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February  2022, 27(2): 659-689. doi: 10.3934/dcdsb.2021060

## Periodicity and stability analysis of impulsive neural network models with generalized piecewise constant delays

 Departamento de Matemática, Facultad de Ciencias Básicas, Universidad Metropolitana de Ciencias de la Educación, José Pedro Alessandri 774, Santiago, Chile

Received  August 2020 Revised  December 2020 Published  February 2022 Early access  February 2021

Fund Project: This research was in part supported by PGI 03-2020 DIUMCE

In this paper, the global exponential stability and periodicity are investigated for impulsive neural network models with Lipschitz continuous activation functions and generalized piecewise constant delay. The sufficient conditions for the existence and uniqueness of periodic solutions of the model are established by applying fixed point theorem and the successive approximations method. By constructing suitable differential inequalities with generalized piecewise constant delay, some sufficient conditions for the global exponential stability of the model are obtained. The methods, which does not make use of Lyapunov functional, is simple and valid for the periodicity and stability analysis of impulsive neural network models with variable and/or deviating arguments. The results extend some previous results. Typical numerical examples with simulations are utilized to illustrate the validity and improvement in less conservatism of the theoretical results. This paper ends with a brief conclusion.

Citation: Kuo-Shou Chiu. Periodicity and stability analysis of impulsive neural network models with generalized piecewise constant delays. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 659-689. doi: 10.3934/dcdsb.2021060
##### References:
 [1] M. U. Akhmet and E. Yımaz, Impulsive Hopfield-type neural network system with piecewise constant argument, Nonlinear Anal. Real World Appl., 11 (2010), 2584-2593.  doi: 10.1016/j.nonrwa.2009.09.003. [2] E. Barone and C. Tebaldi, Stability of equilibria in a neural network model, Math. Meth. Appl. Sci., 23 (2000), 1179-1193.  doi: 10.1002/1099-1476(20000910)23:13<1179::AID-MMA158>3.0.CO;2-6. [3] S. Busenberg and K. Cooke, Vertically Transmitted Diseases: Models and Dynamics in Biomathematics, vol. 23, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-75301-5. [4] Z. Cai, J. Huang and L. Huang, Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks., Discrete and Continuous Dynamical Systems - B, 22 (2017), 3591-3614.  doi: 10.3934/dcdsb.2017181. [5] J. Cao, Global asymptotic stability of neural networks with transmission delays, International Journal of Systems Science, 31 (2000), 1313-1316.  doi: 10.1080/00207720050165807. [6] K.-S. Chiu and M. Pinto, Variation of parameters formula and Gronwall inequality for differential equations with a general piecewise constant argument, Acta Math. Appl. Sin. Engl. Ser., 27 (2011), 561-568.  doi: 10.1007/s10255-011-0107-5. [7] K.-S. Chiu and M. Pinto, Periodic solutions of differential equations with a general piecewise constant argument and applications, E. J. Qualitative Theory of Diff. Equ., 46 (2010), 1-19.  doi: 10.14232/ejqtde.2010.1.46. [8] K.-S. Chiu, M. Pinto and J.-Ch. Jeng, Existence and global convergence of periodic solutions in recurrent neural network models with a general piecewise alternately advanced and retarded argument, Acta Appl. Math., 133 (2014), 133-152.  doi: 10.1007/s10440-013-9863-y. [9] K.-S. Chiu, Existence and global exponential stability of equilibrium for impulsive cellular neural network models with piecewise alternately advanced and retarded argument, Abstract and Applied Analysis, 2013 (2013), Article ID 196139, 13 pages. doi: 10.1155/2013/196139. [10] K.-S. Chiu and J.-Ch. Jeng, Stability of oscillatory solutions of differential equations with general piecewise constant arguments of mixed type, Math. Nachr., 288 (2015), 1085-1097.  doi: 10.1002/mana.201300127. [11] K.-S. Chiu, Exponential stability and periodic solutions of impulsive neural network models with piecewise constant argument, Acta Appl. Math., 151 (2017), 199-226.  