zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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:
34K20Stability theory of functional-differential equations
34K45Functional-differential equations with impulses
37N25Dynamical systems in biology
92B20General theory of neural networks (mathematical biology)
WorldCat.org
Full Text: DOI
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