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DOI: 10.14738/aivp.85.8873
Publication Date: 07th September, 2020
URL: http://dx.doi.org/10.14738/aivp.85.8873
Periodic oscillation for a complex-valued neural network model with
discrete and distributed delays
Chunhua Feng
College of Science, Technology, Engineering and Mathematics, Alabama State University
915 S. Jackson Street, Montgomery, AL, 36104, USA
Abstract: In this paper, a complex-valued neural network model with discrete and distributed
delays is investigated under the assumption that the activation function can be separated into its
real and imaginary parts. Based on the mathematical analysis method, some sufficient conditions to
guarantee the existence of periodic oscillatory solutions are established. Computer simulation is given
to illustrate the validity of the theoretical results.
Keywords: complex-valued neural network model, discrete delay, distributed delay, periodic
oscillation
1 Introduction
In the past two decades, many researchers have studied various complex-valued neural network models
with or without discrete and distributed delays [1-20]. For example, Samidurai et al. have discussed
the delay-dependent stability for some complex-valued neural networks with discrete and distributed
time-varying delays [1,2]. Wang et al. considered the stability for complex-valued stochastic delayed
networks with Markovian switching and impulsive effects [3]. Guo et al. studied the existence,
uniqueness, and exponential stability for complex-valued memristor-based BAM neural networks with
time delays [4]. Popa discussed the global μ-stability of neutral-type impulsive complex-valued BAM
neural networks with leakage delay and unbounded time-varying delays [5]. Many authors focus on
the synchronization of complex-valued networks, such as exponential synchronization of complex- valued delayed coupled systems on networks with aperiodically on-off coupling [7,8]. Synchronization
of impulsive coupled complex-valued neural networks with delay by the matrix measure method,
distributed impulsive method, a direct error method, and a non-reduced order and non-separation
approach [9, 10, 11, 12]. With complex-valued neural networks, a new multi-modal technique for
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Chunhua Feng; Book Review: Periodic oscillation for a complex-valued neural network model with
discrete and distributed delays. European Journal of Applied Sciences, Volume 8 No 5, Oct 2020; pp:1-17
URL: http://dx.doi.org/10.14738/aivp.85.8873 2
assisting visually-impaired people in recognizing objects in public indoor environment was provided
[16]. For a complex-valued Wilson-Cowan neural network with time delay as follows:
v
′
1
(t) = −v1(t) + a1g(v1(t)) + a2g(v2(t − τ )) + P,
v
′
2
(t) = −v2(t) + a3g(v1(t − τ )) + a4g(v2(t)) + Q.
(1)
By using proper translations and coordinate transformations, Ji et al. have decomposed the activation
functions and connection weights into their real and imaginary parts, so as to construct an equivalent
real-valued system. Then, the sufficient conditions for Hopf bifurcation and its directions are provided
through normal form theory and central manifold theorem [19].
Hang et al. have investigated a two-node fractional complex-valued neural network with time delay
as follows [20]:
Dq
z
′
1
(t) = −μ1z1(t) + af(z1(t − τ )) + bf(z2(t − τ )),
Dq
z
′
2
(t) = −μ2z2(t) + cf(z1(t − τ )) + df(z2(t − τ )).
(2)
By using time delay as the bifurcation parameter, the dynamical behaviors that including local asymp- totical stability and Hopf bifurcation were discussed, the conditions of emergence of bifurcation were
also obtained. Furthermore, it reveals that the onset of the bifurcation point can be delayed as the
order increases. In [21], Li et al. extended a real-valued network model to a complex-valued model
with discrete and distributed delays as the following:
z
′
1
(t) = −z1(t) + b11f11(z1(t − τ )) + b12f12(
R t
−∞ F(t − s)z2(s)ds),
z
′
2
(t) = −z2(t) + b21f21(z1(t − τ )) + b22f22(
R t
−∞ F(t − s)z2(s)ds).
