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European Journal of Applied Sciences – Vol. 10, No. 5
Publication Date: October 25, 2022
DOI:10.14738/aivp.105.13237. Feng, C., & Martin, K. (2022). Stability and Periodic Solutions for a Neural Network Model with Multiple Time Delays. European
Journal of Applied Sciences, 10(5). 294-308.
Services for Science and Education – United Kingdom
Stability and Periodic Solutions for a Neural Network Model with
Multiple Time Delays
Chunhua Feng
Department of Mathematics and Computer Science
Alabama State University, Montgomery, AL, USA, 36104
Kimar Martin
Department of Mathematics and Computer Science
Alabama State University, Montgomery, AL, USA, 36104
ABSTRACT
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.
Keywords: neuron network model, delay, stability, periodic solution.
INTRODUCTION
It is known that the bifurcation method is a stronger tool to study neural network models. Many
researchers have studied the stability and the Hopf bifurcation for various neural network
models with delays or without delays [1-18]. For example, Dong et al. have considered a
memristive synaptic Hopfield neural network with a time delay as follows:
!
�!′(�) = −�!(�) + r! tanh(�"(� − �)) + �(� − ��") ∙ (�!(�) − �"(�))
�"
# (�) = −�"(�) + r" tanh6�!(� − �)7 + r$ tanh6�"(�)7 − �(� − ��") ∙ (�!(�) − �"(�))
�# = �!(�) − �"(�) − �
(1)
Some sufficient conditions of zero bifurcation and zero-Hopf bifurcation were obtained by
choosing time delay and coupling strength of memristor as bifurcation parameters [1].
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Feng, C., & Martin, K. (2022). Stability and Periodic Solutions for a Neural Network Model with Multiple Time Delays. European Journal of Applied
Sciences, 10(5). 294-308.
URL: http://dx.doi.org/10.14738/aivp.105.13222
Vaishwar and Yadav have investigated the stability and the Hopf bifurcation for the following
four-dimensional neural network model:
⎩
⎨
⎧ �!′(�) = tanh(�%(�) − �"(�)) − b�!(� − �!)
�"′(�) = tanh(�!(�) + �%(�)) − b�"(� − �")
�$′(�) = tanh(�!(�) + �"(�) − �%(�)) − b�$(�)
�%′(�) = tanh(�$(�) − �"(�)) − b�%(�)
(2)
By means of the normal form theory and center manifold theorem, stability and bifurcating
periodic solutions and direction of the Hopf bifurcation were determined [2]. Kundu et al. have
concerned with the following delayed connections coupled in a star configuration neural
network model:
⎩
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎧�!′(�) = −��!(�) + ��!(� − �!) + ��"(� − �") + ��$(� − �") + ��%(� − �") + ��&(� − �")
+�!�!
$(� − �!) + �!�"
$(� − �") + �!�$
$(� − �") + �!�%
$(� − �") + �!�&
$(� − �")
�"′(�) = −��"(�) + ��!(� − �") + ��$(� − �!) + ��$(� − �") + ��%(� − �") + ��&(� − �")
+�!�!
$(� − �") + �!�"
$(� − �!) + �!�$
$(� − �") + �!�%
$(� − �") + �!�&
$(� − �")
�$′(�) = −��$(�) + ��!(� − �") + ��"(� − �") + ��$(� − �!) + ��%(� − �") + ��&(� − �")
+�!�!
$(� − �") + �!�"
$(� − �!) + �!�$
$(� − �!) + �!�%
$(� − �") + �!�&
$(� − �")
�%′(�) = −��%(�) + ��!(� − �") + ��"(� − �") + ��$(� − �!) + ��%(� − �!) + ��&(� − �")
+�!�!
$(� − �") + �!�"
$(� − �!) + �!�$
$(� − �") + �!�%
$(� − �") + ��&
$(� − �")
�&′(�) = −��&(�) + ��!(� − �") + ��"(� − �") + ��$(� − �!) + ��%(� − �") + ��&(� − �")
+�!�!
$(� − �") + �!�"
$(� − �") + �!�$
$(� − �") + �!�%
$(� − �") + �!�&
$(� − �!)
(3)
Considering the synaptic weight and time delay as parameters of the Hopf bifurcation, steady
state bifurcation, and equivalent steady state bifurcation criteria were given [3]. Wang et al.
have discussed a simplified six-neuron tridiagonal two-layer neural network model:
⎩
⎪⎪
⎨
⎪⎪
⎧ �!
