Abstract
In this paper, we consider a three-dimension discrete neural network model with delay. 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. In addition, it is found that there exist Hopf bifurcations when the parameter passes a sequence of critical values by using Extensional Jury Criterion. Finally, computer simulations are performed to support the theoretical predictions.
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References
Guo, S., Tang, X., Li, H.: Stability and bifurcation in a discrete system of two neurons with delays. Nonlinear Analysis: Real World Applications 9, 1323–1335 (2008)
Zhang, C., Zheng, B.: Stability and bifurcation of a two-dimension discrete neuralnetwork model with multi-delays. Chaos, Solitons and Fractals 31, 1232–1242 (2007)
Wei, J., Zhang, C.: Bifurcation analysis of a class of neural networks with delays. Nonlinear Analysis: Real World Applications 9, 2234–2252 (2008)
Zhang, C., Zheng, B.: Hopf bifurcation in numerical approximation of a n-dimension neural network model with multi-delays. Chaos, Solitons and Fractals 25, 129–146 (2005)
Wei, J., Ruan, S.: Stability and bifurcation in a neural network model with two delays. Physical D 130, 255–272 (1999)
Zhang, C., Liu, M., Zheng, B.: Hopf bifurcation in numerical pproximation of a class delay differential equations. Applied Mathematics and Computation 146, 335–349 (2003)
Wei, J., Zhang, C.: Stability analysis in a first-order complex differential equations with delay. Nonlinear Analysis 59, 657–671 (2004)
Zhao, H., Wang, L., Ma, C.: Hopf bifurcation and stability analysis on discrete-time Hopfield neural network with delay. Nonlinear Analysis: Real World Application 9, 103–113 (2008)
Zheng, B., Liang, L., Zhang, C.: Extended Jury Criterion. Science in China Series A: Mathematics 39, 1239–1260 (2009)
Yuri, A.K.: Elements of Applied Bifurcation Theory. Springer, New York (1995)
Golubitsky, M., Stewart, I.N., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory. Appl. Math. Sci. 2, 69 (1988)
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Yang, W., Zhang, C. (2010). Stability and Bifurcation of a Three-Dimension Discrete Neural Network Model with Delay. In: Zhang, L., Lu, BL., Kwok, J. (eds) Advances in Neural Networks - ISNN 2010. ISNN 2010. Lecture Notes in Computer Science, vol 6063. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13278-0_89
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DOI: https://doi.org/10.1007/978-3-642-13278-0_89
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13277-3
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