The anti-periodic solutions of incommensurate fractional-order Cohen-Grossberg neural network with inertia

Zhiying Li; Wei Liu; Zhiying Li; Wei Liu

doi:10.3934/math.2025147

AIMS Mathematics

2025, Volume 10, Issue 2: 3180-3196. doi: 10.3934/math.2025147

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Research article

The anti-periodic solutions of incommensurate fractional-order Cohen-Grossberg neural network with inertia

Zhiying Li ,
Wei Liu ^,

Department of Mathematics, Shaoxing University Yuanpei College, Qunxian Middle Rd. 2799, Shaoxing, Zhejiang 312000, China

Received: 25 October 2024 Revised: 01 January 2025 Accepted: 24 January 2025 Published: 19 February 2025
MSC : 34K20, 92B20

A class of incommensurate fractional-order Cohen-Grossberg neural networks with inertia was investigated in this paper. First, the sufficient conditions for the boundedness of the solutions of the system were derived using the properties of fractional-order calculus. Second, by constructing a sequence of solutions in the system and using the Ascoli-Arzela theorem, the sufficient conditions for the existence of an anti-period solution and the global asymptotical stability of the system were deduced. Finally, the correctness of theoretical reasoning results was verified by a numerical simulation.
- incommensurate,
- fractional-order,
- Cohen-Grossberg neural networks,
- anti-periodic solution,
- global asymptotical stability
Citation: Zhiying Li, Wei Liu. The anti-periodic solutions of incommensurate fractional-order Cohen-Grossberg neural network with inertia[J]. AIMS Mathematics, 2025, 10(2): 3180-3196. doi: 10.3934/math.2025147

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Abstract

A class of incommensurate fractional-order Cohen-Grossberg neural networks with inertia was investigated in this paper. First, the sufficient conditions for the boundedness of the solutions of the system were derived using the properties of fractional-order calculus. Second, by constructing a sequence of solutions in the system and using the Ascoli-Arzela theorem, the sufficient conditions for the existence of an anti-period solution and the global asymptotical stability of the system were deduced. Finally, the correctness of theoretical reasoning results was verified by a numerical simulation.

