity assumptions. An auxiliary reaction–diffusion model with multiplicative contractive impulses was then analysed by semigroup and Duhamel estimates, and a periodic Fourier-mode formulation was used to characterize transient rate matching between modal and forcing decay rates. Spectral graph estimates further provided a rigorous exponential envelope for the network disagreement energy. Numerical experiments supported these distinctions: The weighted-curvature diagnostic exhibited a fitted decay rate of $ 1.0478 $ with $ R^2 = 0.9914 $, while $ 456 $ of $ 500 $ Monte Carlo realizations satisfied the convergence criterion. The near-matching modal response had a peak ratio of $ 1.2553 $ relative to the selected detuned case, and the topology experiment detected statistically significant differences in both classical-curvature behavior and finite-horizon accumulated curvature. Overall, the results showed that norm stability rigorously controls weighted geometric and graph-energy quantities, whereas classical curvature and topology-dependent geometric effects should be interpreted as empirical finite-horizon diagnostics.

References

[1]	J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. USA, 79 (1982), 2554–2558. https://doi.org/10.1073/pnas.79.8.2554 doi: 10.1073/pnas.79.8.2554
[2]	B. Kosko, Bidirectional associative memories, IEEE Trans. Syst. Man Cybern., 18 (1988), 49–60. https://doi.org/10.1109/21.87054 doi: 10.1109/21.87054
[3]	T. Chen, Global exponential stability of delayed Hopfield-type neural networks, Neural Networks, 14 (2001), 977–980. https://doi.org/10.1016/s0893-6080(01)00059-4 doi: 10.1016/s0893-6080(01)00059-4
[4]	M. A. Cohen, S. Grossberg, Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans. Syst. Man Cybern., SMC-13 (1983), 815–826. https://doi.org/10.1109/TSMC.1983.6313075 doi: 10.1109/TSMC.1983.6313075
[5]	Z. Liu, A. Chen, J. Cao, L. Huang, Existence and global exponential stability of periodic solution for BAM neural networks with periodic coefficients and time-varying delays, IEEE Trans. Circuits Syst. I, 50 (2003), 1162–1173. https://doi.org/10.1109/TCSI.2003.816306 doi: 10.1109/TCSI.2003.816306
[6]	J. Liang, J. Cao, Exponential stability of continuous-time and discrete-time bidirectional associative memory networks with delays, Chaos Soliton. Fract., 22 (2004), 773–785. https://doi.org/10.1016/j.chaos.2004.03.004 doi: 10.1016/j.chaos.2004.03.004
[7]	S. Arik, Global asymptotic stability analysis of bidirectional associative memory neural networks with time delays, IEEE Trans. Neural Networks, 16 (2005), 580–586. https://doi.org/10.1109/TNN.2005.844910 doi: 10.1109/TNN.2005.844910
[8]	A. Chen, L. Huang, J. Cao Existence and stability of almost periodic solutions for BAM neural networks with delays, Appl. Math. Comput., 137 (2003), 177–193. https://doi.org/10.1016/S0096-3003(02)00095-4 doi: 10.1016/S0096-3003(02)00095-4
[9]	C. Zhang, F. Yang, W. Li, D. Wu, J. Yang, Global exponential stability of BAM type Cohen–Grossberg neural networks with delays and impulsive, 2008 IEEE International Conference on Automation and Logistics, 2008, 3055–3059. https://doi.org/10.1109/ICAL.2008.4636703
[10]	F. Yang, C. Zhang, D. Wu, Global stability of impulsive BAM type Cohen–Grossberg neural networks with delays, Appl. Math. Comput., 186 (2007), 932–940. https://doi.org/10.1016/j.amc.2006.08.016 doi: 10.1016/j.amc.2006.08.016