doi: 10.1007/s10440-017-0108-3. [12] K.-S. Chiu, Asymptotic equivalence of alternately advanced and delayed differential systems with piecewise constant generalized arguments, Acta Math. Sci., 38 (2018), 220-236.  doi: 10.1016/S0252-9602(17)30128-5. [13] K.-S. Chiu and T. Li, Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments, Math. Nachr., 292 (2019), 2153-2164.  doi: 10.1002/mana.201800053. [14] K.-S. Chiu and F. Córdova-Lepe, Global exponential periodicity and stability of neural network models with generalized piecewise constant delays, Mathematica Slovaca (2021) appear. [15] K.-S. Chiu, Green's function for impulsive periodic solutions in alternately advanced and delayed differential systems and applications, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 70 (2021), 15-37.  doi: 10.31801/cfsuasmas.785502. [16] L. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits Syst., 35 (1988), 1257-1272.  doi: 10.1109/31.7600. [17] K. Gopalsamy, Stability of artificial neural networks with impulses, Appl. Math. Comput., 154 (2004), 783-813.  doi: 10.1016/S0096-3003(03)00750-1. [18] C.-H. Hsu and S.-Y. Yang, Structure of a class of traveling waves in delayed cellular neural networks, Discrete and Continuous Dynamical Systems - A, 13 (2005), 339-359.  doi: 10.3934/dcds.2005.13.339. [19] Z. K. Huang, X. H. Wang and F. Gao, The existence and global attractivity of almost periodic sequence solution of discrete-time neural networks, Phys. Lett. A, 350 (2006), 182-191.  doi: 10.1016/j.physleta.2005.10.022. [20] O. M. Kwona, S. M. Lee, J. H. Park and E. J. Cha, New approaches on stability criteria for neural networks with interval time–varying delays, Appl. Math. Comput., 218 (2012), 9953-9964.  doi: 10.1016/j.amc.2012.03.082. [21] B. Lisena, Average criteria for periodic neural networks with delay, Discrete and Continuous Dynamical Systems - B, 19 (2014), 761-773.  doi: 10.3934/dcdsb.2014.19.761. [22] T. Li, X. Yao, L. Wu and J. Li, Improved delay–dependent stability results of recurrent neural networks, Appl. Math. Comput., 218 (2012), 9983-9991.  doi: 10.1016/j.amc.2012.03.013. [23] Z. Liu and L. Liao, Existence and global exponential stability of periodic solutions of cellular neural networks with time–varying delays, J. Math. Anal. Appl., 290 (2004), 247-262.  doi: 10.1016/j.jmaa.2003.09.052. [24] X. Y. Lou and B. T. Cui, Novel global stability criteria for high-order Hopfield-type neural networks with time-varying delays, J. Math. Anal. Appl., 330 (2007), 144-158.  doi: 10.1016/j.jmaa.2006.07.058. [25] S. Mohamad and K. Gopalsamy, Exponential stability of continuous-time and discrete-time cellular neural networks with delays, Appl. Math. Comput., 135 (2003), 17-38.  doi: 10.1016/S0096-3003(01)00299-5. [26] S. Novo, R. Obaya and A. M. Sanz, Exponential stability in non-autonomous delayed equations with applications to neural networks, Discrete and Continuous Dynamical Systems - A, 18 (2007), 517-536.  doi: 10.3934/dcds.2007.18.517. [27] J. H. Park, Global exponential stability of cellular neural networks with variable delays, Appl. Math. Comput., 183 (2006), 1214-1219.  doi: 10.1016/j.amc.2006.06.046. [28] M. Pinto, Asymptotic equivalence of nonlinear and quasilinear differential equations with piecewise constant arguments, Math. and Comp. Model., 49 (2009), 1750-1758.  doi: 10.1016/j.mcm.2008.10.001. [29] M. Pinto, Cauchy and Green matrices type and stability in alternately advanced and delayed differential systems, J. Difference Equ. Appl., 17 (2011), 235-254.  doi: 10.1080/10236198.2010.549003. [30] S. M. Shah and J. Wiener, Advanced differential equations with piecewise constant argument deviations, Internat. J. Math.and Math. Sci., 6 (1983), 671-703.  doi: 10.1155/S0161171283000599. [31] T. Su and X. Yang, Finite-time synchronization of competitive neural networks with mixed delays, Discrete and Continuous Dynamical Systems - B, 21 (2016), 3655-3667.  doi: 10.3934/dcdsb.2016115. [32] J.-P. Tseng, Global asymptotic dynamics of a class of nonlinearly coupled neural networks with delays, Discrete and Continuous Dynamical Systems - A, 33 (2013), 4693-4729.  