Stability and bifurcation analysis in a class of two-neuron networks with resonant bilinear terms. (English) Zbl 1218.37122

Summary: A class of two-neuron networks with resonant bilinear terms is considered. The stability of the zero equilibrium and existence of Hopf bifurcation is studied. It is shown that the zero equilibrium is locally asymptotically stable when the time delay is small enough, while change of stability of the zero equilibrium will cause a bifurcating periodic solution as the time delay passes through a sequence of critical values. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Finally, numerical simulations supporting the theoretical analysis are carried out.


37N25 Dynamical systems in biology
92D20 Protein sequences, DNA sequences
Full Text: DOI


[1] J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proceedings of the National Academy of Sciences of the United States of America, vol. 79, no. 8, pp. 2554-2558, 1982. · Zbl 1369.92007 · doi:10.1073/pnas.79.8.2554
[2] J. J. Hopfield, “Neurons with graded response have collective computational properties like those of two-state neurons,” Proceedings of the National Academy of Sciences of the United States of America, vol. 81, no. 10 I, pp. 3088-3092, 1984. · Zbl 1371.92015 · doi:10.1073/pnas.81.10.3088
[3] S. Guo, L. Huang, and L. Wang, “Linear stability and Hopf bifurcation in a two-neuron network with three delays,” International Journal of Bifurcation and Chaos, vol. 14, no. 8, pp. 2799-2810, 2004. · Zbl 1062.34078 · doi:10.1142/S0218127404011016
[4] X. Liao, K.-W. Wong, and Z. Wu, “Bifurcation analysis on a two-neuron system with distributed delays,” Physica D, vol. 149, no. 1-2, pp. 123-141, 2001. · Zbl 0982.34059 · doi:10.1016/S0167-2789(00)00197-4
[5] L. Olien and J. Bélair, “Bifurcations, stability, and monotonicity properties of a delayed neural network model,” Physica D, vol. 102, no. 3-4, pp. 349-363, 1997. · Zbl 0887.34069 · doi:10.1016/S0167-2789(96)00215-1
[6] J. Wei and S. Ruan, “Stability and bifurcation in a neural network model with two delays,” Physica D, vol. 130, no. 3-4, pp. 255-272, 1999. · Zbl 1066.34511 · doi:10.1016/S0167-2789(99)00009-3
[7] B. Wang and J. Jian, “Stability and Hopf bifurcation analysis on a four-neuron BAM neural network with distributed delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 2, pp. 189-204, 2010. · Zbl 1221.37198 · doi:10.1016/j.cnsns.2009.03.033
[8] Z. Yuan, D. Hu, and L. Huang, “Stability and bifurcation analysis on a discrete-time neural network,” Journal of Computational and Applied Mathematics, vol. 177, no. 1, pp. 89-100, 2005. · Zbl 1063.93030 · doi:10.1016/j.cam.2004.09.010
[9] A. Hajihosseini, G. R. R. Lamooki, B. Beheshti, and F. Maleki, “The Hopf bifurcation analysis on a time-delayed recurrent neural network in the frequency domain,” Neurocomputing, vol. 73, no. 4-6, pp. 991-1005, 2010. · doi:10.1016/j.neucom.2009.08.018
[10] S. Guo and Y. Yuan, “Delay-induced primary rhythmic behavior in a two-layer neural network,” Neural Networks, vol. 24, no. 1, pp. 65-74, 2011. · Zbl 1217.92009 · doi:10.1016/j.neunet.2010.09.006
[11] Y. Song, M. Han, and J. Wei, “Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays,” Physica D, vol. 200, no. 3-4, pp. 185-204, 2005. · Zbl 1062.34079 · doi:10.1016/j.physd.2004.10.010
[12] J. Wei and C. Zhang, “Bifurcation analysis of a class of neural networks with delays,” Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, vol. 9, no. 5, pp. 2234-2252, 2008. · Zbl 1156.37325 · doi:10.1016/j.nonrwa.2007.08.008
[13] W. Yu and J. Cao, “Stability and Hopf bifurcation analysis on a four-neuron BAM neural network with time delays,” Physics Letters, Section A, vol. 351, no. 1-2, pp. 64-78, 2006. · Zbl 1234.34047 · doi:10.1016/j.physleta.2005.10.056
[14] X. Yang, M. Yang, H. Liu, and X. Liao, “Bautin bifurcation in a class of two-neuron networks with resonant bilinear terms,” Chaos, Solitons and Fractals, vol. 38, no. 2, pp. 575-589, 2008. · Zbl 1146.34315 · doi:10.1016/j.chaos.2007.01.001
[15] S. Ruan and J. Wei, “On the zeros of transcendental functions with applications to stability of delay differential equations with two delays,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A., vol. 10, no. 6, pp. 863-874, 2003. · Zbl 1068.34072
[16] K. L. Cooke and Z. Grossman, “Discrete delay, distributed delay and stability switches,” Journal of Mathematical Analysis and Applications, vol. 86, no. 2, pp. 592-627, 1982. · Zbl 0492.34064 · doi:10.1016/0022-247X(82)90243-8
[17] 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
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.