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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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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.

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European Journal of Applied Sciences (EJAS) Vol. 9, Issue 4, August-2021

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Preliminaries

For the activation functions �*+ , we assume that �*+ are continuous bounded differentiable

functions, satisfying:

�*+(0) = 0, ��*+(u) > 0 (� ≠ 0) (6)

The general activation functions such as tanh (x), arctan (x) satisfy condition (6). The linearized

system of (5) is the following:

⎧ �!

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

�$

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

�#

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

�%

" (�) = −�%�%(�) + �%!�!(� − �&) + �%$�$(� − �') + �%#�#(� − �() + 4#%�%(� − �)),

�!

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

�$

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

,

�#

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

�%

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

(7)

where �*+ = �*+�*+

" (0), �,� = 1,2, ... ,8. For convenience, let �*(�) = �*(�) (� = 1, ... , 4) and

�&(�) = �!(�), �'(�) = �$(�), �((�) = �#(�), �)(�) = �%(�), then system (7) can be rewritten

as the follows:

⎧ �!

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

�$

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

�#

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

�%

" (�) = −�%�%(�) + �%!�&(� − �&) + �%$�'(� − �') + �%#�((� − �() + 4#%�)(� − �)),

�&

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

�'

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

,

�(

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

�)

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

(8)

The linearized system (8) can be written in a matrix form:

�"

(�) = ��(�) + � �(� − �) (9)

where �(�) = [�!(�), �$(�), ... , �)(�) ]

,, �(� − �) = [�!(� − �!), �$(� − �$), ... , �)(� − �)) ]

, .

� = ���� (−�!, −�$, ... , −�) ), and � is an eight by eight matrix

� = (�*+))×) =

⎡ 0 0 0 0 �!! �!$ �!# �!%

0 0 0 0 �$! �$$ �$# �$%

0

0

�&!

�'!

�(!

�)!

0

0

�&$

�'$

�($

�)$

0

0

�&#

�'#

�(#

�)#

0

0

�&%

�'%

�(%

�)%

�#!

�%!

0

0

0

0

�#$

�%$

0

0

0

0

�##

�%#

0

0

0

0

�#%

�%%

0

0

0

0 ⎦

.

Lemma 1 All solutions of system (5) are bounded.

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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

Proof From condition (6), the activation functions are continuous bounded, and assuming that

_�*+(�)_ ≤ �*+ (� = 1, 2, ... , 8), then

/|1!(3)|

/3 ≤ − �* |�*(�)| + �* , �* = ∑ _�*+_ %

+5! �*+ (� = 1, 2, ... , 8) (10)

From (10), noting that �* (� = 1, 2, ... , 8) are positive real numbers, we obtain

|�*(�)| ≤ 6!

7!

(� = 1, 2, ... , 8) (11)

This system (8) has a unique equilibrium point means that the solutions of system (5) are

uniformly bounded.

Lemma 2 If the matrix � is not a positive definite matrix, then system (5) has a unique

equilibrium point.

Proof First we show that system (8) has a unique equilibrium point. An equilibrium point � ∗ =

[�!

∗, �$

∗ , ... , �)

∗ ]

, is a solution of the following algebraic equation:

⎧ −�!�!

∗ + �!!�&

∗ + �!$�'

∗ + �!#�(

∗ + �!%�)

∗ = 0,

−�$�$

∗ + �$!�&

∗ + �$$�'

∗ + �$#�(

∗ + �$%�)

∗ = 0,

−�#�#

∗ + �#!�&

∗ + �#$�'

∗ + �##�(

∗ + �#%�)

∗ = 0,

−�%�%

∗ + �%!�&

∗ + �%$�'

∗ + �%#�(

∗ + �%%�)

∗ = 0,

−�&�&

∗ + �&!�!

∗ + �&$�$

∗ + �&#�#

∗ + �&%�%

∗ = 0,

−�'�'

∗ + �'!�!

∗ + �'$�$

∗ + �'#�#

∗ + �'%�%

∗ = 0,

,

−�(�(

∗ + �(!�!

∗ + �($�$

∗ + �(#�#

∗ + �(%�%

∗ = 0,

−�)�)

∗ + �)!�!

∗ + �)$�$

∗ + �)#�#

∗ + �)%�%

∗ = 0.