doi: 10.3934/dcds.2013.33.4693. [33] B. Wang, S. Zhong and X. Liu, Asymptotical stability criterion on neural networks with multiple time–varying delays, Appl. Math. Comput., 195 (2008), 809-818.  doi: 10.1016/j.amc.2007.05.027. [34] Z. Wang, J. Cao, Z. Cai and L. Huang, Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks, Discrete and Continuous Dynamical Systems - B, 2020. doi: 10.3934/dcdsb.2020200. [35] J. Wiener, Differential equations with piecewise constant delays, Trends in Theory and Practice of Nonlinear Differential Equations (Arlington, Tex., 1982), Marcel Dekker, New York, 90 (1984), 547–552. [36] J. Wiener, Generalized Solutions of Functional Differential Equations, World Scientific, Singapore, 1993. doi: 10.1142/1860. [37] J. Wiener and V. Lakshmikantham, Differential equations with piecewise constant argument and impulsive equations, Nonlinear Stud., 7 (2000), 60-69. [38] B. Xu, X. Liu and X. Liao, Global exponential stability of high order Hopfield type neural networks, Appl. Math. Comput., 174 (2006), 98-116.  doi: 10.1016/j.amc.2005.03.020. [39] S. Xu, Y. Chu and J. Lu, New results on global exponential stability of recurrent neural networks with time-varying delays, Phys. Lett. A, 352 (2006), 371-379.  doi: 10.1016/j.physleta.2005.12.031. [40] T. H. Yu, D. Q. Cao, S. Q. Liu and H. T. Chen, Stability analysis of neural networks with periodic coefficients and piecewise constant arguments, Journal of the Franklin Institute, 353 (2016), 409-425.  doi: 10.1016/j.jfranklin.2015.11.010. [41] L. Zhou and G. Hu, Global exponential periodicity and stability of cellular neural networks with variable and distributed delays, Appl. Math. Comput., 195 (2008), 402-411.  doi: 10.1016/j.amc.2007.04.114. [42] Y. Zhang, D. Yue and E. Tian, New stability criteria of neural networks with interval time–varying delay: A piecewise delay method, Appl. Math. Comput., 208 (2009), 249-259.  doi: 10.1016/j.amc.2008.11.046.

show all references

##### References:
 [1] M. U. Akhmet and E. Yımaz, Impulsive Hopfield-type neural network system with piecewise constant argument, Nonlinear Anal. Real World Appl., 11 (2010), 2584-2593.  doi: 10.1016/j.nonrwa.2009.09.003. [2] E. Barone and C. Tebaldi, Stability of equilibria in a neural network model, Math. Meth. Appl. Sci., 23 (2000), 1179-1193.  doi: 10.1002/1099-1476(20000910)23:13<1179::AID-MMA158>3.0.CO;2-6. [3] S. Busenberg and K. Cooke, Vertically Transmitted Diseases: Models and Dynamics in Biomathematics, vol. 23, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-75301-5. [4] Z. Cai, J. Huang and L. Huang, Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks., Discrete and Continuous Dynamical Systems - B, 22 (2017), 3591-3614.  doi: 10.3934/dcdsb.2017181. [5] J. Cao, Global asymptotic stability of neural networks with transmission delays, International Journal of Systems Science, 31 (2000), 1313-1316.  doi: 10.1080/00207720050165807. [6] K.-S. Chiu and M. Pinto, Variation of parameters formula and Gronwall inequality for differential equations with a general piecewise constant argument, Acta Math. Appl. Sin. Engl. Ser., 27 (2011), 561-568.  doi: 10.1007/s10255-011-0107-5. [7] K.-S. Chiu and M. Pinto, Periodic solutions of differential equations with a general piecewise constant argument and applications, E. J. Qualitative Theory of Diff. Equ., 46 (2010), 1-19.  doi: 10.14232/ejqtde.2010.1.46. [8] K.-S. Chiu, M. Pinto and J.-Ch. Jeng, Existence and global convergence of periodic solutions in recurrent neural network models with a general piecewise alternately advanced and retarded argument, Acta Appl. Math., 133 (2014), 133-152.  doi: 10.1007/s10440-013-9863-y. [9] K.-S. Chiu, Existence and global exponential stability of equilibrium for impulsive cellular neural network models with piecewise alternately advanced and retarded argument, Abstract and Applied Analysis, 2013 (2013), Article ID 196139, 13 pages. doi: 10.1155/2013/196139. [10] K.-S. Chiu and J.-Ch. Jeng, Stability of oscillatory solutions of differential equations with general piecewise constant arguments of mixed type, Math. Nachr., 288 (2015), 1085-1097.  