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Existence and stability of periodic solutions for Cohen–Grossberg neural networks with multiple delays. (English) Zbl 1048.34118
Summary: We use the continuation theorem of coincidence degree theory and Lyapunov functions to study the existence and stability of periodic solutions for the Cohen-Grossberg neural network with multiple delays.

34K13 Periodic solutions to functional-differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
34K20 Stability theory of functional-differential equations
Full Text: DOI
[1] Cohen, M.; Grossberg, S., Absolute stability and global pattern formation and parallel memory storage by competitive neural networks, IEEE trans. syst. man cybernet., SMC-13, 815-826, (1983) · Zbl 0553.92009
[2] Marcus, C.; Westervelt, R., Stability of analog neural networks with delay, Phys. rev. A, 39, 347-359, (1989)
[3] Wu, J., Introduction to neural dynamics and signal transmission delay, (2001), Walter de Gruyter Berlin · Zbl 0977.34069
[4] Wu, J., Symmetric functional – differential equations and neural networks with memory, Trans. am. math. soc., 350, 4799-4838, (1999) · Zbl 0905.34034
[5] Wu, J.; Zou, X., Patterns of sustained oscillations in neural networks with time delayed interactions, Appl. math. comput., 73, 55-75, (1995) · Zbl 0857.92003
[6] Gopalsamy, K.; He, X., Stability in asymmetric Hopfield nets with transmission delays, Physica D, 76, 344-358, (1994) · Zbl 0815.92001
[7] van den Driessche, P.; Zou, X., Global attractivity in delayed Hopfield neural network models, SIAM J. appl. math., 58, 1878-1890, (1998) · Zbl 0917.34036
[8] Liao, X.X., Stability of the Hopfield neural networks, Sci. China, 23, 5, 523-532, (1993)
[9] Hopfield, J., Neurons with graded response have collective computational properties like those of two-stage neurons, Proc. natl. acad. sci. USA, 81, 3088-3092, (1984) · Zbl 1371.92015
[10] Ye, H.; Michel, A.N.; Wang, K., Global stability and local stability of Hopfield neural networks with delays, Phys. rev. E, 50, 4206-4213, (1994) · Zbl 1345.53034
[11] Ye, H.; Michel, A.N.; Wang, K., Qualitative analysis of cohen – grossberg neural networks with multiple delays, Phys. rev. E, 51, 2611-2618, (1995)
[12] Wang, L.; Zou, X., Harmless delays in cohen – grossberg neural networks, Physica D, 170, 162-173, (2002) · Zbl 1025.92002
[13] Wang, L.; Zou, X., Exponential stability of cohen – grossberg neural networks, Neural networks, 15, 415-422, (2002)
[14] Wei, J.J.; Ruan, S.G., Stability and bifurcation in a neural network model with two delays, Physica D, 130, 255-272, (1999) · Zbl 1066.34511
[15] Gaines, R.E.; Mawhin, J.L., Coincidence degree and nonlinear differential equations, (1977), Springer-Verlag Berlin · Zbl 0326.34021
[16] Mawhin, J., Topological degree methods in nonlinear boundary value problems, () · Zbl 0414.34025
[17] Li, Y.K.; Kuang, Y., Periodic solutions of periodic delay lotka – volterra equations and systems, J. math. anal. appl., 255, 260-280, (2001) · Zbl 1024.34062
[18] Gopalsamy, K., Stability and oscillations in delay differential equations of population, (1992), Kluwer Dordrecht · Zbl 0752.34039
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