(3)
Regarding the discrete time delay as the bifurcating parameter, the problem of Hopf bifurcation in
the newly-proposed complex-valued neural network model was investigated under the assumption that
the activation function can be separated into its real and imaginary parts. Based on the normal form
theory and center manifold theorem, some sufficient conditions which determine the direction of the
Hopf bifurcation and the stability of the bifurcating periodic solutions were established. Zhang et al.
have considered a complex value delayed Hopfield neural networks model, in which a ring topology
consists of two coupling unidirectional rings, each with four oscillators is given [22]. For a real-valued
three-node network as follows:
x
′
1
(t) = −x1(t) + a
R t
−∞ k(t − s)f(x1(s))ds + b[f(x3(t − τ )) + f(x2(t − τ ))],
x
′
2
(t) = −x2(t) + a
R t
−∞ k(t − s)f(x2(s))ds + b[f(x1(t − τ )) + f(x3(t − τ ))],
x
′
3
(t) = −x3(t) + a
R t
−∞ k(t − s)f(x3(s))ds + b[f(x2(t − τ )) + f(x1(t − τ ))].
(4)
In this model, the distributed signal transmission delay has introduced in the self-connection of the
network [23]. The author has discussed the Hopf bifurcation and stability switching of model (4).
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E u r o p e a n J o u r na l o f Ap p l i e d S c i e n c e s , Vo l ume 8 No . 5, O c t 2 0 2 0
S e r v i c e s f o r S c i e n c e a nd E d u c a t i o n , U n i te d K i ng d om 3
Motivated by the above models, In this paper, we extend model (4) to the following complex-valued
network system:
z
′
1
(t) = −l1z1(t) + r11f11(
R t
−∞ F(t − s)z1(s)ds) + r12f12(z2(t − τ )) + r13f13(z3(t − τ )),
z
′
2
(t) = −l2z2(t) + r21f21(z1(t − τ )) + r22f22(
R t
−∞ F(t − s)z2(s)ds) + r23f23(z3(t − τ )),
z
′
3
(t) = −l3z3(t) + r31f31(z1(t − τ )) + r32f32(z2(t − τ )) + r33f33(
R t
−∞ F(t − s)z3(s)ds),
(5)
where F(s) = αe−αs
, s ≥ 0, α > 0 is a weak kernel. lj , rkj (k, j = 1, 2, 3) are complex numbers. Let
z4(t) = R t
−∞ F(t − s)z1(s)ds, z5(t) = R t
−∞ F(t − s)z2(s)ds, z6(t) = R t
−∞ F(t − s)z3(s)ds, then system
(5) changes to the following
z
′
1
(t) = −l1z1(t) + r11f11(z4(t)) + r12f12(z2(t − τ )) + r13f13(z3(t − τ )),
z
′
2
(t) = −l2z2(t) + r21f21(z1(t − τ )) + r22f22(z5(t)) + r23f23(z3(t − τ )),
z
′
3
(t) = −l3z3(t) + r31f31(z1(t − τ )) + r32f32(z2(t − τ )) + r33f33(z6(t)),
z
′
4
(t) = −αz4(t) + αz1(t),
z
′
5
(t) = −αz5(t) + αz2(t),
z
′
6
(t) = −αz6(t) + αz3(t),
(6)
It was emphasized that the bifurcating method is hard to deal with system (6). From [21] one can
see that the authors need to investigate a 6-degree bifurcating equation (see equation (11), page 155).
Similarly we must consider a 12-degree bifurcating equation according to the bifurcating method. It
is quite not easy to deal with a 12-degree algebraic equation. Therefore, in this paper, by means of
the mathematical analysis method, we discuss the periodic oscillation for system (6). For convenience,
let lj = aj + ibj , rkj = pkj + iqkj , fkj (zj (t)) = f
R
kj (xj (t), yj (t)) + ifI
kj
(xj (t), yj (t))(zj (t) = xj (t) +
iyj (t)), k, j = 1, 2, 3. Then the complex-valued system (6) can be expressed by separating it into real
3
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Chunhua Feng; Book Review: Periodic oscillation for a complex-valued neural network model with
