Stability and Periodic Solutions for a Neural Network Model with Multiple Time Delays
DOI:
https://doi.org/10.14738/aivp.105.13222Keywords:
neuron network model, delay, stability, periodic solutionAbstract
A three-triangle neural network model with seven neurons and time delays has been investigated by several researchers. The stability and bifurcating periodic solution were discussed by using the central manifold theorem and the normal form theory. However, by means of the delay as a bifurcation parameter, the authors must make a specific restrictive condition such that the network model with seven delays can be changed to only one delay system. In other words, the stability and the Hopf bifurcation of the three-triangle neural network model were studied under a very specific restrictive condition to the time delays. The present paper also considers the stability and the existence of periodic oscillations for this neural network model. Two theorems are provided to guarantee the stability and the existence of periodic oscillations for this three-triangle neural network model by using of the mathematical analysis method, which is simpler than bifurcation method. Also, our method avoids dealing with a complex bifurcating equation. It does not have any restrictions on the time delays in the model. Thus, our result is an extension of the literature. The criteria for selecting the parameters in this network are provided. Computer simulation examples are presented to demonstrate the correctness of this method. Our computer simulation indicates that the criteria in this paper are only sufficient conditions.
References
Dong T., Gong X., T. Huang, “Zero-Hopf bifurcation of a memristive synaptic Hopfield neural network with time delay,” Neural Networks, vol. 149, pp. 146-156, 2022.
Vaishwar A., B.K. Yadav, “Stability and Hopf-bifurcation analysis of four dimensional minimal neural network model with multiple time delays,” Chinese Journal of Physics, vol. 77, pp. 300-318, 2022.
Kundu A., Das P., A.B. Roy, “Stability, bifurcations and synchronization in a delayed neural network model of $n$-identical neurons,” Mathematics and Computer in Simulation, vol. 121, pp. 12-33, 2016.
Wang T., Cheng Z., Bu R., R. Ma, “Stability and Hopf bifurcation analysis of a simplified six-neuron tridiagonal two-layer neural network model with delay,” Neurocomputing, vol. 332, pp. 201-214, 2019.
Xu C., Tang X., M. Liao, “Stability and bifurcation analysis of a six-neuron BAM neural network model with discrete delays,” Neurocomputing, vol. 74, pp. 689-707, 2011.
Cheng Z., Xie K., Wang T., J. Cao, “Stability and Hopf bifurcation of three-triangle neural networks with delays,” Neurocomputing, vol. 322, pp. 206-215, 2018.
Achouri H., Aouiti C., B.B. Hamed, “Codimension two bifurcation in a coupled FitzHugh-Nagumo system with multiple delays,” Chaos, Solitons and Fractals, vol. 156, article ID 111824, 2022.
Wang Y., Cao J., C. Huang, “Exploration of bifurcation for a fractional-order BAM neural network with n+1 neurons and mixed time delays,” Chaos, Solitons and Fractals, vol. 159, article ID 112117, 2022.
Kaslik E., I.R. Radulescu, “Stability and bifurcations in fractional-order gene regulatory networks,” Applied Mathematics and Computation, vol. 421, article ID 126916, 2022.
Xu C., Tang X., M.X. Liao, “Stability and bifurcation analysis of a six-neuron BAM neural network model with discrete delays,” Neurocomputing, vol. 74, pp. 689-707, 2011.
Wang B., J. Jian, “Stability and Hopf bifurcation analysis on a four-neuron BAM neural network with distributed delays,” Commun Nonlinear Sci Numer Simulat, vol. 15, pp. 189-204, 2010.
Zhou B., Q.K. Song, “Stability and Hopf bifurcation analysis of a tri-neuron BAM neural network with distributed delay,” Neurocomputing, vol. 82, pp. 69-83, 2012.
Tian X., Xu R., Q.T. Gan, “Hopf bifurcation analysis of a BAM neural network with multiple time delays and diffusion,” Applied Mathematics and Computation, vol. 266, pp. 909-926, 2015.
Madhusudanan V., Srinivas M., Nwokoye C., Murthy B., S. Sridhar, “Hopf bifurcation analysis of delayed computer virus model with holling type iii incidence function and treatment,”
Scientific African, vol. 15, article ID e01125, 2022.
Liu M., Xu X., C. Zhang, “Stability and global Hopf bifurcation for neutral BAM neural network,” Neurocomputing, vol. 145, pp. 122-130, 2014.
Huang C., Meng Y., Cao J., Alsaedi A., F. Alsaadi, “New bifurcation results for fractional BAM neural network with leakage delay,” Chaos, Solitons and Fractals, vol. 100, pp. 31-44, 2017.
Xu C., Aouiti C., Z. Liu, “A further study on bifurcation for fractional order BAM neural networks with multiple delays,” Neurocomputing, vol. 417, pp. 501-515, 2020.
Huang C., Wang J., Chen X., J. Cao, “Bifurcations in a fractional-order BAM neural network with four different delays,” Neural Networks, vol. 141, pp. 344-354, 2021.
N. Chafee, “A bifurcation problem for a functional differential equation of finitely retarded type,” Journal of Mathematical Analysis and Applications, vol. 35, pp. 312-348, 1971.
Feng C., R. Plamondon, “An oscillatory criterion for a time delayed neural ring network model.” Neural Networks, vol. 29, pp. 70-79, 2012.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Chunhua Feng, Kimar Martin
This work is licensed under a Creative Commons Attribution 4.0 International License.