[11]	W. Ding, L. Wang, $2^N$ almost periodic attractors for Cohen–Grossberg-type BAM neural networks with variable coefficients and distributed delays, J. Math. Anal. Appl., 373 (2011), 322–342. https://doi.org/10.1016/j.jmaa.2010.06.055 doi: 10.1016/j.jmaa.2010.06.055
[12]	M. Tuz, G. A. Suroğlu, New insights into delay-impulsive interactions and stability in almost periodic Cohen–Grossberg neural networks, Symmetry, 17 (2025), 2063. https://doi.org/10.3390/sym17122063 doi: 10.3390/sym17122063
[13]	T. Ivancevic, L. Jain, J. Patttison, A. Hariz, Nonlinear dynamics and chaos methods in neurodynamics and complex data analysis, Nonlinear Dynam., 56 (2009), 23–44. https://doi.org/10.1007/s11071-008-9376-9 doi: 10.1007/s11071-008-9376-9
[14]	A. Wan, H. Qiao, J. Peng, M. Wang, Delay-independent criteria for exponential stability of generalised Cohen–Grossberg neural networks with discrete delays, Phys. Lett. A, 353 (2006), 151–157. https://doi.org/10.1016/j.physleta.2005.12.085 doi: 10.1016/j.physleta.2005.12.085
[15]	K. Li, Delay-dependent exponential stability analysis for impulsive BAM neural networks with time-varying delays, Comput. Math. Appl., 56 (2008), 2088–2099. https://doi.org/10.1016/j.camwa.2008.03.038 doi: 10.1016/j.camwa.2008.03.038
[16]	M. Luo, J. Jiao, R. Wang, Impulsive control of a nonlinear dynamical network and its application to biological networks, J. Biol. Phys., 45 (2019), 31–44. https://doi.org/10.1007/s10867-018-9513-8 doi: 10.1007/s10867-018-9513-8
[17]	M. Akhmet, M. Tleubergenova, A. Zhamanshin, Z. Nugayeva, Artificial neural networks: Alpha unpredictability and chaotic dynamics, Springer Cham, 2025. https://doi.org/10.1007/978-3-031-68966-6
[18]	J. Hu, Z. Wang, G. P. Liu, Delay compensation-based state estimation for time-varying complex networks with incomplete observations and dynamical bias, IEEE Trans. Cybern., 52 (2022), 12071–12083. https://doi.org/10.1109/TCYB.2020.3043283 doi: 10.1109/TCYB.2020.3043283
[19]	M. Akhmet, M. Tleubergenova, R. Seilova, Z. Nugayeva, Symmetrical impulsive inertial neural networks with unpredictable and Poisson-stable oscillations, Symmetry, 15 (2023), 1812. https://doi.org/10.3390/sym15101812 doi: 10.3390/sym15101812
[20]	H. Bohr, Almost periodic functions, Chelsea Publishing, New York, 1947.
[21]	A. M. Fink, Almost periodic differential equations, Berlin: Springer, 1974. https://doi.org/10.1007/bfb0070324
[22]	G. T. Stamov, Almost periodic solutions of impulsive differential equations, Berlin: Springer, 2012. https://doi.org/10.1007/978-3-642-27546-3
[23]	S. Ponomarev, J. F. Couchouron, M. I. Kamenskii, Almost periodic solutions of evolution equations, Topol. Methods Nonlinear Anal., 50 (2017), 65–87. https://doi.org/10.12775/tmna.2017.012 doi: 10.12775/tmna.2017.012
[24]	Y. Liu, Z. You, L. Cao, On the almost periodic solution of generalised Hopfield neural networks with time-varying delays, Neurocomputing, 69 (2006), 1760–1767. https://doi.org/10.1016/j.neucom.2005.12.117 doi: 10.1016/j.neucom.2005.12.117
[25]	H. Xiang, J. Wang, J. Cao, Almost periodic solution to Cohen–Grossberg-type BAM networks with distributed delays, Neurocomputing, 72 (2009), 3751–3759. https://doi.org/10.1016/j.neucom.2009.05.014 doi: 10.1016/j.neucom.2009.05.014