discrete and distributed delays. European Journal of Applied Sciences, Volume 8 No 5, Oct 2020; pp:1-17
URL: http://dx.doi.org/10.14738/aivp.85.8873 4
and imaginary parts as the following:
x
′
1
(t) = −a1x1(t) + b1y1(t) + p11f
R
11(x4(t), y4(t)) − q11f
I
11(x4(t), y4(t))
+p12f
R
12(x2(t − τ ), y2(t − τ )) − q12f
I
12(x2(t − τ ), y2(t − τ ))
+p13f
R
13(x3(t − τ ), y3(t − τ )) − q13f
I
13(x3(t − τ ), y3(t − τ )),
y
′
1
(t) = −a1y1(t) − b1x1(t) + p11f
I
11(x4(t), y4(t)) + q11f
R
11(x4(t), y4(t))
+p12f
I
12(x2(t − τ ), y2(t − τ )) + q12f
R
12(x2(t − τ ), y2(t − τ ))
+p13f
I
13(x3(t − τ ), y3(t − τ )) + q13f
R
13(x3(t − τ ), y3(t − τ )),
x
′
2
(t) = −a2x2(t) + b2y2(t) + p21f
R
21(x1(t − τ ), y1(t − τ )) − q21f
I
21(x1(t − τ ), y1(t − τ ))
+p22f
R
22(x5(t), y5(t)) − q22f
I
22(x5(t), y5(t))
+p23f
R
23(x3(t − τ ), y3(t − τ )) − q23f
I
23(x3(t − τ ), y3(t − τ )),
y
′
2
(t) = −a2y2(t) − b2x2(t) + p21f
I
21(x1(t − τ ), y1(t − τ )) + q21f
R
21(x1(t − τ ), y1(t − τ ))
+p22f
I
22(x5(t), y5(t)) + q22f
R
22(x5(t), y5(t))
+p23f
I
23(x3(t − τ ), y3(t − τ )) + q23f
R
23(x3(t − τ ), y3(t − τ )),
x
′
3
(t) = −a3x3(t) + b3y3(t) + p31f
R
31(x1(t − τ ), y1(t − τ )) − q31f
I
31(x1(t − τ ), y1(t − τ ))
+p32f
R
32(x2(t − τ ), y2(t − τ )) − q32f
I
32(x2(t − τ ), y2(t − τ ))
+p33f
R
33(x6(t), y6(t)) − q33f
I
33(x6(t), y6(t)),
y
′
3
(t) = −a3y3(t) − b3x3(t) + p31f
I
31(x1(t − τ ), y1(t − τ )) + q31f
R
31(x1(t − τ ), y1(t − τ ))
+p32f
I
32(x2(t − τ ), y2(t − τ )) + q32f
R
32(x2(t − τ ), y2(t − τ ))
+p33f
I
33(x6(t), y6(t)) + q33f
R
33(x6(t), y6(t)),
x
′
4
(t) = −αx4(t) + αx1(t),
y
′
4
(t) = −αy4(t) + αy1(t),
x
′
5
(t) = −αx5(t) + αx2(t),
y
′
5
(t) = −αy5(t) + αy2(t),
x
′
6
(t) = −αx6(t) + αx3(t),
y
′
6
(t) = −αy6(t) + αy3(t).
(7)
Thus, in order to discuss the periodic oscillation of system (6), we only need to consider the periodic
oscillation of system (7). Suppose that the derivative of f
R
kj(x, y) and f
I
kj (x, y) with respect to x and
4
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E u r o p e a n J o u r na l o f Ap p l i e d S c i e n c e s , Vo l ume 8 No . 5, O c t 2 0 2 0
S e r v i c e s f o r S c i e n c e a nd E d u c a t i o n , U n i te d K i ng d om 5
y exist, continuous, and f
R
kj(0, 0) = 0, fI
kj (0, 0) = 0. Then the linearized system of (6) is the following:
x
′
1
(t) = −a1x1(t) + b1y1(t) + c14x4(t) + d14y4(t) + c12x2(t − τ ) + d12y2(t − τ ))
+c13x3(t − τ ) + d13y3(t − τ )),
y
′
1
(t) = −a1y1(t) − b1x1(t) + m14x4(t) + n14y4(t) + m12x2(t − τ ) + n12y2(t − τ ))
+m13x3(t − τ ) + n13y3(t − τ )),
x
′
2
(t) = −a2x2(t) + b2y2(t) + c25x5(t) + d25y5(t) + c21x1(t − τ ) + d21y1(t − τ ))
+c23x3(t − τ ) + d23y3(t − τ )),
y
′
2
(t) = −a2y2(t) − b2x2(t) + m25x5(t) + n25y5(t) + m21x1(t − τ ) + n21y1(t − τ ))
+m23x3(t − τ ) + n23y3(t − τ )),
x
′
3
(t) = −a3x3(t) + b3y3(t) + c36x6(t) + d36y6(t) + c31x1(t − τ ) + d31y1(t − τ ))
+c32x2(t − τ ) + d32y2(t − τ )),
y
′
3
(t) = −a3y3(t) − b3x3(t) + m36x6(t) + n36y6(t) + m31x1(t − τ ) + n31y1(t − τ ))
+m32x2(t − τ ) + n32y2(t − τ )),
x
′
4
(t) = −αx4(t) + αx1(t),
y
′
4
(t) = −αy4(t) + αy1(t),
x
′
5
(t) = −αx5(t) + αx2(t),
y
′
5
(t) = −αy5(t) + αy2(t),
x
′
6
(t) = −αx6(t) + αx3(t),
y
′
6
(t) = −αy6(t) + αy3(t).
