×

zbMATH — the first resource for mathematics

Global dynamics for non-autonomous reaction-diffusion neural networks with time-varying delays. (English) Zbl 1155.68076
Summary: A class of non-autonomous reaction-diffusion neural networks with time-varying delays is considered. Novel methods to study the global dynamical behavior of these systems are proposed. Employing the properties of diffusion operator and the method of delayed inequalities analysis, we investigate global exponential stability, positive invariant sets and global attracting sets of the neural networks under consideration. Furthermore, conditions sufficient for the existence and uniqueness of periodic attractors for periodic neural networks are derived and the existence range of the attractors is estimated. Finally two examples are given to demonstrate the effectiveness of these results.

MSC:
68T05 Learning and adaptive systems in artificial intelligence
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Zhang, Y., Global exponential convergence of recurrent neural networks with variable delays, Theoretical computer science, 312, 281-293, (2004) · Zbl 1070.68132
[2] Lu, H.T.; Shen, R.M.; Chung, F.L., Absolute exponential stability of a class of recurrent neural networks with multiple and variable delays, Theoretical computer science, 344, 103-119, (2005) · Zbl 1079.68088
[3] Zhang, Q.; Wei, X.; Xu, J., Delay-dependent exponential stability of cellular neural networks with time-varying delays, Chaos, solitons & fractals, 23, 1363-1369, (2005) · Zbl 1094.34055
[4] Liang, J.; Cao, J., Exponential stability of continuous-time and discrete-time bidirectional associative memory networks with delays, Chaos, solitons & fractals, 22, 773-785, (2004) · Zbl 1062.68102
[5] Van DenDriessche, P.; Zou, X.F., Global attractivity in delayed Hopfield neural networks models, SIAM journal of applied mathematics, 58, 1878-1890, (1998) · Zbl 0917.34036
[6] Xu, D.Y.; Zhao, H.Y.; Zhu, H., Global dynamics of Hopfield neural networks involving variable delays, Computers and mathematics with applications, 42, 39-45, (2001) · Zbl 0990.34036
[7] Li, X.; Huang, L.; Zhu, H., Global stability of cellular neural networks with constant and variable delays, Nonlinear analysis, 53, 319-333, (2003) · Zbl 1011.92006
[8] Zhang, Y., Global exponential stability and periodic solutions of delay Hopfield neural networks, International journal of systems science, 27, 227-231, (1996) · Zbl 0845.93071
[9] Zeng, Z.G.; Wang, J., Global exponential stability of recurrent neural networks with time-varying delays in the presence of strong external stimuli, Neural networks, 19, 1528-1537, (2006) · Zbl 1178.68479
[10] Forti, M.; Tesi, A., New conditions for global stability of neural networks with application to linear and quadratic programming problems, IEEE transactions on circuits and systems I, 42, 354-366, (1995) · Zbl 0849.68105
[11] Zeng, Z.G.; Wang, J., Improved conditions for global exponential stability of recurrent neural networks with time-varying delays, IEEE transactions on neural networks, 17, 623-635, (2006)
[12] Xu, D.Y.; Zhao, H.Y., Invariant and attracing sets of Hopfield neural networks with delay, International journal of systems science, 32, 863-866, (2001) · Zbl 1003.92002
[13] Mohamad, S.; Gopalsamy, K., Neuronal dynamics in time varying environments: continuous and discrete time models, Discrete and continuous dynamical systems, 6, 841-860, (2000) · Zbl 1007.92008
[14] Jiang, H.; Li, Z.; Teng, Z., Boundedness and stability for nonautonomous cellular neural networks with delay, Physics letters A, 306, 313-325, (2003) · Zbl 1006.68059
[15] He, Q.; Kang, L., Existence and stability of global solution for generalized Hopfield neural network system, Neural parallel & scientific computation, 2, 165-176, (1994) · Zbl 0815.92002
[16] Liao, X.; Fu, Y.; Gao, J.; Zhao, X., Stability of Hopfield neural networks with reaction – diffusition terms, Acta electron sinica, 28, 78-80, (2000), (in Chinese)
[17] Liao, X.; Yang, S.; Cheng, S.; Zhen, Y., Stability of generalized Hopfield neural networks with reaction – diffusition terms, Science in China (series E), 32, 87-94, (2002), (in Chinese)
[18] Wang, L.S.; Xu, D.Y., Global exponential stability of reaction-diffusion Hopfield neural networks with variable delays, Science in China (series E), 33, 488-495, (2003), (in Chinese)
[19] Wu, J., Theory and application of partial functional differential equations, (1996), Springer-Verlag New York
[20] Liao, X., Theory and application of stability for dynamical systems, (2000), Defence Industry Publishing House BeiJing
[21] Zeidler, E., Nonlinear functional analysis and its application, I: fixed-point theorems, (1986), Springer-Verlag New York
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.