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Dynamics of periodic delayed neural networks. (English) Zbl 1082.68101

Summary: This paper formulates and studies a model of periodic delayed neural networks. This model can well describe many practical architectures of delayed neural networks, which is generalization of some additive delayed neural networks such as delayed Hopfied neural networks and delayed cellular neural networks, under a time-varying environment, particularly when the network parameters and input stimuli are varied periodically with time. Without assuming the smoothness, monotonicity and boundedness of the activation functions, the two functional issues on neuronal dynamics of this periodic networks, i.e. the existence and global exponential stability of its periodic solutions, are investigated. Some explicit and conclusive results are established, which are natural extension and generalization of the corresponding results existing in the literature. Furthermore, some examples and simulations are presented to illustrate the practical nature of the new results.

MSC:

68T05 Learning and adaptive systems in artificial intelligence
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[1] Arik, S.; Tavsanoglu, V., Equilibrium analysis of delayed cnns, IEEE transactions on circuits and systems, 45, 168-171, (1998)
[2] Cao, J., On the exponential stability and periodic solution of CNNs with delays, Physics letter A, 267, 312-318, (2000) · Zbl 1098.82615
[3] Cao, J., Periodic solution and exponential stability of delayed cnns, Physics letter A, 270, 157-163, (2000)
[4] Cao, J., A set of stability criteria for delayed neural networks, IEEE transactions on circuits and systems, 48, 494-498, (2001) · Zbl 0994.82066
[5] Cao, J., Global stability conditions for delayed cnns, IEEE transactions on circuits and systems, 48, 1330-1333, (2001) · Zbl 1006.34070
[6] Chen, Y., Matrix analysis, (2000), West-North University of Technology Press Xian, (in Chinese)
[7] Chen, T., Global exponential stability of delayed Hopfield neural networks, Neural networks, 14, 977-980, (2001)
[8] Chen, T.; Amari, S., New theorem on global convergence of some dynamical systems, Neural networks, 14, 251-255, (2001)
[9] Chua, L.O.; Yang, L., Cellular neural networks: theory, IEEE transactions on circuits and systems, 35, 1257-1272, (1988) · Zbl 0663.94022
[10] Chua, L.O.; Yang, L., Cellular neural networks: applications, IEEE transactions on circuits and systems, 35, 1273-1290, (1988)
[11] Cohen, M.A.; Grossberg, S., Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE transactions on systems, man, and cybernetics, 13, 815-826, (1983) · Zbl 0553.92009
[12] Forti, M.; Tesi, A., New conditions for global stability of neural networks with application to linear and quadratic programming problem, IEEE transactions on circuits and systems, 44, 354-366, (1995) · Zbl 0849.68105
[13] Freeman, W.J.; Yau, Y.; Burke, B., Central pattern generating and recognizing in olfactory bulb: a correlation learning rule, Neural networks, 1, 277-288, (1988)
[14] Gain, R.E.; Mawhin, J.L., Coincidence degree and nonlinear differential equations, Lecture notes in mathematics, (1977), Springer Berlin, p. 567
[15] Gopalsamy, K.; He, X., Stability in asymmetric Hopfield nets with transmission delays, Physica D, 76, 344-358, (1994) · Zbl 0815.92001
[16] Gopalsamy, K.; Sariyasa, Time delays and stimulus-dependent pattern formation in periodic environments in isolated neurons, IEEE transactions on neural networks, 13, 551-563, (2002) · Zbl 1017.34073
[17] Grossberg, S., Nonlinear difference-differential equations in prediction and learning theory, Proceedings of the national Academy of science, USA (biophysics), 58, 1329-1334, (1967) · Zbl 0204.20703
[18] Grossberg, S., Learning and energy-entropy dependence in some nonlinear functional-differential system, Bulletin of the American mathematical society, 75, 1238-1242, (1969) · Zbl 0187.17202
[19] Grossberg, S., Pattern learning by functional-differential neural networks with arbitrary path weights, (), 121-160
[20] Guan, Z.H.; Chen, G., On the equilibria, stability, and instability of Hopfield neural networks, IEEE transactions on neural networks, 11, 534-540, (2000)
[21] Hale, J.K., Introduction to functional differential equations, (1977), Springer Berlin · Zbl 0425.34048
[22] Hjelmfelt, A.; Ross, J., Pattern recognition, chaos and multiplicity in neural networks and excitable system, Proceeding of the national Academy of science, USA (biophysics), 91, 63-69, (1994)
[23] Hopfield, J.J., Neural networks and physical systems with emergent collective computational abilities, Proceedings of the national Academy of science, USA (biophysics), 79, 2554-2558, (1982) · Zbl 1369.92007
[24] Hopfield, J.J., Neurons with graded response have collective computational properties like those of two-state neurons, Proceedings of the national Academy of science, USA (biophysics), 81, 3088-3092, (1984) · Zbl 1371.92015
[25] Hsu, C.H.; Li, G., Smale horseshoe of cellular neural networks, International journal of bifurcation and chaos, 10, 2119-2127, (2000) · Zbl 0984.37110
[26] Liang, X.B., Comment on the ‘equilibria, stability, and instability of Hopfield neural networks,’ and the authors’ reply, IEEE transactions on neural networks, 11, 1506-1507, (2000)
[27] Lu, H., Stability criteria for delayed neural networks, Physical review E, 64, 1-13, (2001)
[28] Marcus, C.M.; Westervelt, R.M., Stability of analog neural networks with delay, Physics review A, 39, 347-359, (1989)
[29] 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
[30] Roska, T.; Chua, L.O., Cellular neural networks with nonlinear and delay-type template, International journal of circuit theory and applications, 20, 469-481, (1992) · Zbl 0775.92011
[31] Skarda, A.; Freeman, W.J., How brains make chaos in order to make sense of the world, Brain behavioural science, 10, 161-195, (1987)
[32] Van Den Driessche, P.; Zou, X.F., Global attractivity in delayed Hopfield neural networks models, SIAM journal on applied mathematics, 58, 1878-1890, (1998) · Zbl 0917.34036
[33] Xu, D.; Zhao, H., Invariant and attracting sets of Hopfield neural networks with delay, International journal of systems science, 32, 863-866, (2001) · Zbl 1003.92002
[34] Xu, D.; Zhao, H.; Zhu, H., Global dynamics of Hopfield neural networks involving variably delays, Computers and mathematics with applications, 42, 39-45, (2001) · Zbl 0990.34036
[35] Yau, Y.; Freeman, W.J.; Burke, B.; Yang, Q., Pattern recognition by distributed neural network: an industrial application, Neural networks, 4, 103-121, (1991)
[36] 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
[37] Zhang, Y., Periodic solution and stability of Hopfield neural networks with variable delays, International journal of systems science, 27, 895-901, (1996) · Zbl 0863.34038
[38] Zhang, Y., Absolute periodicity and absolute stability of delayed neural networks, IEEE transactions on circuits and systems, 49, 256-261, (2002) · Zbl 1368.93616
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