July  2022, 15(7): 1651-1667. doi: 10.3934/dcdss.2021160

Positive stability analysis of pseudo almost periodic solutions for HDCNNs accompanying $ D $ operator

1. 

College of Meteorology and Oceanography, National University of Defense Technology, Changsha, Hunan 410073, China

2. 

School of Mathematics and Statistics, Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha 410114, China

*Corresponding author: Chuangxia Huang

Received  August 2021 Revised  September 2021 Published  July 2022 Early access  December 2021

Fund Project: This research was supported by the National Natural Science Foundation of China (Nos. 11971076, 11801562) and the Natural Science Foundation of Hunan Province (No. 2019JJ40142)

In this study, the stable dynamics of a kind of high-order cellular neural networks accompanying $ D $ operators and mixed delays are analyzed. The global existence of bounded positive solutions is substantiated by applying some novel differential inequality analyses. Meanwhile, by exploiting Lyapunov function method, some sufficient criteria are gained to validate the positiveness and globally exponential stability of pseudo almost periodic solutions on the addressed networks. In addition, computer simulations are produced to test the derived analytical findings.

Citation: Lilun Zhang, Le Li, Chuangxia Huang. Positive stability analysis of pseudo almost periodic solutions for HDCNNs accompanying $ D $ operator. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1651-1667. doi: 10.3934/dcdss.2021160
References:
[1]

E. Ait Dads and K. Ezzinbi, Pseudo almost periodic solutions of some delay differential equations, J. Math. Anal. Appl., 201 (1996), 840-850.  doi: 10.1006/jmaa.1996.0287.

[2]

Z. CaiJ. Huang and L. Huang, Periodic orbit analysis for the delayed Filippov system, Proc. Amer. Math. Soc., 146 (2018), 4667-4682.  doi: 10.1090/proc/13883.

[3]

Z. CaiL. Huang and Z. Wang, Mono/multi-periodicity generated by impulses control in time-delayed memristor-based neural networks, Nonlinear Anal. Hybrid Syst., 36 (2020), 100861.  doi: 10.1016/j.nahs.2020.100861.

[4]

Z. CaiJ. HuangL. Yang and L. Huang, Periodicity and stabilization control of the delayed Flippov system with perturbation, Discrete Contin. Dyn. Syst. Ser., 25 (2020), 1439-1467.  doi: 10.3934/dcdsb.2019235.

[5]

Q. Cao and X. Guo, Anti-periodic dynamics on high-order inertial Hopfield neural networks involving time-varying delays, AIMS Math., 5 (2020), 5402-5421.  doi: 10.3934/math.2020347.

[6]

Q. Cao, G. Wang and C. Qian, New results on global exponential stability for a periodic Nicholson's blowflies model involving time-varying delays, Adv. Difference Equ., 2020 (2020), 43, 12 pp. doi: 10.1186/s13662-020-2495-4.

[7]

Y. DengC. Huang and J. Cao, New results on dynamics of neutral type HCNNs with proportional delays, Math. Comput. Simulation, 187 (2021), 51-59.  doi: 10.1016/j.matcom.2021.02.001.

[8]

L. Duan and C. Huang, Existence and global attractivity of almost periodic solutions for a delayed differential neoclassical growth model, Math. Methods Appl. Sci., 40 (2017), 814-822.  doi: 10.1002/mma.4019.

[9]

L. Duan, L. Huang and X. Fang, Finite-time synchronization for recurrent neural networks with discontinuous activations and time-varying delays, Chaos, 27 (2017), 013101, 10 pp. doi: 10.1063/1.4966177.

[10]

L. DuanL. HuangZ. Guo and X. Fang, Periodic attractor for reaction-diffusion high-order Hopfield neural networks with time-varying delays, Comput. Math. Appl., 73 (2017), 233-245.  doi: 10.1016/j.camwa.2016.11.010.

[11]

L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications, Pure and Applied Mathematics (New York). Wiley-Interscience, New York, 2000. doi: 10.1002/9781118033029.

[12]

A. Fink, Almost Periodic Differential Equations, Springer-Verlag, Berlin-New York, 1974.

[13]

X. GuoC. Huang and J. Cao, Nonnegative periodicity on high-order proportional delayed cellular neural networks involving $D$ operator, AIMS Math., 6 (2021), 2228-2243.  doi: 10.3934/math.2021135.

