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Global existence of periodic solutions in a tri-neuron network model with delays. (English) Zbl 1062.34077
Summary: We consider a delayed differential system that models a network of three neurons with memory. Using a global Hopf bifurcation theorem for FDEs due to J. Wu and a Bendixson’s criterion for high-dimensional ODEs due to Li and Muldowney, we obtain a group of sufficient conditions for the system to have multiple periodic solutions when the sum of delays is sufficiently large.

##### MSC:
 34K13 Periodic solutions to functional-differential equations 34K18 Bifurcation theory of functional-differential equations
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##### References:
 [1] Chen, Y.; Wu, J., Minimal instability and unstable set of a phase-locked periodic orbit in a delayed neural network, Physica D, 134, 185-199, (1999) · Zbl 0942.34062 [2] Chen, Y.; Wu, J., Existence and attraction of a phase-locked oscillation in a delayed network of two neurons, Differential integral equations, 14, 1181-1236, (2001) · Zbl 1023.34065 [3] Chen, Y.; Wu, J., Slowly oscillating periodic solutions for a delayed frustrated network of two neurons, J. math. anal. appl., 259, 188-208, (2001) · Zbl 0998.34058 [4] Faria, T., On a planar system modelling a neuron network with memory, J. differential equations, 168, 129-149, (2000) · Zbl 0961.92002 [5] Giamnakopoulos, F.; Zapp, A., Bifurcation in a planar system of differential delay equations modelling neural activity, Physica D, 159, 215-232, (2001) [6] Olien, L.; Bélair, J., Bifurcation stability and monotonicity properties of a delayed neural network model, Physica D, 102, 349-363, (1997) · Zbl 0887.34069 [7] Gopalsamy, K.; Leung, I., Delay induced periodicity in a neural network of excitation and inhibition, Physica D, 89, 395-426, (1996) · Zbl 0883.68108 [8] Majee, N.C.; Roy, A.B., Temporal dynamics of a two-neuron continuous network model with time delay, Appl. math. modeling, 21, 673-679, (1997) · Zbl 0893.68126 [9] Liao, X.; Wong, K.; Wu, Z., Bifurcation analysis on a two-neuron system with distribution delays, Physica D, 149, 123-141, (2001) · Zbl 1348.92035 [10] Wei, J.; Ruan, S., Stability and bifurcation in a neural network model with two delays, Physica D, 130, 255-272, (1999) · Zbl 1066.34511 [11] Baldi, P.; Atiya, A., How delays affect neural dynamics and learning, IEEE. tran. NN, 5, 610-621, (1994) [12] Wu, J., Symmetric functional differential equations and neural networks with memory, Trans. am. math. soc., 350, 4799-4838, (1998) · Zbl 0905.34034 [13] Campbell, S.A., Stability and bifurcation of a simple neural network with multiple time delays, Fields inst. commun., 21, 65-79, (1999) · Zbl 0926.92003 [14] Campbell, S.A.; Ruan, S.; Wei, J., Qualitative analysis of a neural network model with multiple time delays, Int. J. bifu. chaos, 9, 1585-1595, (1999) · Zbl 1192.37115 [15] J. Wei, M. Velarde, Bifurcation analysis and existence of periodic solutions in a simple neural network with delays, Chaos (2004), in press. · Zbl 1080.34064 [16] Ruan, S.; Wei, J., On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretions, IMA. J. math. appl. medi. biol., 18, 41-52, (2001) · Zbl 0982.92008 [17] Li, M.Y.; Muldowney, J., On bendixson’s criterion, J. differential equations, 106, 27-39, (1994) · Zbl 0786.34033 [18] Beretta, E.; Solimano, F.; Tacheuchi, Y., Negative criteria for the existence of periodic solutions in a class of delay- differential equations, Nonlinear anal., 50, 941-966, (2002) · Zbl 1087.34542 [19] Fiedler, M., Additive compound matrices and inequality for eigenvalues of stochastic matrices, Czech. math. J., 99, 392-402, (1974) · Zbl 0345.15013 [20] Muldowney, J.S., Compound matrices and ordinary differential equations, Rocky mountain J. math., 20, 857-871, (1990) · Zbl 0725.34049 [21] Erbe, L.H.; Geba, K.; Krawcewicz, W.; Wu, J., $$S^1$$-degree and global Hopf bifurcation theory of functional differential equations, J. differential equations, 98, 277-298, (1992) · Zbl 0765.34023
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