## 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:

 34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations 34D20 Stability of solutions to ordinary differential equations 92B20 Neural networks for/in biological studies, artificial life and related topics

### Keywords:

almost periodic solutions; stability; neural networks.

Matlab
Full Text:

### 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, 3, 639-662, (2003) · Zbl 1085.68142 [2] Wu, Jianhong, Stable phase-locked periodic solutions in a delay differential system, J differ equat, 194, 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, 2, 256-261, (2002) · Zbl 1368.93616 [4] Huang, Xia; Cao, Jinde, Almost periodic solution of shunting inhibitory cellular neural networks with time-varying delay, Phys lett A, 314, 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, 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, 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, 3, 813-836, (2004) · Zbl 1061.34062 [8] Zhou, Jin; Liu, Zengrong; Chen, Guanrong, Dynamics of periodic delayed neural networks, Neural networks, 17, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 1, 53-61, (2004) [27] Cao, Jinde; Li, Qiong, On the exponential stability and periodic solutions of delayed cellular neural networks, J math anal appl, 252, 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, 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, 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, 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, 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), Springer Berlin
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.