(8)
where for k=1,2,3, j = 4,5,6, ckj = pkk
∂fR
kk(0,0)
∂xj
− qkk
∂f I
kk
(0,0)
∂xj
, dkj = pkk
∂f
R
kk(0,0)
∂yj
− qkk
∂f I
kk
(0,0)
∂yj
, mkj =
pkk
∂f
I
kk(0,0)
∂xj
+qkk
∂fR
kk
(0,0)
∂xj
, nkj = pkk
∂f
I
kk(0,0)
∂yj
+qkk
∂fR
kk
(0,0)
∂yj
. For k, j=1,2,3, ckj = pkj
∂f
R
kj
(0,0)
∂xj
−qkj
∂f I
kj
(0,0)
∂xj
,
dkj = pkj
∂f
R
kj
(0,0)
∂yj
− qkj
∂f I
kj
(0,0)
∂yj
, mkj = pkj
∂f
I
kj (0,0)
∂xj
+ qkj
∂fR
kj
(0,0)
∂xj
, nkj = pkj
∂f
I
kj (0,0)
∂yj
+ qkj
∂fR
kk
(0,0)
∂yj
. The
matrix form of system (8) is the following:
U
′
(t) = AU(t) + BU(t − τ ) (9)
5
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URL: http://dx.doi.org/10.14738/aivp.85.8873 6
where U(t) = [x (t), y (t), · · · , x (t), y (t)] , U(t − τ ) = [x (t − τ ), y (t − τ ), x (t − τ ), y (t − τ ), x3(t −
τ ), y3(t − τ ), 0, 0, 0, 0, 0, 0]T
, Both A = (aij )12×12 and B = (bij )12×12 are 12 × 12 matrices as follows:
A = (aij )12×12 =
−a1 b1 0 0 0 · · · 0 0 0
−b1 −a1 0 0 0 · · · 0 0 0
0 0 −a2 b2 0 · · · d25 0 0
0 0 −b2 −a2 0 · · · n25 0 0
· · · · · · · · · · · · · · · · · · · · · · · · · · ·
0 0 α 0 0 · · · 0 0 0
0 0 0 α 0 · · · −α 0 0
0 0 0 0 α · · · 0 −α 0
0 0 0 0 0 · · · 0 0 −α
,
B = (bij )12×12 =
0 0 c12 d12 c13 · · · 0 0 0
0 0 m12 n12 m13 · · · 0 0 0
c21 d21 0 0 c23 · · · 0 0 0
m21 n21 0 0 m23 · · · 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 · · · 0 0 0
.
2 Preliminaries
Let C is a six by six matrix as follows:
C = (cij )6×6 =
p11 −q11 p12 −q12 p13 −q13
q11 p11 q12 p12 q13 p13
p21 −q21 p22 −q22 p23 −q23
q21 p21 q22 p22 q23 p23
p31 −q31 p32 −q32 p33 −q33
q31 p31 q32 p32 q33 p33
.
Lemma 1 Assume that aj > 0, bj > 0(j = 1, 2, 3), f R
jk(0, 0) = 0, fI
jk(0, 0) = 0, f
R
jk(x, y) >
0, fI
jk(x, y) > 0 when x > 0, y > 0, while f
R
jk(x, y) < 0, fI
jk(x, y) < 0 when x < 0, y < 0 (j, k = 1, 2, 3),
C is not a positive definite matrix, then system (7) has a unique equilibrium.
Proof An equilibrium point x
∗ = [x
∗
1
, y∗
1
, · · · , x∗
6
, y∗
6
]
T of system (7) is a constant solution of the
6
Chunhua Feng; Book Review: Periodic oscillation for a complex-valued neural network model with
discrete and distributed delays. European Journal of Applied Sciences, Volume 8 No 5, Oct 2020; pp:1-17
Page 9 of 17
E u r o p e a n J o u r na l o f Ap p l i e d S c i e n c e s , Vo l ume 8 No . 5, O c t 2 0 2 0
S e r v i c e s f o r S c i e n c e a nd E d u c a t i o n , U n i te d K i ng d om 9
we get xj (t), yj (t) are bounded (j = 1, 2, 3).