[26]	L. Gao, Z. Liu, S. Yang, The input-to-state stability in probability for constrained delay systems with stochastic delayed impulses, IEEE Trans. Autom. Sci. Eng., 22 (2025), 9205–9217. https://doi.org/10.1109/tase.2024.3502656 doi: 10.1109/tase.2024.3502656
[27]	K. Li, H. Zeng, Stability in impulsive Cohen–Grossberg-type BAM neural networks with time-varying delays: a general analysis, Math. Comput. Simul., 80 (2010), 2329–2349. https://doi.org/10.1016/j.matcom.2010.05.012 doi: 10.1016/j.matcom.2010.05.012
[28]	M. Bohner, G. T. Stamov, I. M. Stamova, Almost periodic solutions of Cohen–Grossberg neural networks with time-varying delay and variable impulsive perturbations, Commun. Nonlinear Sci. Numer. Simul., 80 (2020), 104952. https://doi.org/10.1016/j.cnsns.2019.104952 doi: 10.1016/j.cnsns.2019.104952
[29]	A. A. Agrachev, Y. L. Sachkov, Control theory from the geometric viewpoint, Berlin: Springer, 2004. https://doi.org/10.1007/978-3-662-06404-7
[30]	R. L. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly, 82 (1975), 246–251. https://doi.org/10.1080/00029890.1975.11993807 doi: 10.1080/00029890.1975.11993807
[31]	M. P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Englewood Cliffs, 1976.
[32]	J. P. Hespanha, A. S. Morse, Stability of switched systems with average dwell-time, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304), Phoenix, AZ, 1999, 2655–2660. https://doi.org/10.1109/cdc.1999.831330
[33]	K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, Dordrecht: Kluwer Academic Publishers, 1992. https://doi.org/10.1007/978-94-015-7920-9
[34]	F. A. Rihan, Delay differential equations and their applications in biology, Forum for Interdisciplinary Mathematics, Springer, 2021. https://doi.org/10.1007/978-981-16-0626-7
[35]	W. Zhang, J. Huang, Stability analysis of stochastic delayed differential systems with state-dependent-delay impulses: application of neural networks, Cogn. Comput., 14 (2022), 805–813. https://doi.org/10.1007/s12559-021-09967-x doi: 10.1007/s12559-021-09967-x
[36]	W. Yang, W. Yu, J. Cao, F. E. Alsaadi, T. Hayat, Global exponential stability and lag synchronization for delayed memristive fuzzy {Cohen–Grossberg BAM} neural networks with impulses, Neural Networks, 98 (2018), 122–153. https://doi.org/10.1016/j.neunet.2017.11.001 doi: 10.1016/j.neunet.2017.11.001
[37]	M. Akhmet, E. Yılmaz, Neural networks with discontinuous/impact activations, Nonlinear Systems and Complexity, Vol. 9, New York: Springer, 2014. https://doi.org/10.1007/978-1-4614-8566-7
[38]	F. H. Clarke, Optimization and nonsmooth analysis, Philadelphia: SIAM, 1990. https://doi.org/10.1137/1.9781611971309
[39]	P. Li, R. Gao, C. Xu, Y. Li, A. Akgül, D. Baleanu, Dynamics exploration for a fractional-order delayed zooplankton–phytoplankton system, Chaos Soliton. Fract., 166 (2023), 112975. https://doi.org/10.1016/j.chaos.2022.112975 doi: 10.1016/j.chaos.2022.112975
[40]	F. R. K. Chung, Spectral graph theory, CBMS Regional Conference Series in Mathematics, Vol. 92, Providence: AMS, 1997. https://doi.org/10.1090/cbms/092
[41]	Q. Ma, S. Xu, Intentional delay can benefit consensus of second-order multi-agent systems, Automatica, 147 (2023), 110750. https://doi.org/10.1016/j.automatica.2022.110750 doi: 10.1016/j.automatica.2022.110750

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