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)
Multi-stability and almost periodic solutions of a class of recurrent neural networks. (English) Zbl 1136.34311
The paper studies a class of reccurent neural networks described by the equations $$ \dot x_i(t)=-a_i x_i(t)+\sum_{j=1}^n w_{ij} f(x_j(t))+c_i\,,\quad f(x)\in (-1,\,1)\quad i=1,\dots,n. $$ Using Lyapunov functions, a sufficient condition for the complete stability is obtained. On this base applying the Mawhin coincidence degree theory, many sufficient conditions guaranteeing the existence of at least one almost periodic solution are obtained. These conditions are derived for an arbitrary activation function $f$. Few simulations done by Matlab illustrate that the simulation results fit well the theoretic analysis.

MSC:
34C27Almost and pseudo-almost periodic solutions of ODE
34D20Stability of ODE
92B20General theory of neural networks (mathematical biology)
Software:
Matlab
WorldCat.org
Full Text: DOI
References:
[1] Yi, Zhang; Tan, K. K.; Lee, T. H.: Multi-stability analysis for recurrent neural networks with unsaturating piecewise linear transfer functions. Neural comput 15, No. 3, 639-662 (2003) · Zbl 1085.68142
[2] Wu, Jianhong: Stable phase-locked periodic solutions in a delay differential system. J differ equat 194, No. 2, 237-286 (2003) · Zbl 1044.34024
[3] Yi, Zhang; Heng, Pheng Ann; Vadakkepat, Prahlad: Absolute periodicity and absolute stability of delayed neural networks. IEEE trans circ syst I: Fundam theory appl 49, No. 2, 256-261 (2002)
[4] Huang, Xia; Cao, Jinde: Almost periodic solution of shunting inhibitory cellular neural networks with time-varying delay. Phys lett A 314, No. 3, 222-231 (2003) · Zbl 1052.82022
[5] Chen, Anping; Cao, Jinde: Almost periodic solution of shunting inhibitory cnns with delays. Phys lett A 298, No. 2 -- 3, 161-170 (2002) · Zbl 0995.92003
[6] Huang, He; Ho, Daniel W. C.; Cao, Jinde: Analysis of global exponential stability and periodic solutions of neural networks with time-varying delays. Neural networks 18, No. 2, 161-170 (2005) · Zbl 1078.68122
[7] Zhu, Huiyan; Huang, Lihong; Dai, Binxiang: Convergence and periodicity of solutions for a neural network of two neurons. Appl math comput 155, No. 3, 813-836 (2004) · Zbl 1061.34062
[8] Zhou, Jin; Liu, Zengrong; Chen, Guanrong: Dynamics of periodic delayed neural networks. Neural networks 17, No. 1, 87-101 (2004) · Zbl 1082.68101
[9] Zhao, Hongyong: Existence and global attractivity of almost periodic solution for cellular neural network with distributed delays. Appl math comput 154, No. 3, 683-695 (2004) · Zbl 1057.34099
[10] Liu, Zhigang; Liao, Liusheng: Existence and global exponential stability of periodic solution of cellular neural networks with time-varying delays. J math anal appl 290, No. 1, 247-262 (2004) · Zbl 1055.34135
[11] Liu, Zhigang; Chen, Anping; Huang, Lihong: Existence and global exponential stability of periodic solution to self-connection BAM neural networks with delays. Phys lett A 328, No. 2 -- 3, 127-143 (2004) · Zbl 1134.34329
[12] Chen, Anping; Huang, Lihong; Cao, Jinde: Existence and stability of almost periodic solution for BAM neural networks with delays. Appl math comput 137, No. 1, 177-193 (2003) · Zbl 1034.34087
[13] Dong, Qinxi; Matsui, K.; Huang, Xiankai: Existence and stability of periodic solutions for Hopfield neural network equations with periodic input. Nonlinear anal 49, No. 4, 471-479 (2002) · Zbl 1004.34065