(12)

System (12) can be written as a matrix form

�� ∗ = −�� ∗ (13)

Obviously, there is a trivial solution of system (12), implying that zero is an equilibrium point

of system (8).Noting that – � is a positive definite matrix, so system (8) has a unique

equilibrium point if B is not a positive definite matrix. Now for system (5), zero also is an

equilibrium point of system (5) since �*+ (0) = 0. Meanwhile, condition ��*+ (u) > 0 (� ≠ 0)

ensures that there exists a unique equilibrium point and it is exactly the zero point of system

(5).

We adopt the following norms of vectors and matrices in this paper [22]: ‖�(�)‖ = ∑ |�*(�)| )

*5! ,

‖�‖ = max!9+9) ∑ _�*+_, )

*5! the measure �(�) of a matrix B is defined by �(�) = lim

:→<=

‖?=:@‖A!

: ,

which for the chosen norms reduces to �(�) = max!9+9)(�++ + ∑ _�*+_ ). )

*5!,*C+ B > 0 which

indicates that B is a positive definite matrix.

Definition 1 The trivial solution of system (5) is unstable, if there exists at least one component

of the trivial solution which is unstable.

Existence of periodic solutions

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European Journal of Applied Sciences (EJAS) Vol. 9, Issue 4, August-2021

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Theorem 1 Assume that system (5) has a unique equilibrium point for a set of selected

parameters. If the following condition holds

9(‖�‖) �∗

$ �∗ exp(−�∗ �∗) > 5 (14)

where �∗= min {�!, �$, ... , �) }, and �∗ = min {�!, �$, ... , �) }. Then the unique equilibrium point

of the system (5) is unstable and it generates a limit cycle, namely, a periodic solution.

Proof We must prove that the unique equilibrium point of system (5) is unstable. Obviously, if

the trivial solution of linearized system (7) or equivalent system (8) is unstable, then the trivial

solution of system (5) is unstable. Thus we only need to consider system (8) and prove that the

unique equilibrium point of system (8) which is exactly the zero point is unstable. Consider a

special case of system (8) as follows:

�"

(�) = ��(�) + � �(� − �∗ ) (15)

where �∗= min {�!, �$, ... , �) }, and �(� − �∗ ) = [�!(� − �∗), �$(� − �∗), ... , �)(� − �∗) ]

,. Noting

that |�*(�)| = �*(�) as �*(�) > 0 and |�*(�)| = −�*(�) as �*(�) < 0 (� = 1, 2, ... , 8). From (15),

when each �*(�) > 0 we have

/|D(3)|

/3 = ��(�) + � �(� − �∗ ) (16)

and for each �*(�) < 0 one can obtain

/|D(3)|

/3 = −��(�) − � �(� − �∗ ) (17)

Noting that −�* < 0 (� = 1, 2, ... , 8). Therefore we have

/(∑ | #

!$% 1!(3)|

/3 ≤ −�∗ ∑ | )

*5! �*(�)| + ‖�‖ ∑ | )

*5! �*(� − �∗)| (18)

Specially, for the scalar time delay differential equation

/F(3)

/3 = −�∗�(�) + ‖�‖ �(� − �∗) (19)

where �(�) = ∑ | )

*5! �*(�)|. If the unique equilibrium point of system (19) is table, then the

characteristic equation associated with (19) given by

� = −�∗ + ‖�‖ �AGH∗ (20)

will have a real negative root say �<, and we have from (20)

|�<| ≥ ‖�‖ �|G'|H∗ − �∗ (21)

Using the formula �I ≥ J

&

�$ for � ≥ 0 one can get

1 ≥ ‖�‖ �|G'|H∗

�∗ + |�<| = ‖�‖ �∗�A7∗H∗�(7∗=|G'|)H∗

(�∗ + |�<|)�∗

≥ (‖�‖ �∗�A7∗H∗ )

9(�∗ + |�<|)�∗

5

≥ (‖�‖ �∗�A7∗H∗ ) J7∗H∗

& = J

& (‖�‖) �∗

$ �∗ exp(−�∗ �∗) (22)

The last inequality contradicts (14). Hence, our claim regarding the instability of the

equilibrium point of system (15) is valid. Based on the comparison theorem of differential

equation, we have ∑ | )

*5! �*(�)| ≤ �(�). According to the definition of the instability of the trivial

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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

solution, for an arbitrary � > 0, there exists a sequence {�K} such that |�(�K)| > �, where

�(�) represents the trivial solution of system (15). Since ∑ | )

*5! �*(�)| ≤ �(�), this means that

there exists a subsequence v�K(

w of the sequence {�K} such that ∑ | )

*5! �* x�K(

y | ≤ �(�K(

) .

