Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays. (English) Zbl 1062.34079

Summary: A delay-differential equation modelling a bidirectional associative memory (BAM) neural network with three neurons is investigated. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. By analyzing the associated characteristic equation, its linear stability is investigated and Hopf bifurcations are demonstrated. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold reduction. Numerical simulation results are given to support the theoretical predictions.


34K18 Bifurcation theory of functional-differential equations
34K19 Invariant manifolds of functional-differential equations
34K17 Transformation and reduction of functional-differential equations and systems, normal forms
34K13 Periodic solutions to functional-differential equations
34K20 Stability theory of functional-differential equations
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[1] Babcock, K.L.; Westervelt, R.M., Dynamics of simple electronic neural networks, Physica D, 28, 305-316, (1987)
[2] Baldi, P.; Atiya, A., How delays affect neural dynamics and learning, IEEE trans. neural netw., 5, 610-621, (1994)
[3] BĂ©lair, J.; Campbell, S.A.; van den Driessche, P., Frustration, stability, and delay-induced oscillations in a neural network model, SIAM J. appl. math., 56, 245-255, (1996) · Zbl 0840.92003
[4] Campbell, S.A., Stability and bifurcation of a simple neural network with multiple time delays, Fields inst. commun., 21, 65-67, (1999) · Zbl 0926.92003
[5] Campbell, S.A.; Ruan, S.; Wei, J., Qualitative analysis of a neural network model with multiple time delays, Int. J. bifur. chaos, 9, 1585-1595, (1999) · Zbl 1192.37115
[6] Cao, J.D., On stability analysis in delayed celler neural networks, Phys. rev. E, 59, 5940-5944, (1999)
[7] Cao, J.D., Periodic oscillation solution of bidrectional associative memory networks with delays, Phys. rev. E, 59, 1825-1828, (2000)
[8] Chen, Y.; Wu, J., Minimal instability and unstable set of a phaselocked periodic orbit in a delayed neural network, Physica D, 134, 185-199, (1999) · Zbl 0942.34062
[9] Faria, T., On a planar system modeling a neuron network with memory, J. diff. eqs., 168, 129-149, (2000) · Zbl 0961.92002
[10] Giannakopoulos, F.; Zapp, A., Bifurcations in a planar system of differential delay equations modeling neural activity, Physica D, 159, 215-232, (2001) · Zbl 0984.92505
[11] Gopalsamy, K.; Leung, I., Delay induced periodicity in a neural netlet of excitation and inhibition, Physica D, 89, 395-426, (1996) · Zbl 0883.68108
[12] Gopalsamy, K.; He, X., Delay-independent stability in bi-directional associative memory networks, IEEE trans. neural netw., 5, 998-1002, (1994)
[13] Guo, S.; Huang, L., Hopf bifurcating periodic orbits in a ring of neurons with delays, Physica D, 183, 19-44, (2003) · Zbl 1041.68079
[14] Hassard, B.D.; Kazarinoff, N.D.; Wan, Y.H., Theory and applications of Hopf bifurcation, (1981), Cambridge University Press Cambridge · Zbl 0474.34002
[15] Hopfield, J.J., Neurons with graded response have collective computational properties like those of two-state neurons, Proc. nat. acad. sci. USA, 81, 3088-3092, (1984) · Zbl 1371.92015
[16] Huang, L.; Wu, J., Nonlinear waves in networks of neurons with delayed feedback: pattern formation and continuation, SIAM J. math. anal., 34, 836-860, (2003) · Zbl 1038.34076
[17] Kelly, D.G., Stability in contractive nonlinear neural networks, IEEE trans. biomed., 37, 231-242, (1990)
[18] Kosto, B., Bi-directional associative memories, IEEE trans. syst. man cybernet, 18, 49, (1988)
[19] Kosko, B., Unsupervised learing in nose, IEEE trans. neural netw., 1, 1-12, (1990)
[20] Liao, X., Stability of the Hopfield neural networks, Sci. chin., 23, 523-532, (1993)
[21] Marcus, C.M.; Westervelt, R.M., Stability of analong neural network with delay, Phys. rev. A, 39, 347-359, (1989)
[22] Mohamad, S., Global exponential stability in continuous-time and discrete-time delayed bidirectional neural networks, Physica D, 159, 233-251, (2001) · Zbl 0984.92502
[23] Ncube, I.; Campbell, S.A.; Wu, J., Change in criticality of synchronous Hopf bifurcation in a multiple-delayed neural system, Fields inst. commun., 36, 179-193, (2003) · Zbl 1162.92301
[24] Olien, L.; Blair, J., Bifurcations, stability, and monotonicity properties of a delayed neural network model, Physica D, 102, 349-363, (1997) · Zbl 0887.34069
[25] Ruan, S.; Wei, J., On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion, IMA J. math. appl. med. biol., 18, 41-52, (2001) · Zbl 0982.92008
[26] Ruan, S.; Wei, J., On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. contin. discrete impuls. syst. ser. A math. anal., 10, 863-874, (2003) · Zbl 1068.34072
[27] Shayer, P.L.; Campbell, S.A., Stability, bifurcation, and multistability in a system of two coupled neurons with multiple time delays, SIAM J. appl. math., 61, 673-700, (2000) · Zbl 0992.92013
[28] van den Driessche, P.; Zou, X., Global attractivity in delayed Hopfield neural network models, SIAM J. appl. math., 58, 1878-1890, (1998) · Zbl 0917.34036
[29] van den Driessche, P.; Wu, J.; Zou, X., Stabilization role of inhibitory self-connections in a delayed neural network, Physica D, 150, 84-90, (2001) · Zbl 1007.34072
[30] Wang, L.; Zou, X., Harmless delays in Cohen-Grossberg neural networks, Physica D, 170, 163-173, (2002) · Zbl 1025.92002
[31] 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
[32] Wei, J.; Ruan, S., Stability and bifurcation in a neural network model with two delays, Physica D, 130, 255-272, (1999) · Zbl 1066.34511
[33] Wei, J.; Li, M.Y., Global exstance of periodic solutions in a tri-neuron model with delays, Physica D, 198, 1-2, 106-119, (2004) · Zbl 1062.34077
[34] Wei, J.; Velarde, M., Bifurcation analysis and existence of periodic solutions in a simple neural network with delays, Chaos, 3, 940-953, (2004) · Zbl 1080.34064
[35] Wu, J., Symmetric functional-differential equations and neural networks with memory, Trans. am. math. soc., 350, 4799-4838, (1998) · Zbl 0905.34034
[36] Wu, J., Introduction to neural dynamics and signal transmission delay, (2001), Walther de Gruyter Berlin · Zbl 0977.34069
[37] Wu, J., Stable phase-locked periodic solutions in a delay differential system, J. diff. eqs., 194, 237-286, (2003) · Zbl 1044.34024
[38] Ye, H.; Michel, A.; Wang, K., Global stability and local stability of Hopfield neural networks, Phys. rev. E, 50, 4206-4213, (1994) · Zbl 1234.82017
[39] Ye, H.; Michel, A.; Wang, K., Qualitative analysis of Cohen-Grossberg neural networks with multiple delays, Phys. rev. E, 51, 2611-2618, (1995)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.