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Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, 1473. Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.

[15]

C. HuangB. LiuC. Qian and J. Cao, Stability on positive pseudo almost periodic solutions of HPDCNNs incorporating operator, Math. Comput. Simulation, 190 (2021), 1150-1163.  doi: 10.1016/j.matcom.2021.06.027.

[16]

C. HuangX. Long and J. Cao, Stability of anti-periodic recurrent neural networks with multi-proportional delays, Math. Methods Appl. Sci., 43 (2020), 6093-6102.  doi: 10.1002/mma.6350.

[17]

C. HuangX. LongL. Huang and S. Fu, Stability of almost periodic Nicholson's blowflies model involving patch structure and mortality terms, Canad. Math. Bull., 63 (2020), 405-422.  doi: 10.4153/S0008439519000511.

[18]

C. HuangR. SuJ. Cao and S. Xiao, Asymptotically stable high-order neutral cellular neural networks with proportional delays and D operators, Math. Comput. Simulation, 171 (2020), 127-135.  doi: 10.1016/j.matcom.2019.06.001.

[19]

C. Huang, J. Wang and L. Huang, Asymptotically almost periodicity of delayed Nicholson-type system involving patch structure, Electronic Journal of Differential Equations, 2020 (2020), 1–17. Available from: http://ejde.math.txstate.edu.

[20]

C. HuangS. Wen and L. Huang, Dynamics of anti-periodic solutions on shunting inhibitory cellular neural networks with multi-proportional delays, Neurocomputing, 357 (2019), 47-52. 

[21]

C. HuangH. Yang and J. Cao, Weighted pseudo almost periodicity of multi-proportional delayed shunting inhibitory cellular neural networks with D operator, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 1259-1272.  doi: 10.3934/dcdss.2020372.

[22]

C. HuangX. Yang and J. Cao, Stability analysis of Nicholson's blowflies equation with two different delays, Math. Comput. Simulation, 171 (2020), 201-206.  doi: 10.1016/j.matcom.2019.09.023.

[23]

C. HuangZ. YangT. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Differential Equations, 256 (2014), 2101-2114.  doi: 10.1016/j.jde.2013.12.015.

[24]

C. Huang and H. Zhang, Periodicity of non-autonomous inertial neural networks involving proportional delays and non-reduced order method, Int. J. Biomath., 12 (2019), 1950016.  doi: 10.1142/S1793524519500165.

[25]

C. HuangH. Zhang and L. Huang, Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term, Commun. Pure Appl. Anal., 18 (2019), 3337-3349.  doi: 10.3934/cpaa.2019150.

[26]

C. HuangX. ZhaoJ. Cao and F. E. Alsaadi, Global dynamics of neoclassical growth model with multiple pairs of variable delays, Nonlinearity, 33 (2020), 6819-6834.  doi: 10.1088/1361-6544/abab4e.

[27]

M. IswaryaR. RajaG. RajchakitJ. CaoJ. Alzabut and C. Huang, Existence, uniqueness and exponential stability of periodic solution for discrete-time delayed BAM neural networks based on coincidence degree theory and graph theoretic method, Mathematics, 7 (2019), 1055.  doi: 10.3390/math7111055.

[28]

J. LiJ. Ying and D. Xie, On the analysis and application of an ion size-modified Poisson-Boltzmann equation, Nonlinear Anal. Real World Appl., 47 (2019), 188-203.  doi: 10.1016/j.nonrwa.2018.10.011.

[29]

X. LiJ. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Appl. Math. Comput., 329 (2018), 14-22.  doi: 10.1016/j.amc.2018.01.036.

[30]

X. LiX. Yang and T. Huang, Persistence of delayed cooperative models: Impulsive control method, Appl. Math. Comput., 342 (2019), 130-146.  doi: 10.1016/j.amc.2018.09.003.

[31]

Y. LiX. Meng and L. Xiong, Pseudo almost periodic solutions for neutral type high-order Hopfield neural networks with mixed time-varying delays and leakage delays on time scales, International Journal of Machine Learning and Cybernetics, 8 (2017), 1915-1927.  doi: 10.1007/s13042-016-0570-7.

[32]

W. Lu and T. Chen, Global exponential stability of almost periodic solutions for a large class of delayed dynamical systems, Sci. China Ser., 48 (2005), 1015-1026.  doi: 10.1360/04ys0076.