References

[1]	V. Lakshmikantham, A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal.: Theory, Methods Appl., 699 (2008), 2677–2682. https://doi.org/10.1016/j.na.2007.08.042 doi: 10.1016/j.na.2007.08.042
[2]	D. Shantanu, Functional fractional calculus, Berlin Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-642-20545-3
[3]	B. B. Mandelbrot, The fractal geometry of nature, New York: W. H Freeman and Company, 1982. https://doi.org/10.1119/1.13295
[4]	R. Hilfer, Applications of fractional calculus in physics, New Jersey: World Scientific, 2001. https://doi.org/10.1142/3779
[5]	B. J. West, M. Bologna, P. Grigolini, Physics of fractional operators, New York: Springer, 2003. https://doi.org/10.1007/978-0-387-21746-8
[6]	A. A. Kilbas, S. A. Marzan S A, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Diff. Equat., 41 (2005), 84–89. https://doi.org/10.1007/s10625-005-0137-y doi: 10.1007/s10625-005-0137-y
[7]	S. Dadras, H. R. Momeni, Control of a fractional-order economical system via sliding mode, Physica A: Stat. Mech. Appl., 389 (2010), 2434–2442. https://doi.org/10.1016/j.physa.2010.02.025 doi: 10.1016/j.physa.2010.02.025
[8]	K. Babcock, R. Westervelt, Stability and dynamics of simple electronic neural networks with added inertia, Physica D., 23 (1986), 464–469. https://doi.org/10.1016/0167-2789(86)90152-1 doi: 10.1016/0167-2789(86)90152-1
[9]	A. Mauro, F. Conti, F. Dodge, R. Schor, Subthreshold behavior and phenomenological impendance of the squid giant axion, J. Gen. Physiol, 55 (1970), 497–523. https://doi.org/10.1085/jgp.55.4.497 doi: 10.1085/jgp.55.4.497
[10]	D. Angelaki, M. Correia, Models of membrance resonance in pigeon semicircular canal type Ⅱ hair cells, Biolo. Cybern., 65 (1991), 1–10. https://doi.org/10.1007/BF00197284 doi: 10.1007/BF00197284
[11]	C. Koch, Cable theory in neurons with active, linearized membranes, Biolo. Cyber., 50 (1984), 15–33. https://doi.org/10.1007/BF00317936 doi: 10.1007/BF00317936
[12]	R. Rakkiyappan, E. Kumari, A. Chandrasekar, R. Krishnasamy, Synchronization and periodicity of coupled inertial memristive neural networks with supremums, Neurocomputing, 214 (2016), 739–749. https://doi.org/10.1016/j.neucom.2016.06.061 doi: 10.1016/j.neucom.2016.06.061
[13]	Q. Fu, S. Zhong, K. Shi, Exponential synchronization of memristive neural networks with inertial and nonlinear coupling terms: Pinning impulsive control approache, Appl. Math. Comput., 402 (2021). https://doi.org/10.1016/j.amc.2021.126169 doi: 10.1016/j.amc.2021.126169
[14]	L. Ke, Mittag-Leffler stability and aysmptotic $\omega$-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
[15]	L. Wang, M. Ge, J. Hu, Global stability and stabilization for inertial memristive neural networks with unbounded distributed delays, Nonlinear Dyn., 95 (2019), 943–955. https://doi.org/10.1007/s11071-018-4606-2 doi: 10.1007/s11071-018-4606-2
[16]	C. Huang, S. Mo, H. Liu, J. Cao, Bifurcation analysis of a fractional-order Cohen-Grossberg neural network with three delays, Chinese J. Phys., 88 (2024), 360–379. https://doi.org/10.1016/j.cjph.2023.12.031 doi: 10.1016/j.cjph.2023.12.031
[17]	L. Li, Y. Lu, F. Wang, X. Liu, Mittag-Leffler stability and synchronization of multi-delayed fractional neural networks via Halanay inequality, Circuits Syst. Signal Proc., 44 (2024), 862–887. https://doi.org/10.1007/s00034-024-02883-z doi: 10.1007/s00034-024-02883-z
[18]	M. Roohi, S. Mirzajani, A. R. Haghighi, A. Basse-O'Connor, Robust design of two-level non-integer SMC based on deep soft actor-critic for synchronization of chaotic fractional order memristive neural networks, Fract. Fractional, 8 (2024), 548. https://doi.org/10.3390/fractalfract8090548 doi: 10.3390/fractalfract8090548
[19]	Z. Li, Y. Zhang, The boundedness and the global mittag-leffler synchronization of fractional-order inertial cohen-grossberg neural networks with time delay, Neural Proc. Lett., 54 (2022), 597–611. https://doi.org/10.1007/s11063-021-10648-x doi: 10.1007/s11063-021-10648-x
[20]	Z. Li, W. Jiang, Dynamic analysis of fractional-order neural networks with inertia, AIMS Math., 7 (2022), 16889–16906. https://doi.org/10.3934/math.2022927 doi: 10.3934/math.2022927
[21]	S. Zhang, M. Tang, X. Liu, Synchronization of a Riemann-Liouville fractional time-delayed neural network with two inertial terms, Circuits Sys. Signal Proce., 51002 (2021). https://doi.org/10.1007/s00034-021-01717-6 doi: 10.1007/s00034-021-01717-6
[22]	Y. Liu, Y. Sun, L. liu, Stability analysis and synchronization control of fractional-order inertial neural networks with time-varying delay, ACCESS, 3178123 (2022), 56081–560093. https://doi.org/10.1109/ACCESS.2022.3178123 doi: 10.1109/ACCESS.2022.3178123
[23]	Y. Gu, H. Wang, Y. 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
[24]	Y. Zhang, Z. Li, W. Liu, The stability of anti-periodic solutions for fractional-oeder inertial BAM neural networks with time-delays, AIMS Math., 8 (2023), 6176–6190. https://doi.org/10.3934/math.2023312 doi: 10.3934/math.2023312
[25]	D. Xu, Z. Li, Mittag-Leffler stabilization of anti-periodic solutions for fractional-order neural networks with time-varying delays, AIMS Math., 8 (2023), 1610–1619. https://doi.org/10.3934/math.2023081 doi: 10.3934/math.2023081
[26]	T. Munevver, G. Surolu, Anti-periodic solutions for fractional-order bidirectional associative memory neural networks with delays, Thermal Scien., 23 (2019), 2169–2177. https://doi.org/10.2298/TSCI190805406T doi: 10.2298/TSCI190805406T
[27]	T. Munevver, Global asymptotic stability of anti-periodic solutions of time-delayed fractional bam neural networks, Neural Proce. Lett., 56 (2024). https://doi.org/10.1007/s11063-024-11561-9 doi: 10.1007/s11063-024-11561-9
[28]	Y. Ke, C. Miao, Anti-periodic solutions of inertial neural networks with time delays, Neural Proce. Lett., 45 (2017), 523–538. https://doi.org/10.1007/s11063-016-9540-z doi: 10.1007/s11063-016-9540-z

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