doi: 10.1002/mana.201300127. [11] K.-S. Chiu, Exponential stability and periodic solutions of impulsive neural network models with piecewise constant argument, Acta Appl. Math., 151 (2017), 199-226.  doi: 10.1007/s10440-017-0108-3. [12] K.-S. Chiu, Asymptotic equivalence of alternately advanced and delayed differential systems with piecewise constant generalized arguments, Acta Math. Sci., 38 (2018), 220-236.  doi: 10.1016/S0252-9602(17)30128-5. [13] K.-S. Chiu and T. Li, Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments, Math. Nachr., 292 (2019), 2153-2164.  doi: 10.1002/mana.201800053. [14] K.-S. Chiu and F. Córdova-Lepe, Global exponential periodicity and stability of neural network models with generalized piecewise constant delays, Mathematica Slovaca (2021) appear. [15] K.-S. Chiu, Green's function for impulsive periodic solutions in alternately advanced and delayed differential systems and applications, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 70 (2021), 15-37.  doi: 10.31801/cfsuasmas.785502. [16] L. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits Syst., 35 (1988), 1257-1272.  doi: 10.1109/31.7600. [17] K. Gopalsamy, Stability of artificial neural networks with impulses, Appl. Math. Comput., 154 (2004), 783-813.  doi: 10.1016/S0096-3003(03)00750-1. [18] C.-H. Hsu and S.-Y. Yang, Structure of a class of traveling waves in delayed cellular neural networks, Discrete and Continuous Dynamical Systems - A, 13 (2005), 339-359.  doi: 10.3934/dcds.2005.13.339. [19] Z. K. Huang, X. H. Wang and F. Gao, The existence and global attractivity of almost periodic sequence solution of discrete-time neural networks, Phys. Lett. A, 350 (2006), 182-191.  doi: 10.1016/j.physleta.2005.10.022. [20] O. M. Kwona, S. M. Lee, J. H. Park and E. J. Cha, New approaches on stability criteria for neural networks with interval time–varying delays, Appl. Math. Comput., 218 (2012), 9953-9964.  doi: 10.1016/j.amc.2012.03.082. [21] B. Lisena, Average criteria for periodic neural networks with delay, Discrete and Continuous Dynamical Systems - B, 19 (2014), 761-773.  doi: 10.3934/dcdsb.2014.19.761. [22] T. Li, X. Yao, L. Wu and J. Li, Improved delay–dependent stability results of recurrent neural networks, Appl. Math. Comput., 218 (2012), 9983-9991.  doi: 10.1016/j.amc.2012.03.013. [23] Z. Liu and L. Liao, Existence and global exponential stability of periodic solutions of cellular neural networks with time–varying delays, J. Math. Anal. Appl., 290 (2004), 247-262.  doi: 10.1016/j.jmaa.2003.09.052. [24] X. Y. Lou and B. T. Cui, Novel global stability criteria for high-order Hopfield-type neural networks with time-varying delays, J. Math. Anal. Appl., 330 (2007), 144-158.  doi: 10.1016/j.jmaa.2006.07.058. [25] S. Mohamad and K. Gopalsamy, Exponential stability of continuous-time and discrete-time cellular neural networks with delays, Appl. Math. Comput., 135 (2003), 17-38.  doi: 10.1016/S0096-3003(01)00299-5. [26] S. Novo, R. Obaya and A. M. Sanz, Exponential stability in non-autonomous delayed equations with applications to neural networks, Discrete and Continuous Dynamical Systems - A, 18 (2007), 517-536.  doi: 10.3934/dcds.2007.18.517. [27] J. H. Park, Global exponential stability of cellular neural networks with variable delays, Appl. Math. Comput., 183 (2006), 1214-1219.  doi: 10.1016/j.amc.2006.06.046. [28] M. Pinto, Asymptotic equivalence of nonlinear and quasilinear differential equations with piecewise constant arguments, Math. and Comp. Model., 49 (2009), 1750-1758.  doi: 10.1016/j.mcm.2008.10.001. [29] M. Pinto, Cauchy and Green matrices type and stability in alternately advanced and delayed differential systems, J. Difference Equ. Appl., 17 (2011), 235-254.  doi: 10.1080/10236198.2010.549003. [30] S. M. Shah and J. Wiener, Advanced differential equations with piecewise constant argument deviations, Internat. J. Math.and Math. Sci., 6 (1983), 671-703.  doi: 10.1155/S0161171283000599. [31] T. Su and X. Yang, Finite-time synchronization of competitive neural networks with mixed delays, Discrete and Continuous Dynamical Systems - B, 21 (2016), 3655-3667.  