# (�) = −��!(�) + �!�%(� − �) + �!�&(� − �)
�"
# (�) = −��"(�) + �!�%(� − �) + �"�&(� − �) + �"�'(� − �)
�$
# (�) = −��$(�) + �"�&(� − �) + �$�'(� − �)
�%
# (�) = −��%(�) + �!�!(�) + �!�"(�)
�&
# (�) = −��&(�) + �!�!(�) + �"�"(�) + �"�$(�)
�'
# (�) = −��'(�) + �"�"(�) + �$�$(�)
(4)
By matrix decomposition method, the Hopf bifurcation analysis is simplified. Some sufficient
conditions for the stability and bifurcation were exhibited which are simpler and more practice
than those obtained by the Hurwitz discriminant method [4]. Xu et al. studied a six dimensional
delayed BAM network system:
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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 5, October-2022
Services for Science and Education – United Kingdom
⎩
⎪
⎪
⎨
⎪
⎪
⎧�!′(�) = −�!�!(�) + �!!�!!6�!(� − �%)7 + �!"�!"6�"(� − �%)7 + �!$�!$6�$(� − �%)7
�"′(�) = −�"�"(�) + �"!�"!6�!(� − �&)7 + �""�""6�"(� − �&)7 + �"$�"$6�$(� − �&)7
�$′(�) = −�$�$(�) + �$!�$!6�!(� − �')7 + �$"�$"6�"(� − �')7 + �$$�$$6�$(� − �')7
�!′(�) = −�%�!(�) + �%!�%!6�!(� − �!)7 + �%"�%"6�"(� − �")7 + �%$�%$6�$(� − �$)7
�"′(�) = −�&�"(�) + �&!�&!6�!(� − �!)7 + �&"�&"6�"(� − �")7 + �&$�&$6�$(� − �$)7
�$′(�) = −�'�$(�) + �'!�'!6�!(� − �!)7 + �'"�'"6�"(� − �")7 + �'$�'$6�$(� − �$)7
(5)
By analyzing the associated characteristic transcendental equation, the linear stability of the
model and the Hopf bifurcation were demonstrated by using the normal form method and
center manifold theory [5]. Chen et al. have discussed a seven neurons and time delayed neural
network model as the follows:
⎩
⎪
⎪
⎨
⎪
⎪
⎧�!′(�) = −�!�!(�) + �$ �$(�$(� − �!)) + �& �&(�&(� − �!)) + �) �)(�)(� − �!))
�"′(�) = −�"�!(�) + �! �!(�!(� − �"))
�$′(�) = −�$�$(�) + �" �"(�"(� − �$))
�%′(�) = −�%�%(�) + �! �!(�!(� − �%))
�&′(�) = −�&�&(�) + �% �%(�%(� − �&))
�'′(�) = −�'�'(�) + �! �!(�!(� − �'))
�)′(�) = −�)�)(�) + �' �'(�'(� − �)))
(6)
where constants �* > 0 (� = 1,2, ... . ,7). By analyzing the corresponding characteristic equation
and using the delay as bifurcation parameter, the critical value of bifurcation was given. The
stability and bifurcation periodic solution were discussed by using the central manifold
theorem and the norm form [6].
However, we point out that by using bifurcating method to discuss the existence of bifurcating
periodic solution, it is necessary to make some restrictive conditions such that the neural
network contains only one delay. For example, in model(5) the authors assumed that �! + �% =
�" + �& = �$ + �' = � , and let �!(�) = �!(� − �!), �"(�) = �"(� − �"), �$(�) = �$(� −
�$), �%(�) = �!(�), �&(�) = �"(�), �'(�) = �$(�), and then model (5) changes to only one
delay system as the follows:
⎩
⎪
⎪
⎨
⎪
⎪
⎧�!′(�) = −�!�!(�) + �!!�!!6�%(� − �)7 + �!"�!"6�&(� − �)7 + �!$�!$6�'(� − �)7
�"′(�) = −�"�"(�) + �"!�"!6�%(� − �)7 + �""�""6�&(� − �)7 + �"$�"$6�'(� − �)7
�$′(�) = −�$�$(�) + �$!�$!6�%(� − �)7 + �$"�$"6�&(� − �)7 + �$$�$$6�'(� − �)7
�%′(�) = −�%�%(�) + �%!�%!6�!(�)7 + �%"�%"6�"(�)7 + �%$�%$6�$(�)7
�&′(�) = −�&�&(�) + �&!�&!6�!(�)7 + �&"�&"6�"(�)7 + �&$�&$6�$(�)7
�'′(�) = −�'�'(�) + �'!�'!6�!(�)7 + �'"�'"6�"(�)7 + �'$�'$6�$(�)7
(7)
Thus, the bifurcating equation can be discussed. In model (6), the authors assumed that �! +
�' + �) = �! + �% + �& = �! + �" + �$ = � , then setting �!(�) = �!(� − �" − �$ − �% − �& −
�' − �)), �"(�) = �"(� − �$ − �% − �& − �' − �)), �$(�) = �$(� − �% − �& − �' − �)), �%(�) =