Guo, Shangjiang; Tang, Xianhua; Huang, Lihong Stability and bifurcation in a discrete system of two neurons with delays. (English) Zbl 1154.37381 Nonlinear Anal., Real World Appl. 9, No. 4, 1323-1335 (2008). Summary: We consider a simple discrete two-neuron network model with three delays. The characteristic equation of the linearized system at the zero solution is a polynomial equation involving very high order terms. We derive some sufficient and necessary conditions on the asymptotic stability of the zero solution. Regarding the eigenvalues of connection matrix as the bifurcation parameters, we also consider the existence of three types of bifurcations: Fold bifurcations, Flip bifurcations, and Neimark-Sacker bifurcations. The stability and direction of these three kinds of bifurcations are studied by applying the normal form theory and the center manifold theorem. Our results are a very important generalization to the previous works in this field. Cited in 24 Documents MSC: 37N25 Dynamical systems in biology 39A30 Stability theory for difference equations 39A28 Bifurcation theory for difference equations 92C20 Neural biology 39A12 Discrete version of topics in analysis Keywords:delay; bifurcation; neural network; stability PDFBibTeX XMLCite \textit{S. Guo} et al., Nonlinear Anal., Real World Appl. 9, No. 4, 1323--1335 (2008; Zbl 1154.37381) Full Text: DOI References: [1] Gopalsamy, K.; Leung, I., Delay induced periodicity in a neural network of excitation and inhibition, Physica D, 89, 395-426 (1996) · Zbl 0883.68108 [2] Guo, S.; Chen, Y., Stability and bifurcation of a discrete-time three-neuron system with delays, Int. J. Appl. Math. Eng. Sci., 1, 103-115 (2007) [3] Guo, S.; Huang, L.; Wang, L., Linear stability and Hope bifurcation in a two-neuron network with three delays, Int. J. Bifur. Chaos, 14, 2799-2810 (2004) · Zbl 1062.34078 [4] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H., Theory and Applications of Hopf Bifurcation (1981), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0474.34002 [5] Marcus, C. M.; Waugh, F. R.; Westervelt, R. M., Nonlinear dynamics and stability of analog neural networks, Physica D, 51, 234-247 (1991) · Zbl 0800.92059 [6] Mohamad, S.; Gopalsamy, K., Dynamics of a class of discrete-time neural networks and their continuous-time counterparts, Math. Comput. Simul., 53, 1-39 (2000) [7] Olien, L.; Bélair, J., Bifurcations, stability and monotonicity properties of a delayed neural network model, Physica D, 102, 349-363 (1997) · Zbl 0887.34069 [8] Stuart, A.; Humphries, A., Dynamical Systems and Numerical Analysis (1996), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0869.65043 [9] Yuan, Z.; Hu, D.; Huang, L., Stability and bifurcation analysis on a discrete-time neural network, J. Comput. Appl. Math., 177, 89-100 (2005) · Zbl 1063.93030 [10] Zhang, C.; Zheng, B., Hopf bifurcation in numerical approximation of a \(n\)-dimension neural network model with multi-delays, Chaos, Solitons & Fractals, 25, 129-146 (2005) · Zbl 1136.65326 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.