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## 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

* Corresponding author: Kuo-Shou Chiu

Received  August 2020 Revised  December 2020 Published  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021060
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##### References:
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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