[14] Li, Yongkun; Liu, Ping: Existence and stability of positive periodic solution for BAM neural networks with delays. Math comput modell 40, No. 7 -- 8, 757-770 (2004) · Zbl 1197.34125
[15] Sun, Changyin; Feng, Chun-Bo: Exponential periodicity and stability of delayed neural networks. Math comput simul 66, No. 6, 469-478 (2004) · Zbl 1057.34097
[16] Wei, Junjie; Li, Michael Y.: Global existence of periodic solutions in a tri-neuron network model with delays. Physica D: Nonlinear phenom 198, No. 1 -- 2, 106-119 (2004) · Zbl 1062.34077
[17] Guo, Shangjiang; Huang, Lihong; Dai, Binxiang; Zhang, Zhongzhi: Global existence of periodic solutions of BAM neural networks with variable coefficients. Phys lett A 317, No. 1 -- 2, 97-106 (2003) · Zbl 1046.68090
[18] Chen, Boshan; Wang, Jun: Global exponential periodicity and global exponential stability of a class of recurrent neural networks. Phys lett A 329, No. 1 -- 2, 36-48 (2004) · Zbl 1208.81063
[19] Li, Yongkun; Lu, Linghong: Global exponential stability and existence of periodic solution of Hopfield-type neural networks with impulses. Phys lett A 333, No. 1 -- 2, 62-71 (2004) · Zbl 1123.34303
[20] Dong, Meifang: Global exponential stability and existence of periodic solutions of cnns with delays. Phys lett A 300, No. 1, 49-57 (2002) · Zbl 0997.34067
[21] He, H.; Cao, J.; Wang, J.: Global exponential stability and periodic solutions of recurrent neural networks with delays. Phys lett 298, No. 5 -- 6, 393-404 (2002) · Zbl 0995.92007
[22] Zhao, Hongyong: Global exponential stability and periodicity of cellular neural networks with variable delays. Phys lett A 336, No. 4 -- 5, 331-341 (2005) · Zbl 1136.34348
[23] Yongkun, Li; Chunchao, Liu; Lifei, Zhu: Global exponential stability of periodic solution for shunting inhibitory cnns with delays. Phys lett A 337, No. 1 -- 2, 46-54 (2005) · Zbl 1135.34338
[24] Beretta, Edoardo; Solimano, Fortunata; Takeuchi, Yasuhiro: Negative criteria for the existence of periodic solutions in a class of delay-differential equations. Nonlinear anal 50, No. 7, 941-966 (2002) · Zbl 1087.34542
[25] Cao, Jinde: New results concerning exponential stability and periodic solutions of delayed cellular neural networks. Phys lett A 307, No. 2 -- 3, 136-147 (2003) · Zbl 1006.68107
[26] Sun, Changyin; Feng, Chun-Bo: On robust exponential periodicity of interval neural networks with delays. Neural process lett 20, No. 1, 53-61 (2004) · Zbl 1057.34097
[27] Cao, Jinde; Li, Qiong: On the exponential stability and periodic solutions of delayed cellular neural networks. J math anal appl 252, No. 1, 50-64 (2000) · Zbl 0976.34067
[28] Guo, Shangjiang; Huang, Lihong: Periodic oscillation for a class of neural networks with variable coefficients. Nonlinear anal: real world appl 6, No. 3, 545-561 (2005) · Zbl 1080.34051
[29] Guo, Shangjiang; Huang, Lihong: Periodic solutions in an inhibitory two-neuron network. J comput appl math 161, No. 1, 217-229 (2003) · Zbl 1044.34034
[30] Zurada, Jacek M.; Cloete, Ian; Van Der Poel, Etienne: Generalized Hopfield networks for associative memories with multi-valued stable states. Neurocomputing 13, No. 2 -- 4, 135-149 (1996)
[31] Liu, Yiguang; You, Zhisheng; Cao, Liping: Dynamical behaviors of Hopfield neural network with multilevel activation functions. Chaos, solitons & fractals 25, No. 5, 1141-1153 (2005) · Zbl 1067.92005
[32] Gain, R. E.; Mawhin, J. L.: Coincidence degree and nonlinear differential equations. Lecture notes in mathematics (1977)