[33]

G. M. NGuérékata, Almost Automorphic Functions and Almost Periodic Functions in Abstract Spaces, Kluwer Academic, Plenum Publishers, New York, 2001. doi: 10.1007/978-1-4757-4482-8.

[34]

C. Qian and Y. Hu, Novel stability criteria on nonlinear density-dependent mortality Nicholson's blowflies systems in asymptotically almost periodic environments, J. Inequal. Appl., 2020 (2020), 13, 18 pp. doi: 10.1186/s13660-019-2275-4.

[35]

W. ShenX. Zhang and Y. Wang, Stability analysis of high order neural networks with proportional delays, Neurocomputing, 372 (2020), 33-39.  doi: 10.1016/j.neucom.2019.09.019.

[36]

H. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995.

[37]

Y. Tan, Dynamics analysis of Mackey-Glass model with two variable delays, Math. Biosci. Eng., 17 (2020), 4513-4526.  doi: 10.3934/mbe.2020249.

[38]

R. WeiJ. Cao and C. Huang, Lagrange exponential stability of quaternion-valued memristive neural networks with time delays, Math. Methods Appl. Sci., 43 (2020), 7269-7291.  doi: 10.1002/mma.6463.

[39]

T. WeiX. Xie and X. Li, Persistence and periodicity of survival red blood cells model with time-varying delays and impulses, Mathematical Modelling and Control, 1 (2021), 12-25.  doi: 10.3934/mmc.2021002.

[40]

B. Xiao and H. Meng, Existence and exponential stability of positive almost periodic solutions for high-order Hopfield neural networks, Appl. Math. Model., 33 (2009), 532-542.  doi: 10.1016/j.apm.2007.11.027.

[41]

Y. Xu, Convergence on non-autonomous inertial neural networks with unbounded distributed delays, Journal of Experimental & Theoretical Artificial Intelligence, 32 (2020), 503-513.  doi: 10.1080/0952813X.2019.1652941.

[42]

Y. Xu, Exponential stability of weighted pseudo almost periodic solutions for HCNNs with mixed delays, Neural Processing Letters, 46 (2017), 507-519.  doi: 10.1007/s11063-017-9595-5.

[43]

Y. Xu, Exponential stability of pseudo almost periodic solutions for neutral type cellular neural networks with D operator, Neural Processing Letters, 46 (2017), 329-342.  doi: 10.1007/s11063-017-9584-8.

[44]

Y. XuQ. Cao and X. Guo, Stability on a patch structure Nicholson's blowflies system involving distinctive delays, Appl. Math. Lett., 105 (2020), 106340.  doi: 10.1016/j.aml.2020.106340.

[45]

Y. Xu and J. Zhong, Convergence of neutral type proportional-delayed HCNNs with D operators, Int. J. Biomath., 12 (2019), 1950002.  doi: 10.1142/S1793524519500025.

[46]

D. YangX. Li and J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Anal. Hybrid Syst., 32 (2019), 294-305.  doi: 10.1016/j.nahs.2019.01.006.

[47]

G. Yang, Exponential stability of positive recurrent neural networks with multi-proportional delays, Neural Processing Letters, 49 (2019), 67-78.  doi: 10.1007/s11063-018-9802-z.

[48]

H. Yang, Weighted pseudo almost periodicity on neutral type CNNs involving multi-proportional delays and D operator, AIMS Math., 6 (2021), 1865-1879.  doi: 10.3934/math.2021113.

[49]

L. Yao, Global convergence of CNNs with neutral type delays and D operator, Neural Computing and Applications, 29 (2018), 105-109.  doi: 10.1007/s00521-016-2403-8.

[50]

C. Zhang, Pseudo almost periodic solutions of some differential equations II, J. Math. Anal. Appl., 192 (1995), 543-561.  doi: 10.1006/jmaa.1995.1189.

[51]

C. Zhang, Almost Periodic Type Functions and Ergodicity, Science Press Beijing, Beijing; Kluwer Academic Publishers, Dordrecht, 2003.

[52]

H. Zhang, Q. Cao and H. Yang, Asymptotically almost periodic dynamics on delayed Nicholson-type system involving patch structure, J. Inequal. Appl., 2020 (2020), 102, 27 pp. doi: 10.1186/s13660-020-02366-0.

