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Berezansky, Leonid; Braverman, Elena; Idels, Lev

New global exponential stability criteria for nonlinear delay differential systems with applications to BAM neural networks. (English) Zbl 1335.92007

Appl. Math. Comput. 243, 899-910 (2014).
Summary: We consider a nonlinear non-autonomous system with time-varying delays \[ \dot{x_i}(t) = - a_i(t) x_i(h_i(t)) + \sum_{j = 1}^m F_{ij} (t, x_j(g_{ij}(t))), \quad i = 1, \dots, m \] which has a large number of applications in the theory of artificial neural networks. Via the \(M\)-matrix method, easily verifiable sufficient stability conditions for the nonlinear system and its linear version are obtained. Application of the main theorem requires just to check whether a matrix, which is explicitly constructed using the system’s parameters, is an \(M\)-matrix. Comparison with the tests obtained by K. Gopalsamy [J. Math. Anal. Appl. 325, No. 2, 1117–1132 (2007; Zbl 1116.34058)] and B. Liu [Nonlinear Anal., Real World Appl. 14, No. 1, 559–566 (2013; Zbl 1260.34138)] for BAM neural networks illustrates novelty of the stability theorems. Some open problems conclude the paper.

Cited in 28 Documents

MSC:

92B20 Neural networks for/in biological studies, artificial life and related topics
37N25 Dynamical systems in biology

Keywords:

systems of nonlinear delay differential equations; time-varying delays; global stability; \(M\)-matrix; BAM neural network; leakage delays

Citations:

Zbl 1116.34058; Zbl 1260.34138
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References:

[1] Berezansky, L.; Braverman, E., Stability and linearization for differential equations with a distributed delay, Funct. Diff. Equ., 19, 27-43 (2012) · Zbl 1318.34105
[2] Berezansky, L.; Braverman, E., On nonoscillation and stability for systems of differential equations with a distributed delay, Automatica, 48, 612-618 (2012) · Zbl 1239.93049
[3] Berezansky, L.; Braverman, E., On the existence of positive solutions for systems of differential equations with a distributed delay, Comput. Math. Appl., 63, 1256-1265 (2012) · Zbl 1247.34114
[4] Berezansky, L.; Braverman, E., Linearized oscillation theory for a nonlinear equation with a distributed delay, Math. Comput. Model., 48, 287-304 (2008) · Zbl 1145.45303
[5] Berman, A.; Plemmons, R., Nonnegative Matrices in the Mathematical Sciences, Computer Science and Applied Mathematics (1979), Academic Press: Academic Press New York-London · Zbl 0484.15016
[6] Chen, Z.; Meng, J., Exponential convergence for cellular neural networks with time-varying delays in the leakage terms, Abstr. Appl. Anal., 10 (2012), Art. ID 941063 · Zbl 1253.93092
[7] Diblík, J.; Khusainov, D.; Mamedova, V., Stability of systems of differential equations with pure delay, Vi-sn. Kiiv. Uni-v. Ser. Fi-z.-Mat. Nauki, 2, 201-210 (2002), (Ukrainian. English, Ukrainian summary) · Zbl 1026.34083
[8] Faria, T.; Oliveira, J., General criteria for asymptotic and exponential stabilities of neural network models with unbounded delays, Appl. Math. Comput., 217, 9646-9658 (2011) · Zbl 1229.34113
[10] Gopalsamy, K., Leakage delays in BAM, J. Math. Anal. Appl., 325, 1117-1132 (2007) · Zbl 1116.34058
[11] Györi, I.; Hartung, F., Fundamental solution and asymptotic stability of linear delay differential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 13, 261-287 (2006) · Zbl 1099.34067
[12] Haykin, S., Neural Networks: A Comprehensive Foundation (1998), Prentice Hall: Prentice Hall NJ · Zbl 0828.68103
[13] Krasnosel’skij, M. A.; Lifshits, Je. A.; Sobolev, A. V., Positive Linear Systems. The Method of Positive Operators. Positive Linear Systems. The Method of Positive Operators, Sigma Series in Applied Mathematics, 5 (1989), Heldermann Verlag: Heldermann Verlag Berlin, (Translated from Russian by Jurgen Appell) · Zbl 0674.47036
[14] Kosko, B., Bidirectional associative memories, IEEE Trans. Syst. Man Cybernet., 18, 49-60 (1988)
[15] Li, C.; Huang, T., On the stability of nonlinear systems with leakage delay, J. Franklin Inst., 346, 366-377 (2009) · Zbl 1166.93367
[16] Li, X.; Fu, X., Effect of leakage time-varying delay on stability of nonlinear differential systems, J. Franklin Inst., 350, 1335-1344 (2013) · Zbl 1293.34065
[17] Li, X.; Fu, X.; Balasubramaniam, P.; Rakkiyappan, R., Existence, uniqueness and stability analysis of recurrent neural networks with time delay in the leakage term under impulsive perturbations, Nonlinear Anal. Real World Appl., 11, 4092-4108 (2010) · Zbl 1205.34108
[18] Liu, B., Global exponential stability for BAM neural networks with time-varying delays in the leakage terms, Nonlinear Anal. Real World Appl., 14, 559-566 (2013) · Zbl 1260.34138
[19] Oliveira, J., Global asymptotic stability for neural network models with distributed delays, Math. Comput. Model., 50, 81-91 (2009) · Zbl 1185.34107
[20] Peng, S., Global attractive periodic solutions of BAM neural networks with continuously distributed delays in the leakage terms, Nonlinear Anal. Real World Appl., 11, 2141-2151 (2010) · Zbl 1239.34081
[21] Senan, S.; Arik, S.; Liu, D., New robust stability results for bidirectional associative memory neural networks with multiple time delays, Appl. Math. Comput., 218, 11472-11482 (2012) · Zbl 1277.93066
[22] Wang, Y.; Zheng, C.; Feng, E., Stability analysis of mixed recurrent neural networks with time delay in the leakage term under impulsive perturbations, Neurocomputing, 119, 454-461 (2013)
[23] Wang, H.; Li, C.; Xu, H., Existence and global exponential stability of periodic solution of cellular neural networks with impulses and leakage delay, Int. J. Bifur. Chaos Appl. Sci. Eng., 19, 831-842 (2009) · Zbl 1167.34391
[24] Xiong, W.; Meng, J., Exponential convergence for cellular neural networks with continuously distributed delays in the leakage terms, Electron. J. Qual. Theory Diff. Equ., 10, 12 (2013) · Zbl 1340.34282
[25] Xia, Y.; Wong, P., Global exponential stability of a class of retarded impulsive differential equations with applications, Chaos Solitons Fractals, 39, 440-453 (2009) · Zbl 1197.34146
[26] So, J. W.-H.; Tang, X.; Zou, X., Stability in a linear delay system without instantaneous negative feedback, SIAM J. Math. Anal., 33, 6, 1297-1304 (2002) · Zbl 1019.34074
[27] So, J. W.-H.; Tang, X.; Zou, X., Global attractivity for non-autonomous linear delay systems, Funkcialaj Ekvacioj, 47, 25-40 (2004) · Zbl 1122.34342
[28] Zeng, Z.; Yu, P.; Pei; Liao, X., A new comparison method for stability theory of differential systems with time-varying delays, Int. J. Bifur. Chaos Appl. Sci. Eng., 18, 169-186 (2008) · Zbl 1158.34341
[29] Zeng, Z.; Wang, Z.; Liao, X., The exponential stability of neural network with purely delays, Dynam. Syst. Appl., 13, 259-271 (2004) · Zbl 1094.34054
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