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European Journal of Applied Sciences – Vol. 9, No. 4

Publication Date: August 25, 2021

DOI:10.14738/aivp.94.10718. Feng, C. (2021). Periodic Oscillation for an Eight-Neuron Bam Neural Network Model With Discrete Delays. European Journal of

Applied Sciences, 9(4). 258-270.

Services for Science and Education – United Kingdom

Periodic Oscillation for an Eight-Neuron BAM Neural Network

Model with Discrete Delays

Chunhua Feng

Department of Mathematics and Computer Science,

Alabama State University, Montgomery, AL, USA, 36104

ABSTRACT

This paper investigates the existence of periodic oscillations for an eight-neuron

BAM neural network model with discrete delays. Two theorems are provided to

guarantee the existence of periodic oscillations for this model by using

mathematical analysis method, which is simpler than bifurcation method. The

criteria for selecting of the parameters in this network are provided. Computer

simulation examples are presented to demonstrate the correctness of this method.

Keywords: eight-neuron BAM network model, delay, instability, periodic solution

INTRODUCTION

Due to various applications of neural networks, the dynamics behaviors of neural networks

with delays have attracted great attention of many researchers [1-21]. In particular, there are

extensive literatures on simplified bidirectional associative memory (BAM) neural networks.

Various interesting results on periodic solution have been reported. For example, in 2006, Yu

and Cao have considered the Hope bifurcation for the following four-neuron BAM neural

network model with discrete delays [1]:

⎩

⎨

⎧ �!

" (�) = −�!�!(�) + �!!�!!(�!(� − �#)) + �!$�!$(�$(� − �#)),

�$

" (�) = −�$�$(�) + �$!�$!(�!(� − �%)) + �$$�$$(�$(� − �%)),

�!

" (�) = −�#�!(�) + �!!�!!(�!(� − �!)) + �!$�!$(�$(� − �$)),

�$

" (�) = −�%�$(�) + �$!�$!(�!(� − �!)) + �$$�$$(�$(� − �$)).

(1)

The existence of Hopf bifurcation, a formula for determining of the Hopf bifurcation and the

stability of bifurcating periodic solution have been obtained. Ge and Xu have investigated a five- neuron BAM network model with delays as follows [2]:

⎩

⎪

⎨

⎪

⎧�!

" (�) = −��!(�) + �!!�(�!(� − �#)) + �!$�(�$(� − �#)) + �!#�(�#(� − �#)),

�$

" (�) = −��$(�) + �$!�(�!(� − �%)) + �$$�(�$(� − �%)) + �$#�(�#(� − �%)),

�!

"(�) = −��!(�) + �!!�(�!(� − �!)) + �!$�(�$(� − �$)),

�$

" (�) = −��$(�) + �$!�(�!(� − �!)) + �$$�(�$(� − �$),

�#

" (�) = −��#(�) + �#!�(�!(� − �!)) + �#$�(�$(� − �$).

(2)

Some sufficient conditions for the synchronization and bifurcation were exhibited. The global

attractively of the trivial solution were also established. Xu et al. extended model (1) to six

dimensional delayed BAM network system [3]:

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259

Feng, C. (2021). Periodic Oscillation for an Eight-Neuron Bam Neural Network Model With Discrete Delays. European Journal of Applied Sciences,

9(4). 258-270.

URL: http://dx.doi.org/10.14738/aivp.94.10718

⎩

⎪⎪

⎨

⎪⎪

⎧�!

" (�) = −�!�!(�) + �!!�!!(�!(� − �%)) + �!$�!$(�$(� − �%)) + �!#�!#(�#(� − �%)),

�$

" (�) = −�$�$(�) + �$!�$!(�!(� − �&)) + �$$�$$(�$(� − �&)) + �$#�$#(�#(� − �&)),

�#

" (�) = −�#�#(�) + �#!�#!(�!(� − �')) + �#$�#$(�$(� − �')) + �##�##(�#(� − �')),

�!

"(�) = −�%�!(�) + �%!�%!(�!(� − �!)) + �%$�%$(�$(� − �$)) + �%#�%#(�#(� − �#)),

�$

" (�) = −�&�$(�) + �&!�&!(�!(� − �!)) + �&$�&$(�$(� − �$)) + �&#�&#(�#(� − �#)),

�#

" (�) = −�'�!(�) + �'!�'!(�!(� − �!)) + �'$�'$(�$(� − �$)) + �'#�'#(�#(� − �#)).