Therefore, there exists at least one �*(�) , and without loss of generality, we assume that

|�! x�K(

y | > L

)

. Since � is an arbitrary sufficiently small positive number, L

) is also an arbitrary

sufficiently small positive number. Thus, �!(�) is unstable. According to the definition 1, the

instability of the component �!(�) implies that the trivial solution of system (15) is unstable.

Now we prove that the trivial solution of system (9) is also unstable. System (15) is a special

case of (9). Obviously, �*(� − �∗) is equivalent to �*(� − �*) (� = 1, 2, ... , 8) as � is sufficiently

large. So, we still have

|�! x�K(

y | > L

)

as �K( is sufficiently large. This means that the trivial solution of system (9) is

unstable. We can then prove that the trivial solution of system (5) is unstable. Indeed, system

(9) is a linearized version of system (5), in other words, system (5) is a disturbing system of (9).

Thus, the instability of the trivial solution of system (9) implies that the instability of the trivial

solution of the original system (5). Since all solutions of system (5) are bounded, the instability

of the unique equilibrium point together with boundedness of the solutions lead system (5) to

generate a limit cycle, namely, a periodic solution based on

Chafee’s criterion [23] and the appendix of [24].

Theorem 2 Assume that system (5) has a unique equilibrium point for a set of selected

parameters. If the following condition holds

‖�‖ − �∗ > 0 (23)

then the unique equilibrium point of the system (5) is unstable. System (5) generates a limit

cycle, namely, a periodic solution.

Proof We still prove that the trivial solution of system (15) is unstable. The characteristic

equation of system (15) is the following

� = −�∗ + ‖�‖ �AGH∗ (24)

namely

� + �∗ − ‖�‖ �AGH∗ = 0 (25)

and there exists a positive characteristic root of equation (25) under the restrictive condition

(23). Indeed, let �(�) = � + �∗ − ‖�‖ �AGH∗ , then �(�) is a continuous function of �. Obviously,

�(0) = �∗ − ‖�‖ = −(‖�‖ − �∗) < 0, and while as � (> 0) is sufficiently large, �AGH∗ will be

sufficiently small, Therefore,

there exists a �̅(> 0) such that �{�̅

| = �̅+ �∗ − ‖�‖ �AGMH∗ > 0. According to the well known

the Intermediate Value Theorem, there exists a positive value of � say �!, �! ∈ (0, �̅

) such that

�(�!) = �! + �∗ − ‖�‖ �AG%H∗ = 0. In other words, equation (25) has a positive characteristic

root. Therefore, the trivial solution of system (15) is unstable. Similar to the proof of Theorem

1, the unique equilibrium point of system (5) is unstable. According to the generalized Chafee’s

criterion, system (5) has a limit cycle, namely, a periodic solution.

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Simulation result

These simulations are performed by using the system (5). First, the activation function is

selected as �*+(�) = tanh(�). Then �"

(�) = 1 − tanh$(�), so �"

(0) = 1 . The parameters are

selected as �! = 0.45,

�$ = 0.64, �# = 0.48, �% = 0.55, �& = 0.62, �' = 0.58, �( = 0.52, �) = 0.65; �!! = 1.25, �!$

= 1.15,

�!# = 0.85, �!% = 0.75, �$! = −2.15, �$$ = 0.55, �$# = −1.05, �$% = 0.45, �#! = 1.35, �#$ =

−0.48,

�## = 0.52, �#% = −0.25, �%! = 0.75, �%$ = 1.25, �%# = −0.35, �%% = 0.36, �&! = −0.95, �&$ =

0.85,

�&# = −1.16, �&% = 0.88, �'! = −2.65, �'$ = −0.75, �'# = 0.82, �'% = 0.45, �(! = −0.45, �($ =

−0.32, �(# = 0.64, �(% = −0.25, �)! = −0.42, �)$ = 0.36, �)# = −0.12, �)% = −0.85; time delays

are selected as �! = 1.12, �$ = 1.16, �# = 1.25, �% = 1.22, �& = 1.15, �' = 1.24, �( = 1.26, �) =

1.18. The characteristic values of matrix � are −0.7177, 0.7177, −0.3376 ± 1.9017 �, 0.3376 ±

1.9017 �, ±0.1104 �. Thus, � is not a positive definite matrix. From Lemma 2, this system has a

unique equilibrium point, namely, the zero point. We have �∗ = 1.12, �∗ = 0.45, ‖�‖ = 5.5 and

