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Bifurcation analysis of a class of neural networks with delays. (English) Zbl 1156.37325
Summary: A class of $n$-dimensional neural network model with multi-delay is considered. Firstly, a group of sufficient conditions for the existence of Hopf bifurcation are obtained via employing the polynomial theorem to analyze the distribution of the roots of the associated characteristic equation. Secondly, the stability and direction of the Hopf bifurcation are determined by applying the normal form method and center manifold theorem. Finally, the multiple stability is investigated and pitchfork bifurcation is also found. The results are illustrated by some numerical simulations.

37N25Dynamical systems in biology
92B20General theory of neural networks (mathematical biology)
34K18Bifurcation theory of functional differential equations
Full Text: DOI
[1] Arik, S.: Global asymptotic stability analysis of bidirectional associative memory neural networks with time delays, IEEE trans. Neural networks 16, 580-586 (2005)
[2] Babcock, K. L.; Westervelt, R. M.: Dynamics of simple electronic neural networks, Physica D: Nonlinear phenom. 28, 305-359 (1987)
[3] Baldi, P.; Atiya, A. F.: How delays affect neural dynamics and learning, IEEE trans. Neural networks 5, 612-621 (1994)
[4] Berns, D. W.; Moiola, J. L.; Chen, G.: Predicting period-doubling bifurcations and multiple oscillations in nonlinear time-delayed feedback systems, IEEE trans. Circuits syst. I --- fund. Theory appl. 45, 759-763 (1998) · Zbl 0952.94024 · doi:10.1109/81.703844
[5] 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
[6] Campbell, S. A.; Ncube, I.; Wu, J.: Multistability and stable asynchronous periodic oscillations in a multiple-delayed neural system, Physica D: Nonlinear phenom. 214, 101-119 (2006) · Zbl 1100.34054 · doi:10.1016/j.physd.2005.12.008
[7] Cao, J.; Wang, L.: Exponential stability and periodic oscillatory solution in BAM networks with delays, IEEE trans. Neural networks 13, 457-463 (2002)
[8] Cao, J.; Wang, J.: Global asymptotic stability of a general class of recurrent neural networks with time-varying delays, IEEE trans. Circuits syst. I 50, 34-44 (2003)
[9] 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 · doi:10.1006/jmaa.2000.7410
[10] Faria, T.: On a planar system modelling a neuron network with memory, J. differential equations 168, 129-149 (2001) · Zbl 0961.92002 · doi:10.1006/jdeq.2000.3881
[11] Giannakopoulos, F.; Zapp, A.: Bifurcations in a planar system of differential delay equations modeling neural activity, Physica D: Nonlinear phenom. 159, 215-232 (2001) · Zbl 0984.92505 · doi:10.1016/S0167-2789(01)00337-2
[12] Guo, S.; Huang, L.: Periodic solutions in an inhibitory two-neuron network, J. comput. Appl. math. 161, 217-229 (2003) · Zbl 1044.34034 · doi:10.1016/j.cam.2003.08.002
[13] Guo, S.; Huang, L.: Stability analysis of Cohen -- Grossberg neural networks, IEEE trans. Neural networks 17, 106-117 (2006)
[14] Guo, S.; Huang, L.; Wu, J.: Regular dynamics in a delayed network of two neurons with all-or-none activation functions, Physica D: Nonlinear phenom. 206, 32-48 (2005) · Zbl 1081.34069 · doi:10.1016/j.physd.2003.09.049
[15] Hale, J.: Theory of functional differential equations, (1977) · Zbl 0352.34001
[16] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H.: Theory and applications of Hopf bifurcation, (1981) · Zbl 0474.34002
[17] Hopfield, J.: Neurons with graded response have collective computational properties like those of two-state neurons, Proc. natl. Acad. sci. USA 81, 3088-3092 (1984)
[18] Jiang, H.; Zhang, L.; Teng, Z.: Existence and global exponential stability of almost periodic solution for cellular neural networks with variable coefficients and time-varying delays, IEEE trans. Neural networks 16, 1340-1351 (2005)