3 Main Results
In order to discuss the instability of the trivial solution of system (7) we only need to discuss the
instability of the trivial solution of system (8).
Theorem 1 Assume that Lamma1 and Lemma 2 hold. For selected parameters values of aj , bj , pjk
and qjk(1 ≤ j, k ≤ 3). Let the eigenvalues of matrix A + B = D be δi(i = 1, 2, · · · , 12). If there
exists at least one eigenvalue δk, k ∈ {1, 2, · · · , 12} such that δk > 0 or Re(δk) > 0, then the unique
equilibrium point of system (8) is unstable, implying that system (7) generates an oscillatory solution.
Proof Obviously, if the trivial solution of system (8) is unstable, then the trivial solution of system
(7) is also unstable. Therefore, we only need to consider the instability of the trivial solution of system
(8). When τ = 0, system (8) reduced to
U
′
(t) = AU(t) + BU(t) = DU(t) (13)
The characteristic equation corresponding to system (13) is
det[λIij − D] = 0. (14)
where Iij is the 12 by 12 identity matrix. Since δi(i = 1, 2, · · · , 12) are eigenvalues of matrix D,
therefore we have from (14)
Π
12
j=1(λ − δj ) = 0. (15)
Since there exists some δk > 0 or Re(δk) > 0, this means that system (14) has a positive real eigenvalue
or an eigenvalue which has a positive real part. Therefore the trivial solution of system (13) is unstable
according to the basic result of differential equation. Now we prove that the trivial solution of system
(8) also is unstable. Suppose that U(t) is a trivial solution of system (13). It is known that U(t)
∼ U(t − τ ) when t is sufficiently large. According to the definition of instability of the trivial solution,
for any ε > 0 one can find a sequence {t1, t2, · · · , tn, · · ·}, where t1(> τ ) is sufficiently large such that
|U(tk)| > ε. When t1 is sufficiently large, we have U(tk) ∼ U(tk − τ ), means that U(tk) also is the
solution of system (8). Thus, the trivial solution of system (8) is unstable, implying that the trivial
solution of system (7) is unstable. Since system (7) has a unique equilibrium point and all solutions
are bounded, this instability of the trivial solution will force system (7) to generate a limit cycle,
namely, a periodic oscillation [24], and the appendix of [25].
9
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URL: http://dx.doi.org/10.14738/aivp.85.8873 10
Set μ
P | |}, k k = max P12
Theorem 2 Assume that Lamma1 and Lemma 2 hold. For selected parameters values of aj , bj , pjk
and qjk(1 ≤ j, k ≤ 3). If
μ(A) + kBk > 0. (16)
Then the unique equilibrium point of system (8) is unstable, implying that system (7) generates an
oscillatory solution.
Proof Similar to Theorem 1, we show that the trivial solution of system (8) is unstable, then the
trivial solution of system (7) also is unstable. The characteristic equation corresponding to system (8)
is
λ = μ(A) + kBke
−λτ
. (17)
or
λ − μ(A) − kBke
−λτ = 0. (18)
Equation (18) is a transcendental equation. We prove that there is a positive characteristic root
of system (18). Let f(λ) = λ − μ(A) − kBke
−λτ
, then f(λ) is a continuous function of λ. f(0) =
−μ(A) − kBk = −(μ(A) + kBk) < 0. Noting that limλ→∞ e
−λτ = 0. So, there exists a λ0 > 0 such
that f(λ0) = λ0 − μ(A) − kBke
−λ0τ > 0. According to the Intermediate Value Theorem, there exists
a λ1 ∈ (0, λ0) such that f(λ1) = λ1 − μ(A) − kBke
−λ1τ = 0. This means that equation (18) has a
positive characteristic root. Therefore, the trivial solution of system (8) is unstable, implying that the
trivial solution of system (7) is unstable. This instability of the trivial solution will force system (7)
to generate a limit cycle, namely, a periodic oscillatory solution.