doi: 10.3934/dcdsb.2016115. [32] J.-P. Tseng, Global asymptotic dynamics of a class of nonlinearly coupled neural networks with delays, Discrete and Continuous Dynamical Systems - A, 33 (2013), 4693-4729.  doi: 10.3934/dcds.2013.33.4693. [33] B. Wang, S. Zhong and X. Liu, Asymptotical stability criterion on neural networks with multiple time–varying delays, Appl. Math. Comput., 195 (2008), 809-818.  doi: 10.1016/j.amc.2007.05.027. [34] Z. Wang, J. Cao, Z. Cai and L. Huang, Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks, Discrete and Continuous Dynamical Systems - B, 2020. doi: 10.3934/dcdsb.2020200. [35] J. Wiener, Differential equations with piecewise constant delays, Trends in Theory and Practice of Nonlinear Differential Equations (Arlington, Tex., 1982), Marcel Dekker, New York, 90 (1984), 547–552. [36] J. Wiener, Generalized Solutions of Functional Differential Equations, World Scientific, Singapore, 1993. doi: 10.1142/1860. [37] J. Wiener and V. Lakshmikantham, Differential equations with piecewise constant argument and impulsive equations, Nonlinear Stud., 7 (2000), 60-69. [38] B. Xu, X. Liu and X. Liao, Global exponential stability of high order Hopfield type neural networks, Appl. Math. Comput., 174 (2006), 98-116.  doi: 10.1016/j.amc.2005.03.020. [39] S. Xu, Y. Chu and J. Lu, New results on global exponential stability of recurrent neural networks with time-varying delays, Phys. Lett. A, 352 (2006), 371-379.  doi: 10.1016/j.physleta.2005.12.031. [40] T. H. Yu, D. Q. Cao, S. Q. Liu and H. T. Chen, Stability analysis of neural networks with periodic coefficients and piecewise constant arguments, Journal of the Franklin Institute, 353 (2016), 409-425.  doi: 10.1016/j.jfranklin.2015.11.010. [41] L. Zhou and G. Hu, Global exponential periodicity and stability of cellular neural networks with variable and distributed delays, Appl. Math. Comput., 195 (2008), 402-411.  doi: 10.1016/j.amc.2007.04.114. [42] Y. Zhang, D. Yue and E. Tian, New stability criteria of neural networks with interval time–varying delay: A piecewise delay method, Appl. Math. Comput., 208 (2009), 249-259.  doi: 10.1016/j.amc.2008.11.046.
Some trajectories uniformly convergent to the unique exponentially stable $\pi$/2-periodic solution of the ICNN models with IDEGPCD system (33)
Phase plots of state variable ($x_1$, $x_2$, $x_3$) in the ICNN models with IDEGPCD system (33) with the initial condition (7, 6, 3)
Phase plots of state variable ($x_1$, $x_2$, $x_3$) in the ICNN models with IDEGPCD system (33) with the initial condition (6.7897, 6.0565, 4.6992)
Phase plots of state variable ($t$, $x_1$, $x_2$) in the ICNN models with IDEGPCD system (33)
Phase plots of state variable ($t$, $x_1$, $x_3$) in the ICNN models with IDEGPCD system (33)
Phase plots of state variable ($t$, $x_2$, $x_3$) in the ICNN models with IDEGPCD system (33)
$\pi/2$-periodic solution of the CNN models with DEGPCD system (33a) for $t\in [0, 6\pi]$ with the initial value (4.9228, 4.5238, 3.6121)
Trajectories uniformly convergent to the unique exponentially stable $\pi$/2-periodic solution of the CNN models with DEGPCD system (33a) with the initial value (5.0, 4.3, 3.65)
Phase plots of state variable ($x_1$, $x_2$, $x_3$) in the CNN models with DEGPCD system (33a) with the initial condition (4.9228, 4.5238, 3.6121)
Some trajectories uniformly convergent to the unique $1$-periodic solution of the ICNN models with IDEGPCD system (37)
Exponential convergence of two trajectories towards a $1$-periodic solution of the ICNN models with IDEGPCD system (37). Initial conditions: ($i$) (3, 6) in red and ($ii$) (4, 6) in blue
Phase plots of state variable ($t$, $x_1$, $x_2$) in the ICNN models with IDEGPCD system (37)
Unique asymptotically stable solution of the CNN models with DEGPCD system (37a)
Unique asymptotically stable solution of the CNN models with DEGPCD system (37a)
Some trajectories uniformly convergent to the unique asymptotically stable solution of the CNN models with DEGPCD system (37a)
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