Dynamics and bifurcations of a discrete-time neural network model with a single delay

[1] J. B´elair, S.A. Campbell, P.V.D. Driessche, Frustration, stability, and delay-induced oscillations in
a neural network model, SIAM J. Appl. Math. 56 (1996) 245–255.
[2] J. Cao, L. Wang, Periodic oscillatory solution of bidirectional associative memory networks with
delays, Phys. Rev. E. 61 (2000) 1825.
[3] J. Carr, Applications of Center Manifold Theory, Springer-Verlag: Newyork, Heidelberg, Berlin,
1981.
[4] A. Dhooge, W. Govaerts, Y.A. Kuznetsov, H.G.E. Meijer, B. Sautois, New features of the soft-
ware MatCont for bifurcation analysis of dynamical systems, Math. Comput. Model. Dyn. Syst. 14(2008) 147–175.
[5] P.V.D. Driessche, X. Zou, Global attractivity in delayed Hopfield neural network models, SIAM J.
Appl. Math. 58 (1998) 1878–1890.
[6] Z. Eskandari, J. Alidousti, Z. Avazzadeh, R.K. Ghaziani, Dynamics and bifurcations of a discrete
time neural network with self connection, Eur. J. Control. 66 (2022) 100642.
[7] K. Gopalsamy, I. Leung, Delay induced periodicity in a neural network of excitation and inhibition,
Phys. D. 89 (1996) 395–426.
[8] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vec-
tor Fields, Springer Science & Business Media, 2013.
[9] S. Guo, L. Huang, Periodic oscillation for a class of neural networks with variable coefficients,
Nonlinear Anal. Real World Appl. 6 (2005) 545–561.
[10] S. Guo, L. Huang, L. Wang, Linear stability and Hopf bifurcation in a two-neuron network with
three delays, Int. J. Bifurc. Chaos Appl. Sci. Engrg. 14 (2004) 2799–2810.
[11] W. Govaerts, R.K. Ghaziani, Y.A. Kuznetsov, H.G. Meijer, Numerical methods for two-parameter
local bifurcation analysis of maps, SIAM J. Sci. Comput. 29 (2007) 2644–2667.
[12] J. Hadadi, J. Alidousti, R. Khoshsiar Ghaziani, Z. Eskandari, Bifurcations and complex dynamics
of a two dimensional neural network model with delayed discrete time, Math. Methods Appl. Sci.
46 (2023) 1–12.
[13] J.J. Hopfield, Neural networks and physical systems with emergent collective computational abili-
ties, Proc. Natl. Acad. Sci. USA. 79 (1982) 2554–2558.
[14] J.J. Hopfield, Neurons with graded response have collective computational properties like those of
two-state neurons, Proc. Natl. Acad. Sci. USA. 81 (1984) 3088–3092.
[15] J.J. Hopfield, D.W. Tank, Computing with neural circuits: A model, Science 233 (1986) 625–633.
[16] E. Kaslik, S. Balint, Bifurcation analysis for a two-dimensional delayed discrete-time Hopfield
neural network, Chaos Solit. Fractals 34 (2007) 1245–1253.
[17] X. Liao, K.W. Wong, Z. Wu, Bifurcation analysis on a two-neuron system with distributed delays,
Phys. D. 149 (2001) 123–141.
[18] Y.A. Kuznetsov, Numerical Analysis of Bifurcations. In Elements of Applied Bifurcation Theory,
Springer, New York, NY, 2004.
[19] I.A. Kuznetsov, Y.A. Kuznetsov, H.G. Meijer, Numerical Bifurcation Analysis of Maps, Cambridge
University Press, 2019.
[20] C.M. Marcus, R.M. Westervelt, Stability of analog neural networks with delay, Phys. Rev. A 39
(1989) 347.

[21] L. Olien, J. Belair, Bifurcations, stability, and monotonicity properties of a delayed neural network
model, Phys. D. 102 (1997) 349–363.
[22] H. Shu, J. Wei, Bifurcation analysis in a discrete BAM network model with delays, J. Differ. Equ.
Appl. 17 (2011) 69–84.
[23] Y. Song, J. Wei, M. Han, Local and global Hopf bifurcation in a delayed hematopoiesis model,Int.
J. Bifurc. Chaos Appl. Sci. Engrg. 14 (2004) 3909–3919.
[24] S. Wiggins, S. Wiggins, M. Golubitsky, Introduction to Applied Nonlinear Dynamical Systems and
Chaos , New York: Springer, 2003.
[25] J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay, Walter de Gruyter, 2011.
[26] M. Xiao, J. Cao, Stability and Hopf bifurcation in a delayed competitive web sites model, Phys.
Lett. A. 353 (2006) 138–150.
[27] W. Yu, J. Cao, Stability and Hopf bifurcation analysis on a four-neuron BAM neural network with
time delays, Phys. Lett. A. 351 (2006) 64–78.