(3)

By analyzing the associated characteristic transcendental equation, the linear stability of the

model and Hopf bifurcation were demonstrated by using the normal form method and center

manifold theory. However, it was pointed out that by using the 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 (3) the authors

assumed that,

�! + �% = �$ + �& = �# + �' = �, and let �!(�) = �!(� − �!), �$(�) = �$(� − �$), �#(�) =

�#(� − �#), �%(�) = �!(�), �&(�) = �$(�), �'(�) =

�#(�), and then model (3) changes to only one delay system as the follows:

⎩

⎪⎪

⎨

⎪⎪

⎧�!

" (�) = −�!�!(�) + �!!�!!(�%(� − �)) + �!$�!$(�&(� − �)) + �!#�!#(�'(� − �)),

�$

" (�) = −�$�$(�) + �$!�$!(�%(� − �)) + �$$�$$(�&(� − �)) + �$#�$#(�'(� − �)),

�#

" (�) = −�#�#(�) + �#!�#!(�%(� − �)) + �#$�#$(�&(� − �)) + �##�##(�'(� − �)),

�&

" (�) = −�%�%(�) + �%!�%!(�!(�)) + �%$�%$(�$(�)) + �%#�%#(�#(�)),

�&

" (�) = −�&�&(�) + �&!�&!(�!(�)) + �&$�&$(�$(�)) + �&#�&#(�#(�)),

�'

" (�) = −�'�'(�) + �'!�'!(�!(�)) + �'$�'$(�$(�)) + �'#�'#(�#(�)).

(4)

Thus, the bifurcating equation can be discussed. In this paper, we extend model (3) to an eight- neuron BAM system:

⎩

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎧ �!

" (�) = −�!�!(�) + �!!�!!(�!(� − �&)) + �!$�!$(�$(� − �')) + �!#�!#(�#(� − �()) + �!%�!%(�%(� − �))),

�$

" (�) = −�$�$(�) + �$!�$!(�!(� − �&)) + �$$�$$(�$(� − �')) + �$#�$#(�#(� − �()) + �$%�$%(�%(� − �))),

�#

" (�) = −�#�#(�) + �#!�#!(�!(� − �&)) + �#$�#$(�$(� − �')) + �##�##(�#(� − �()) + �#%�#%(�%(� − �))),

�%

" (�) = −�%�%(�) + �%!�%!(�!(� − �&)) + �%$�%$(�$(� − �')) + �%#�%#(�#(� − �()) + �%%�%%(�%(� − �))),

�!

"(�) = −�&�!(�) + �&!�&!(�!(� − �!)) + �&$�&$(�$(� − �$)) + �&#�&#(�#(� − �#)) + �&%�&%(�%(� − �%))

�$

" (�) = −�'�$(�) + �'!�'!(�!(� − �!)) + �'$�'$(�$(� − �$)) + �'#�'#(�#(� − �#)) + �'%�'%(�%(� − �%)),

,

�#

" (�) = −�(�#(�) + �(!�(!(�!(� − �!)) + �($�($(�$(� − �$)) + �(#�(#(�#(� − �#)) + �(%�(%(�%(� − �%)),

�%

"(�) = −�)�%(�) + �)!�)!(�!(� − �!)) + �)$�)$(�$(� − �$)) + �)#�)#(�#(� − �#)) + �)%�)%(�%(� − �%)).

(5)

where −�* < 0 (� = 1, 2, ... , 8). Our goal is to consider the dynamical behavior of model (5).

Noting that system (5) has eight delays. If those delays are different positive numbers, then the

bifurcating method is extremely hard to deal with model (5). Because one cannot solve the

bifurcating equation which is an eight variables transcendental equation. In order to discuss

the existence of periodic solutions for system (5), we adopt the generalized Chafee's criterion

[23], and the appendix of [24]. and this particular instability of the unique equilibrium point

and the boundedness of the solutions will force system (5) to generate a limit cycle, namely, a

periodic solution. In this paper, by using the mathematical analysis method, the existence of

periodic solutions has been established.