[53]

J. Zhang and C. Huang, Dynamics analysis on a class of delayed neural networks involving inertial terms, Adv. Difference Equ., 2020 (2020), 12 pp. doi: 10.1186/s13662-020-02566-4.

[54]

Q. Zhou, Weighted pseudo anti-periodic solutions for cellular neural networks with mixed delays, Asian J. Control, 19 (2017), 1557-1563.  doi: 10.1002/asjc.1468.

show all references

References:
[1]

E. Ait Dads and K. Ezzinbi, Pseudo almost periodic solutions of some delay differential equations, J. Math. Anal. Appl., 201 (1996), 840-850.  doi: 10.1006/jmaa.1996.0287.

[2]

Z. CaiJ. Huang and L. Huang, Periodic orbit analysis for the delayed Filippov system, Proc. Amer. Math. Soc., 146 (2018), 4667-4682.  doi: 10.1090/proc/13883.

[3]

Z. CaiL. Huang and Z. Wang, Mono/multi-periodicity generated by impulses control in time-delayed memristor-based neural networks, Nonlinear Anal. Hybrid Syst., 36 (2020), 100861.  doi: 10.1016/j.nahs.2020.100861.

[4]

Z. CaiJ. HuangL. Yang and L. Huang, Periodicity and stabilization control of the delayed Flippov system with perturbation, Discrete Contin. Dyn. Syst. Ser., 25 (2020), 1439-1467.  doi: 10.3934/dcdsb.2019235.

[5]

Q. Cao and X. Guo, Anti-periodic dynamics on high-order inertial Hopfield neural networks involving time-varying delays, AIMS Math., 5 (2020), 5402-5421.  doi: 10.3934/math.2020347.

[6]

Q. Cao, G. Wang and C. Qian, New results on global exponential stability for a periodic Nicholson's blowflies model involving time-varying delays, Adv. Difference Equ., 2020 (2020), 43, 12 pp. doi: 10.1186/s13662-020-2495-4.

[7]

Y. DengC. Huang and J. Cao, New results on dynamics of neutral type HCNNs with proportional delays, Math. Comput. Simulation, 187 (2021), 51-59.  doi: 10.1016/j.matcom.2021.02.001.

[8]

L. Duan and C. Huang, Existence and global attractivity of almost periodic solutions for a delayed differential neoclassical growth model, Math. Methods Appl. Sci., 40 (2017), 814-822.  doi: 10.1002/mma.4019.

[9]

L. Duan, L. Huang and X. Fang, Finite-time synchronization for recurrent neural networks with discontinuous activations and time-varying delays, Chaos, 27 (2017), 013101, 10 pp. doi: 10.1063/1.4966177.

[10]

L. DuanL. HuangZ. Guo and X. Fang, Periodic attractor for reaction-diffusion high-order Hopfield neural networks with time-varying delays, Comput. Math. Appl., 73 (2017), 233-245.  doi: 10.1016/j.camwa.2016.11.010.

[11]

L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications, Pure and Applied Mathematics (New York). Wiley-Interscience, New York, 2000. doi: 10.1002/9781118033029.

[12]

A. Fink, Almost Periodic Differential Equations, Springer-Verlag, Berlin-New York, 1974.

[13]

X. GuoC. Huang and J. Cao, Nonnegative periodicity on high-order proportional delayed cellular neural networks involving $D$ operator, AIMS Math., 6 (2021), 2228-2243.  doi: 10.3934/math.2021135.

[14]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, 1473. Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.

[15]

C. HuangB. LiuC. Qian and J. Cao, Stability on positive pseudo almost periodic solutions of HPDCNNs incorporating operator, Math. Comput. Simulation, 190 (2021), 1150-1163.  doi: 10.1016/j.matcom.2021.06.027.

[16]

C. HuangX. Long and J. Cao, Stability of anti-periodic recurrent neural networks with multi-proportional delays, Math. Methods Appl. Sci., 43 (2020), 6093-6102.  doi: 10.1002/mma.6350.

[17]

C. HuangX. LongL. Huang and S. Fu, Stability of almost periodic Nicholson's blowflies model involving patch structure and mortality terms, Canad. Math. Bull., 63 (2020), 405-422.  doi: 10.4153/S0008439519000511.

[18]

C. HuangR. SuJ. Cao and S. Xiao, Asymptotically stable high-order neutral cellular neural networks with proportional delays and D operators, Math. Comput. Simulation, 171 (2020), 127-135.  doi: 10.1016/j.matcom.2019.06.001.

