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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:
⎩
⎪⎪
⎨
⎪⎪
⎧�!
" (�) = −�!�!(�) + �!!�!!(�%(� − �)) + �!$�!$(�&(� − �)) + �!#�!#(�'(� − �)),
�$
" (�) = −�$�$(�) + �$!�$!(�%(� − �)) + �$$�$$(�&(� − �)) + �$#�$#(�'(� − �)),
�#
" (�) = −�#�#(�) + �#!�#!(�%(� − �)) + �#$�#$(�&(� − �)) + �##�##(�'(� − �)),
�&
" (�) = −�%�%(�) + �%!�%!(�!(�)) + �%$�%$(�$(�)) + �%#�%#(�#(�)),
�&
" (�) = −�&�&(�) + �&!�&!(�!(�)) + �&$�&$(�$(�)) + �&#�&#(�#(�)),
�'
" (�) = −�'�'(�) + �'!�'!(�!(�)) + �'$�'$(�$(�)) + �'#�'#(�#(�)).
(4)
Thus, the bifurcating equation can be discussed. In this paper, we extend model (3) to an eight- neuron BAM system:
⎩
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎧ �!
" (�) = −�!�!(�) + �!!�!!(�!(� − �&)) + �!$�!$(�$(� − �')) + �!#�!#(�#(� − �()) + �!%�!%(�%(� − �))),
�$
" (�) = −�$�$(�) + �$!�$!(�!(� − �&)) + �$$�$$(�$(� − �')) + �$#�$#(�#(� − �()) + �$%�$%(�%(� − �))),
�#
" (�) = −�#�#(�) + �#!�#!(�!(� − �&)) + �#$�#$(�$(� − �')) + �##�##(�#(� − �()) + �#%�#%(�%(� − �))),
�%
" (�) = −�%�%(�) + �%!�%!(�!(� − �&)) + �%$�%$(�$(� − �')) + �%#�%#(�#(� − �()) + �%%�%%(�%(� − �))),
�!
"(�) = −�&�!(�) + �&!�&!(�!(� − �!)) + �&$�&$(�$(� − �$)) + �&#�&#(�#(� − �#)) + �&%�&%(�%(� − �%))
�$
" (�) = −�'�$(�) + �'!�'!(�!(� − �!)) + �'$�'$(�$(� − �$)) + �'#�'#(�#(� − �#)) + �'%�'%(�%(� − �%)),
,
�#
" (�) = −�(�#(�) + �(!�(!(�!(� − �!)) + �($�($(�$(� − �$)) + �(#�(#(�#(� − �#)) + �(%�(%(�%(� − �%)),
�%
"(�) = −�)�%(�) + �)!�)!(�!(� − �!)) + �)$�)$(�$(� − �$)) + �)#�)#(�#(� − �#)) + �)%�)%(�%(� − �%)).
(5)
where −�* < 0 (� = 1, 2, ... , 8). Our goal is to consider the dynamical behavior of model (5).
Noting that system (5) has eight delays. If those delays are different positive numbers, then the
bifurcating method is extremely hard to deal with model (5). Because one cannot solve the
bifurcating equation which is an eight variables transcendental equation. In order to discuss
the existence of periodic solutions for system (5), we adopt the generalized Chafee's criterion
[23], and the appendix of [24]. and this particular instability of the unique equilibrium point
and the boundedness of the solutions will force system (5) to generate a limit cycle, namely, a
periodic solution. In this paper, by using the mathematical analysis method, the existence of
periodic solutions has been established.
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European Journal of Applied Sciences (EJAS) Vol. 9, Issue 4, August-2021
Services for Science and Education – United Kingdom
Preliminaries
For the activation functions �*+ , we assume that �*+ are continuous bounded differentiable
functions, satisfying:
�*+(0) = 0, ��*+(u) > 0 (� ≠ 0) (6)
The general activation functions such as tanh (x), arctan (x) satisfy condition (6). The linearized
system of (5) is the following:
⎩
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎧ �!
" (�) = −�!�!(�) + �!!�!(� − �&) + �!$�$(� − �') + �!#�#(� − �() + �!%�%(� − �)),
�$
" (�) = −�$�$(�) + �$!�!(� − �&) + �$$�$(� − �') + �$#�#(� − �() + �$%�%(� − �)),
�#
" (�) = −�#�#(�) + �#!�!(� − �&) + �#$�$(� − �') + �##�#(� − �() + �#%�%(� − �)),
�%
" (�) = −�%�%(�) + �%!�!(� − �&) + �%$�$(� − �') + �%#�#(� − �() + 4#%�%(� − �)),
�!
"(�) = −�&�!(�) + �&!�!(� − �!) + �&$�$(� − �$) + �&#�#(� − �#) + �&%�%(� − �%)
�$
" (�) = −�'�$(�) + �'!�!(� − �!) + �'$�$(� − �$) + �'#�#(� − �#) + �'%�%(� − �%),
,
�#
" (�) = −�(�#(�) + �(!�!(� − �!) + �($�$(� − �$) + �(#�#(� − �#) + �(%�%(� − �%),
�%
"(�) = −�)�%(�) + �)!�!(� − �!) + �)$�$(� − �$) + �##�#(� − �#) + �)%�%(� − �%).
(7)
where �*+ = �*+�*+
" (0), �,� = 1,2, ... ,8. For convenience, let �*(�) = �*(�) (� = 1, ... , 4) and
�&(�) = �!(�), �'(�) = �$(�), �((�) = �#(�), �)(�) = �%(�), then system (7) can be rewritten
as the follows:
⎩
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎧ �!
" (�) = −�!�!(�) + �!!�&(� − �&) + �!$�'(� − �') + �!#�((� − �() + �!%�)(� − �)),
�$
" (�) = −�$�$(�) + �$!�&(� − �&) + �$$�'(� − �') + �$#�((� − �() + �$%�)(� − �)),
�#
" (�) = −�#�#(�) + �#!�&(� − �&) + �#$�'(� − �') + �##�((� − �() + �#%�)(� − �)),
�%
" (�) = −�%�%(�) + �%!�&(� − �&) + �%$�'(� − �') + �%#�((� − �() + 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
Services for Science and Education – United Kingdom
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
Page 6 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
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).
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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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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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