[19] Li, C.; Chen, G.: Local stability and bifurcation in small-world delayed networks, Chaos solitons fractals 20, 353-361 (2004) · Zbl 1045.34047 · doi:10.1016/S0960-0779(03)00405-3
[20] Li, X.; Wei, J.: Stability and bifurcation analysis on a delayed neural network model, Int. J. Bifur. chaos 15, 2883-2893 (2005) · Zbl 1093.92008 · doi:10.1142/S0218127405013757
[21] Liao, X.; Li, S.; Chen, G.: Bifurcation analysis on a two-neuron system with distributed delays in the frequency domain, Neural networks 17, 545-561 (2004) · Zbl 1073.68715 · doi:10.1016/j.neunet.2003.10.001
[22] Liao, X.; Yu, J.; Chen, G.: Novel stability criteria for bidirectional associative memory neural networks with time delays, Int. J. Circuit theory appl. 30, 519-546 (2002) · Zbl 1014.93036 · doi:10.1002/cta.206
[23] Lu, H.; Shen, R.; Fu, C.: Global exponential convergence of Cohen -- Grossberg neural networks with time delays, IEEE trans. Neural networks 16, 1694-1696 (2005)
[24] Ruan, S.; Filfil, R. S.: Dynamics of a two-neuron system with discrete and distributed delays, Physica D: Nonlinear phenom. 191, 323-342 (2004) · Zbl 1049.92004 · doi:10.1016/j.physd.2003.12.004
[25] Ruan, S.; Wei, J.: Periodic solutions of planar systems with two delays, Proc. R. Soc. Edinburgh 129A, 1017-1032 (1999) · Zbl 0946.34062 · doi:10.1017/S0308210500031061
[26] Ruan, S.; Wei, J.: On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. continuous discrete impulsive syst. Ser. A: math. Anal. 10, 863-874 (2003) · Zbl 1068.34072
[27] Wang, L.; Zou, X.: Harmless delays in Cohen -- Grossberg neural networks, Physica D: Nonlinear phenom. 170, 162-173 (2002) · Zbl 1025.92002 · doi:10.1016/S0167-2789(02)00544-4
[28] Wang, L.; Zou, X.: Hopf bifurcation in bidirectional associative memory neural networks with delays: analysis and computation, J. comput. Appl. math. 167, 73-90 (2004) · Zbl 1054.65076 · doi:10.1016/j.cam.2003.09.047
[29] Wei, J.; Li, M. Y.: Global existence of periodic solutions in a tri-neuron network model with delays, Physica D: Nonlinear phenom. 198, 106-119 (2004) · Zbl 1062.34077 · doi:10.1016/j.physd.2004.08.023
[30] Wei, J.; Ruan, S.: Stability and bifurcation in a neural network model with two delays, Physica D: Nonlinear phenom. 130, 255-272 (1999) · Zbl 1066.34511 · doi:10.1016/S0167-2789(99)00009-3
[31] Wei, J.; Velarde, M.: Bifurcation analysis and existence of periodic solutions in a simple neural network with delays, Chaos 14, 940-953 (2004) · Zbl 1080.34064 · doi:10.1063/1.1768111
[32] Wei, J.; Yuan, Y.: Synchronized Hopf bifurcation analysis in a neural network model with delays, J. math. Anal. appl. 312, 205-229 (2005) · Zbl 1085.34058 · doi:10.1016/j.jmaa.2005.03.049
[33] Wei, J.; Zhang, C.; Li, X.: Bifurcation in a two-dimensional neural network model with delay, Appl. math. Mech. 26, 210-217 (2005) · Zbl 1144.34366 · doi:10.1007/BF02438244
[34] Wu, J.: Introduction to neural dynamics and signal transmission delay, (2001) · Zbl 0977.34069
[35] Yan, X. P.: Hopf bifurcation and stability for a delayed tri-neuron network model, J. comput. Appl. math. 196, 579-595 (2006) · Zbl 1175.37086 · doi:10.1016/j.cam.2005.10.012
[36] Yi, Z.; Tan, K. K.: Multistability analysis of discrete-time recurrent neural networks with unsaturating piecewise linear activation functions, IEEE trans. Neural networks 15, 329-336 (2004)
[37] Yi, Z.; Tan, K. K.; Lee, T. H.: Multistability analysis for recurrent neural networks with unsaturating piecewise linear transfer functions, Neural comput. 15, 639-662 (2003) · Zbl 1085.68142 · doi:10.1162/089976603321192112
[38] Yuan, Y.: Dynamics in a delayed-neural network, Chaos solitons fractals 33, No. 2, 443-454 (2007) · Zbl 1135.34039 · doi:10.1016/j.chaos.2006.01.018
[39] Zhang, Z.; Guo, S.: Periodic oscillation for a three-neuron network with delays, Appl. math. Lett. 16, 1251-1255 (2003) · Zbl 1056.34067 · doi:10.1016/S0893-9659(03)90125-X