[19]

C. Huang, J. Wang and L. Huang, Asymptotically almost periodicity of delayed Nicholson-type system involving patch structure, Electronic Journal of Differential Equations, 2020 (2020), 1–17. Available from: http://ejde.math.txstate.edu.

[20]

C. HuangS. Wen and L. Huang, Dynamics of anti-periodic solutions on shunting inhibitory cellular neural networks with multi-proportional delays, Neurocomputing, 357 (2019), 47-52. 

[21]

C. HuangH. Yang and J. Cao, Weighted pseudo almost periodicity of multi-proportional delayed shunting inhibitory cellular neural networks with D operator, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 1259-1272.  doi: 10.3934/dcdss.2020372.

[22]

C. HuangX. Yang and J. Cao, Stability analysis of Nicholson's blowflies equation with two different delays, Math. Comput. Simulation, 171 (2020), 201-206.  doi: 10.1016/j.matcom.2019.09.023.

[23]

C. HuangZ. YangT. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Differential Equations, 256 (2014), 2101-2114.  doi: 10.1016/j.jde.2013.12.015.

[24]

C. Huang and H. Zhang, Periodicity of non-autonomous inertial neural networks involving proportional delays and non-reduced order method, Int. J. Biomath., 12 (2019), 1950016.  doi: 10.1142/S1793524519500165.

[25]

C. HuangH. Zhang and L. Huang, Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term, Commun. Pure Appl. Anal., 18 (2019), 3337-3349.  doi: 10.3934/cpaa.2019150.

[26]

C. HuangX. ZhaoJ. Cao and F. E. Alsaadi, Global dynamics of neoclassical growth model with multiple pairs of variable delays, Nonlinearity, 33 (2020), 6819-6834.  doi: 10.1088/1361-6544/abab4e.

[27]

M. IswaryaR. RajaG. RajchakitJ. CaoJ. Alzabut and C. Huang, Existence, uniqueness and exponential stability of periodic solution for discrete-time delayed BAM neural networks based on coincidence degree theory and graph theoretic method, Mathematics, 7 (2019), 1055.  doi: 10.3390/math7111055.

[28]

J. LiJ. Ying and D. Xie, On the analysis and application of an ion size-modified Poisson-Boltzmann equation, Nonlinear Anal. Real World Appl., 47 (2019), 188-203.  doi: 10.1016/j.nonrwa.2018.10.011.

[29]

X. LiJ. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Appl. Math. Comput., 329 (2018), 14-22.  doi: 10.1016/j.amc.2018.01.036.

[30]

X. LiX. Yang and T. Huang, Persistence of delayed cooperative models: Impulsive control method, Appl. Math. Comput., 342 (2019), 130-146.  doi: 10.1016/j.amc.2018.09.003.

[31]

Y. LiX. Meng and L. Xiong, Pseudo almost periodic solutions for neutral type high-order Hopfield neural networks with mixed time-varying delays and leakage delays on time scales, International Journal of Machine Learning and Cybernetics, 8 (2017), 1915-1927.  doi: 10.1007/s13042-016-0570-7.

[32]

W. Lu and T. Chen, Global exponential stability of almost periodic solutions for a large class of delayed dynamical systems, Sci. China Ser., 48 (2005), 1015-1026.  doi: 10.1360/04ys0076.

[33]

G. M. NGuérékata, Almost Automorphic Functions and Almost Periodic Functions in Abstract Spaces, Kluwer Academic, Plenum Publishers, New York, 2001. doi: 10.1007/978-1-4757-4482-8.

[34]

C. Qian and Y. Hu, Novel stability criteria on nonlinear density-dependent mortality Nicholson's blowflies systems in asymptotically almost periodic environments, J. Inequal. Appl., 2020 (2020), 13, 18 pp. doi: 10.1186/s13660-019-2275-4.

[35]

W. ShenX. Zhang and Y. Wang, Stability analysis of high order neural networks with proportional delays, Neurocomputing, 372 (2020), 33-39.  doi: 10.1016/j.neucom.2019.09.019.

[36]

H. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995.

[37]

Y. Tan, Dynamics analysis of Mackey-Glass model with two variable delays, Math. Biosci. Eng., 17 (2020), 4513-4526.  doi: 10.3934/mbe.2020249.

