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## 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 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 & Continuous Dynamical Systems - S, 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.  Google Scholar [2] Z. Cai, J. 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.  Google Scholar [3] Z. Cai, L. 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.  Google Scholar [4] Z. Cai, J. Huang, L. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] Y. Deng, C. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] L. Duan, L. Huang, Z. 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.  Google Scholar [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.  Google Scholar [12] A. Fink, Almost Periodic Differential Equations, Springer-Verlag, Berlin-New York, 1974.  Google Scholar [13] X. Guo, C. 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.  Google Scholar [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.  Google Scholar [15] C. Huang, B. Liu, C. 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.  Google Scholar [16] C. Huang, X. 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.  Google Scholar [17] C. Huang, X. Long, L. 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.  Google Scholar [18] C. Huang, R. Su, J. 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.  Google Scholar [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.  Google Scholar [20] C. Huang, S. Wen and L. Huang, Dynamics of anti-periodic solutions on shunting inhibitory cellular neural networks with multi-proportional delays, Neurocomputing, 357 (2019), 47-52.   Google Scholar [21] C. Huang, H. 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.  Google Scholar [22] C. Huang, X. 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.  Google Scholar [23] C. Huang, Z. Yang, T. 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.  Google Scholar [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.  Google Scholar [25] C. Huang, H. 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.  Google Scholar [26] C. Huang, X. Zhao, J. 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.  Google Scholar [27] M. Iswarya, R. Raja, G. Rajchakit, J. Cao, J. 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.  Google Scholar [28] J. Li, J. 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.  Google Scholar [29] X. Li, J. 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.  Google Scholar [30] X. Li, X. 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.  Google Scholar [31] Y. Li, X. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [35] W. Shen, X. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [38] R. Wei, J. 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.  Google Scholar [39] T. Wei, X. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [44] Y. Xu, Q. 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.  Google Scholar [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.  Google Scholar [46] D. Yang, X. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [51] C. Zhang, Almost Periodic Type Functions and Ergodicity, Science Press Beijing, Beijing; Kluwer Academic Publishers, Dordrecht, 2003.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [2] Z. Cai, J. 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.  Google Scholar [3] Z. Cai, L. 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.  Google Scholar [4] Z. Cai, J. Huang, L. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] Y. Deng, C. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] L. Duan, L. Huang, Z. 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.  Google Scholar [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.  Google Scholar [12] A. Fink, Almost Periodic Differential Equations, Springer-Verlag, Berlin-New York, 1974.  Google Scholar [13] X. Guo, C. 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.  Google Scholar [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.  Google Scholar [15] C. Huang, B. Liu, C. 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.  Google Scholar [16] C. Huang, X. 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.  Google Scholar [17] C. Huang, X. Long, L. 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.  Google Scholar [18] C. Huang, R. Su, J. 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.  Google Scholar [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.  Google Scholar [20] C. Huang, S. Wen and L. Huang, Dynamics of anti-periodic solutions on shunting inhibitory cellular neural networks with multi-proportional delays, Neurocomputing, 357 (2019), 47-52.   Google Scholar [21] C. Huang, H. 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.  Google Scholar [22] C. Huang, X. 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.  Google Scholar [23] C. Huang, Z. Yang, T. 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.  Google Scholar [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.  Google Scholar [25] C. Huang, H. 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.  Google Scholar [26] C. Huang, X. Zhao, J. 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.  Google Scholar [27] M. Iswarya, R. Raja, G. Rajchakit, J. Cao, J. 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.  Google Scholar [28] J. Li, J. 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.  Google Scholar [29] X. Li, J. 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.  Google Scholar [30] X. Li, X. 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.  Google Scholar [31] Y. Li, X. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [35] W. Shen, X. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [38] R. Wei, J. 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.  Google Scholar [39] T. Wei, X. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [44] Y. Xu, Q. 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.  Google Scholar [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.  Google Scholar [46] D. Yang, X. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [51] C. Zhang, Almost Periodic Type Functions and Ergodicity, Science Press Beijing, Beijing; Kluwer Academic Publishers, Dordrecht, 2003.  Google Scholar [52] H. Zhang, Q. Cao and H. 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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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