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Global dynamic analysis of general Cohen-Grossberg neural networks with impulse. (English) Zbl 1142.34045
This paper is concerned with the global exponential stability of an impulsive version of the Cohen-Grossberg neural networks model [{\it M. Cohen} and {\it S. Grossberg}, IEEE Trans. Syst. Man. Cybern. 13, 815--826 (1993; Zbl 0553.92009)]. More precisely, the authors consider the following problem $$\multline\frac{dx(t)}{dt}=-a_i(x_i(t))\left[b_i(x_i(t))-\sum_{j=1}^nc_{ij}f_j(x_i(t))\right. \\\left. -\sum_{j=1}^n \int_0^{+\infty}g_j(x_j(t-\tau_{ij}(t)-s)d_sK_{ij}(s)+I_i\right],\ t\not=t_k,\endmultline$$ $$x_i(t_k^+)-x_i(t_k)=I_{ik}(x_i(t_k)),\ t=t_k.$$ The principal result of this paper is based on the method of Lyapunov function. Several consequences are given.

##### MSC:
 34K20 Stability theory of functional-differential equations 34K45 Functional-differential equations with impulses 37N25 Dynamical systems in biology 92B20 General theory of neural networks (mathematical biology)
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##### References:
 [1] Cohen, M.; Grossberg, S.: Absolute stability and global pattern formation and parallel memory storage by competitive neural networks. IEEE trans syst man cybernet 13, 815-826 (1983) · Zbl 0553.92009 [2] Michel, A.; Wang, K.: Qualitative analysis of Cohen-Grossberg neural networks with multiple delays. Phys rev E 51, 2611-2618 (1995) [3] Wang, L.; Zou, X. F.: Exponential stability of Cohen-Grossberg neural networks. Neural networks 15, 415-422 (2002) [4] Chen, T. P.; Rong, L. B.: Delay-independent stability analysis of Cohen-Grossberg neural networks. Phys lett A 317, 436-449 (2003) · Zbl 1030.92002 [5] Wang, L.; Zou, X. F.: Harmless delays in Cohen-Grossberg neural network. Physica D 170, 162-173 (2002) · Zbl 1025.92002 [6] Liao, X. F.; Li, C. G.; Wong, K.: Criteria for exponential stability of Cohen-Grossberg neural networks. Neural networks 17, 1401-1414 (2004) · Zbl 1073.68073 [7] Li, Y. K.: Existence and stability of periodic solutions for Cohen-Grossberg neural networks with multiple delays. Chaos, solitons & fractals 20, 459-466 (2004) · Zbl 1048.34118 [8] Chen, Z.; Ruan, J.: Global stability analysis of impulsive Cohen-Grossberg neural networks with delay. Phys lett A 345, 101-111 (2005) · Zbl 05314183 [9] Lakshimikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations. (1989) [10] Liu, X. Z.: Further extension of the direct method and stability of impulsive systems. Nonlinear world 1, 341-354 (1994) · Zbl 0809.34066 [11] Fu, X. L.; Qi, J. G.; Liu, Y. S.: General comparison principle for impulsive variable time differential equations with applications. Nonlinear anal 42, 1421-1429 (2000) · Zbl 0985.34011 [12] Fu, X. L.; Liu, X. Z.; Sivaloganathan, S.: Oscillation criteria for impulsive parabolic differential equations with delay. J math anal appl 268, 647-664 (2002) · Zbl 1160.35429 [13] Yu, J. S.: Stability for nonlinear delay differential equations of unstable type under impulsive perturbations. Appl math lett 14, 849-857 (2001) · Zbl 0992.34055 [14] Liu, X. Z.; Ballinger, G.: Uniform asymptotic stability of impulsive delay differential equations. Comput math appl 41, 903-915 (2001) · Zbl 0989.34061 [15] Guan, Z. H.; Chen, G. R.: On delayed impulsive Hopfield neural networks. Neural networks 12, 273-280 (1999) [16] Guan, Z. H.; Lam, James; Chen, G. R.: On impulsive autoassociative neural networks. Neural networks 13, 63-69 (2000) [17] Qian, T. H.; Guan, Z. H.; Chen, G. R.: Robust stability for uncertain impulsive autoassociative neural networks. Dyn contin discrete impulsive syst ser B 10, 69-79 (2003) · Zbl 1021.92003 [18] Jin, Z.; Ma, Z. E.; Han, M. A.: The existence of periodic solutions of the n-species Lotka-Volterra competition systems with impulsive. Chaos, solitons & fractals 22, 181-188 (2004) · Zbl 1058.92046 [19] Haydar, A.; Rajai, A.; Valery, C.; Zlatinka, C.; Eada, A. -Z.: Continuous-time additive Hopfield-type neural networks with impulses. J math anal appl 290, 436-451 (2004) · Zbl 1057.68083 [20] Xu, D. Y.; Yang, Z. C.: Impulsive delay differential inequality and stability of neural networks. J math anal appl 305, 107-120 (2005) · Zbl 1091.34046 [21] Li, Y. K.: Global exponential stability of BAM neural networks with delays and impulses. Chaos, solitons & fractals 24, 279-285 (2005) · Zbl 1099.68085 [22] Lu, W. L.; Chen, T. P.: On periodic dynamical systems. Chin ann math ser B 4, 455-462 (2004) · Zbl 1074.34065