[38]

R. WeiJ. Cao and C. Huang, Lagrange exponential stability of quaternion-valued memristive neural networks with time delays, Math. Methods Appl. Sci., 43 (2020), 7269-7291.  doi: 10.1002/mma.6463.

[39]

T. WeiX. Xie and X. Li, Persistence and periodicity of survival red blood cells model with time-varying delays and impulses, Mathematical Modelling and Control, 1 (2021), 12-25.  doi: 10.3934/mmc.2021002.

[40]

B. Xiao and H. Meng, Existence and exponential stability of positive almost periodic solutions for high-order Hopfield neural networks, Appl. Math. Model., 33 (2009), 532-542.  doi: 10.1016/j.apm.2007.11.027.

[41]

Y. Xu, Convergence on non-autonomous inertial neural networks with unbounded distributed delays, Journal of Experimental & Theoretical Artificial Intelligence, 32 (2020), 503-513.  doi: 10.1080/0952813X.2019.1652941.

[42]

Y. Xu, Exponential stability of weighted pseudo almost periodic solutions for HCNNs with mixed delays, Neural Processing Letters, 46 (2017), 507-519.  doi: 10.1007/s11063-017-9595-5.

[43]

Y. Xu, Exponential stability of pseudo almost periodic solutions for neutral type cellular neural networks with D operator, Neural Processing Letters, 46 (2017), 329-342.  doi: 10.1007/s11063-017-9584-8.

[44]

Y. XuQ. Cao and X. Guo, Stability on a patch structure Nicholson's blowflies system involving distinctive delays, Appl. Math. Lett., 105 (2020), 106340.  doi: 10.1016/j.aml.2020.106340.

[45]

Y. Xu and J. Zhong, Convergence of neutral type proportional-delayed HCNNs with D operators, Int. J. Biomath., 12 (2019), 1950002.  doi: 10.1142/S1793524519500025.

[46]

D. YangX. Li and J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Anal. Hybrid Syst., 32 (2019), 294-305.  doi: 10.1016/j.nahs.2019.01.006.

[47]

G. Yang, Exponential stability of positive recurrent neural networks with multi-proportional delays, Neural Processing Letters, 49 (2019), 67-78.  doi: 10.1007/s11063-018-9802-z.

[48]

H. Yang, Weighted pseudo almost periodicity on neutral type CNNs involving multi-proportional delays and D operator, AIMS Math., 6 (2021), 1865-1879.  doi: 10.3934/math.2021113.

[49]

L. Yao, Global convergence of CNNs with neutral type delays and D operator, Neural Computing and Applications, 29 (2018), 105-109.  doi: 10.1007/s00521-016-2403-8.

[50]

C. Zhang, Pseudo almost periodic solutions of some differential equations II, J. Math. Anal. Appl., 192 (1995), 543-561.  doi: 10.1006/jmaa.1995.1189.

[51]

C. Zhang, Almost Periodic Type Functions and Ergodicity, Science Press Beijing, Beijing; Kluwer Academic Publishers, Dordrecht, 2003.

[52]

H. Zhang, Q. Cao and H. Yang, Asymptotically almost periodic dynamics on delayed Nicholson-type system involving patch structure, J. Inequal. Appl., 2020 (2020), 102, 27 pp. doi: 10.1186/s13660-020-02366-0.

[53]

J. Zhang and C. Huang, Dynamics analysis on a class of delayed neural networks involving inertial terms, Adv. Difference Equ., 2020 (2020), 12 pp. doi: 10.1186/s13662-020-02566-4.

[54]

Q. Zhou, Weighted pseudo anti-periodic solutions for cellular neural networks with mixed delays, Asian J. Control, 19 (2017), 1557-1563.  doi: 10.1002/asjc.1468.

Figure 1.  Numerical characteristics for solutions $ x(t) $ of HDCNNs (4.1) incorporating distinct initial values: ($ -0.3\cos 2t $, $ 3+\sin 3t $), ($ 2.3\cos 2t $, $ 2+1.8\sin 3t $), ($ 0.5+\cos 3t $, $ -0.8\sin 2t $)
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Chuangxia Huang, Hedi Yang, Jinde Cao. Weighted pseudo almost periodicity of multi-proportional delayed shunting inhibitory cellular neural networks with D operator. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1259-1272. doi: 10.3934/dcdss.2020372

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