9(‖�‖) �∗

$ �∗ exp(−�∗ �∗) = 9 × 5.5 × 1.12 × 1.12 × 0.45 × exp(−1.12 × 0.45) = 16.8799 >

5. Based on Theorem 1, there is a periodic solution (see figure 1). Then we change time delays

as �! = 0.52, �$ = 0.56, �# = 0.58, �% = 0.62, �& = 0.65, �' = 0.64, �( = 0.46, �) = 0.48, the

other parameters are the same as in figure 1. Obviously, the conditions of Theorem 2 are

satisfied, there exists a periodic solution (see figure 2). However, we see that

9(‖�‖) �∗

$ �∗ exp(−�∗ �∗) = 9 × 5.5 × 0.46 × 0.46 × 0.45 × exp(−0.46 × 0.45) = 3.8321 < 5.

The condition of Theorem 1 is not satisfied. This means that the Theorem 1 is a stronger

sufficient condition. Then we select the activation function as �*+(�) = arctan(�). Keeping all

parameters similar to those used to generate figure 2, we see that the oscillatory frequency and

amplitude remain almost the same (see figure 3), implying that the oscillatory behavior is just

a little effected by the activation functions. However, when we increase the time delays, the

oscillatory frequency changes greatly (see figure 4). In order to see the effect of �*, we change

�! = 1.25, �$ = 1.42, �# = 1.48, �% = 1.35, �& = 1.42, �' = 1.52, �( = 1.46, �) = 1.38 , the

other parameters are the same as in figure 4, time delays are selected as �! = 0.72, �$ =

0.76, �# = 0.78, �% = 0.82, �& = 0.85, �' = 0.84, �( = 0.86, �) = 0.88, and �! = 1.52, �$ =

1.56, �# = 1.58, �% = 1.62, �& = 1.65, �' = 1.64, �( = 1.46, �) = 1.48, respectively. We see that

the oscillatory frequency changes very much (see figure 5 and figure 6). This means that time

delays and �* effect the oscillatory frequency. Finally we select another set of parameters as

�! = 0.85, �$ = 0.82, �# = 0.75, �% = 0.72, �& = 0.82, �' = 0.78, �( = 0.76, �) = 0.84; �!! =

−0.45, �!$ = 1.25, �!# = 0.35, �!% = 0.55, �$! = −1.65, �$$ = 0.85, �$# = 1.45, �$% = 0.75,

�#! = 1.15, �#$ = −0.68, �## = 0.22, �#% = −0.58, �%! = 0.15, �%$ = 1.65, �%# = −0.95, �%% =

0.46, �&! = −0.75, �&$ = 0.45, �&# = −1.76, �&% = 0.38, �'! = −2.35, �'$ = −0.92, �'# =

0.42, �'% = 0.75, �(! = −0.85, �($ = −0.72, �(# = 0.24, �(% = −0.85, �)! = −0.32, �)$ = 0.96,

�)# = −0.42, �)% = −0.15. We see that the characteristic values of matrix � are − 1.2444,

1.2444, ±2.2159 �, ±1.2138 �, ±0.2593 � , and ‖�‖ = 4.27. Therefore, � is not a positive

definite matrix. From Lemma 2, the system has a unique equilibrium point, namely, the zero

point. When time delays are selected as �! = 0.85, �$ = 0.82, �# = 0.75, �% = 0.72, �& =

0.82, �' = 0.78, �( = 0.76, �) = 0.80, andtime delays are selected as �! = 1.24, �$ = 1.26, �# =

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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

1.28, �% = 1.32, �& = 1.35, �' = 1.34, �( = 1.38, �) = 1.36, respectivery, there exist periodic

oscillations based on the Theorem 2 (see figure 7 and figure 8).

Page 9 of 13

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European Journal of Applied Sciences (EJAS) Vol. 9, Issue 4, August-2021

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Page 10 of 13

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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

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European Journal of Applied Sciences (EJAS) Vol. 9, Issue 4, August-2021

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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

CONCLUSION

This paper considers an eight-neuron BAM network with discrete delays. Two sufficient

conditions to guarantee the existence of periodic solutions are provided. A specific selection of

parameters is used to demonstrate the results. Our simple criteria to ensure the existence of

permanent oscillations are easy to check, as compared to predicting the regions of bifurcation.

The computer simulation suggests that our theorems are only sufficient conditions.

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