Global existence of periodic solutions in a simplified four-neuron BAM neural network model with multiple delays. (English) Zbl 1100.92015
Summary: We consider a simplified bidirectional associated memory (BAM) neural network model with four neurons and multiple time delays. The global existence of periodic solutions bifurcating from Hopf bifurcations is investigated by applying the global Hopf bifurcation theorem due to J. Wu [Trans. Am. Math. Soc. 350, No. 12, 4799–4838 (1998; Zbl 0905.34034)] and Bendixson’s criterion for high-dimensional ordinary differential equations due to Y. Li and J. S. Muldowney [ J. Differ. Equations 106, No. 1, 27–39 (1993; Zbl 0786.34033)]. It is shown that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of the sum of two delays. Numerical simulations supporting the theoretical analysis are also included.
MSC:
| 92C20 | Neural biology |
| 34C25 | Periodic solutions to ordinary differential equations |
| 37N25 | Dynamical systems in biology |
References:
| [1] | S. A. Campbell, S. Ruan, and J. Wei, “Qualitative analysis of a neural network model with multiple time delays,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 9, no. 8, pp. 1585-1595, 1999. · Zbl 1192.37115 · doi:10.1142/S0218127499001103 |
| [2] | T. Faria, “On a planar system modelling a neuron network with memory,” Journal of Differential Equations, vol. 168, no. 1, pp. 129-149, 2000. · Zbl 0961.92002 · doi:10.1006/jdeq.2000.3881 |
| [3] | K. Gopalsamy and X.-Z. He, “Delay-independent stability in bidirectional associative memory networks,” IEEE Transactions on Neural Networks, vol. 5, no. 6, pp. 998-1002, 1994. · doi:10.1109/72.329700 |
| [4] | K. Gopalsamy and I. Leung, “Delay induced periodicity in a neural netlet of excitation and inhibition,” Physica D: Nonlinear Phenomena, vol. 89, no. 3-4, pp. 395-426, 1996. · Zbl 0883.68108 · doi:10.1016/0167-2789(95)00203-0 |
| [5] | B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1981. · Zbl 0474.34002 |
| [6] | Y. Li and J. S. Muldowney, “On Bendixson’s criterion,” Journal of Differential Equations, vol. 106, no. 1, pp. 27-39, 1993. · Zbl 0786.34033 · doi:10.1006/jdeq.1993.1097 |
| [7] | S. Ruan and J. Wei, “On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion,” IMA Journal of Mathematics Applied in Medicine and Biology, vol. 18, no. 1, pp. 41-52, 2001. · Zbl 0982.92008 · doi:10.1093/imammb/18.1.41 |
| [8] | J. Wei and M. Y. Li, “Global existence of periodic solutions in a tri-neuron network model with delays,” Physica D: Nonlinear Phenomena, vol. 198, no. 1-2, pp. 106-119, 2004. · Zbl 1062.34077 · doi:10.1016/j.physd.2004.08.023 |
| [9] | J. Wei and S. Ruan, “Stability and bifurcation in a neural network model with two delays,” Physica D: Nonlinear Phenomena, vol. 130, no. 3-4, pp. 255-272, 1999. · Zbl 1066.34511 · doi:10.1016/S0167-2789(99)00009-3 |
| [10] | J. Wei and M. G. Velarde, “Bifurcation analysis and existence of periodic solutions in a simple neural network with delays,” Chaos, vol. 14, no. 3, pp. 940-953, 2004. · Zbl 1080.34064 · doi:10.1063/1.1768111 |
| [11] | J. Wu, “Symmetric functional-differential equations and neural networks with memory,” Transactions of the American Mathematical Society, vol. 350, no. 12, pp. 4799-4838, 1998. · Zbl 0905.34034 · doi:10.1090/S0002-9947-98-02083-2 |
| [12] | X.-P. Yan and W.-T. Li, “Stability and bifurcation in a simplified four-neuron BAM neural network with multiple delays,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 32529, pp. 1-29, 2006. · Zbl 1116.34059 · doi:10.1155/DDNS/2006/32529 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