4 Simulation Results
This simulation is based on system (7). We first select the parameters as α = 0.24, time delay τ = 0.8,
a1 = 0.0065, a2 = 0.0055, a3 = 0.0075; b1 = 0.095, b2 = 0.085, b3 = 0.075; the other parameter values as
p11 = 0.5, p12 = 0.4, p13 = 0.6, p21 = −0.075, p22 = 0.086, p23 = −0.065,, p31 = 0.55, p32 = 0.45, p33 =
0.65; q11 = 0.95, q12 = −0.92, q13 = 0.96, q21 = 0.25, q22 = 0.35, q23 = −0.15,, q31 = 0.15, q32 =
0.12, q33 = −0.16. The activation function as f
R
jk(x, y) = f
I
jk(x, y) = (tanh(x) + tanh(y)) + i(tanh(x) +
tanh(y))(j, k = 1, 2, 3), thus ∂fR
jk(0,0)
∂x =
∂fR
jk(0,0)
∂y = 1, and ∂f I
jk(0,0)
∂x =
∂f I
jk(0,0)
∂y = 1 (j, k = 1, 2, 3). We
10
Chunhua Feng; Book Review: Periodic oscillation for a complex-valued neural network model with
discrete and distributed delays. European Journal of Applied Sciences, Volume 8 No 5, Oct 2020; pp:1-17
Page 11 of 17
E u r o p e a n J o u r na l o f Ap p l i e d S c i e n c e s , Vo l ume 8 No . 5, O c t 2 0 2 0
S e r v i c e s f o r S c i e n c e a nd E d u c a t i o n , U n i te d K i ng d om 11
have c14 = d14 = p11 − q11 = −0.4, c12 = d12 = p12 − q12 = 1.32, c13 = d13 = p13 − q13 = −0.36;
m14 = n14 = p11 + q11 = 1.45, m12 = n12 = p12 + q12 = −0.52, m13 = n13 = p13 + q13 = 1.56;
c25 = d25 = p22 − q22 = −0.264, c21 = d21 = p21 − q21 = −0.0175, c23 = d23 = p23 − q23 = 0.085; m25 =
n25 = p22 + q22 = 0.436, m21 = n21 = p21 + q21 = 0.175, m23 = n23 = p23 + q23 = −0.215; c36 = d36 =
p33−q33 = 0.81, c31 = d31 = p31−q31 = 0.4, c32 = d32 = p32−q32 = 0.33; m36 = n36 = p33+q33 = 0.49,
m31 = n31 = p31 + q31 = 0.7, m32 = n32 = p32 + q32 = 0.57. Thus, the eigenvalues of matrix C are
1.1537 ± 1.1291i, 0.4928 ± 0.2051i, −0.4105 ± 0.2161i. This means that C is not a positive definite
matrix. The eigenvalues of matrix A + B are 1.3757, 0.0208, −1.3757, −0.4354, −0.2400, −0.2306,
0.0818 ± 0.1350i, −0.2114 ± 0.0760i, −0.2400 ± 0.0001i. Obviously, 1.3757 is a positive eigenvalue.
Based on Theorem 1, there exists a periodic oscillatory solution (see Fig. 1). In order to see the effect
of the parameter α, we change α = 0.86, the other parameters are the same as in figure 1, we see the
oscillatory frequency and amplitude are almost the some as in figure 1 (see Fig. 2). Then we change the
activation function as f
R
jk(x, y) = f
I
jk
(x, y) = (arctan(x) + arctan(y)) + i(arctan(x) + arctan(y))(j, k =
1, 2, 3), thus we still have ∂fR
jk(0,0)
∂x =
∂f
R
jk
(0,0)
∂y = 1, and ∂f I
jk
(0,0)
∂x =
∂f
I
jk
(0,0)
∂y = 1 (j, k = 1, 2, 3). The
parameters are the same as in figure 2, we see the oscillatory frequency almost the same as in figure
2. However, the oscillatory amplitude changes too much (see Fig. 3). Now we select the parameters
as α = 1.6, time delay τ = 0.5, a1 = 0.15, a2 = 0.25, a3 = 0.18; b1 = 1.15, b2 = 1.05, b3 = 1.18;p11 =
0.72, p12 = 0.25, p13 = 0.48, p21 = 0.175, p22 = −0.16, p23 = 0.65, p31 = 0.15, p32 = −0.24, p33 = 0.36;
q11 = −0.25, q12 = 0.24, q13 = −0.36, q21 = 0.18, q22 = 0.15, q23 = −0.35, q31 = 0.28, q32 = −0.25, q33 =
0.4. We have c14 = d14 = p11 − q11 = 0.97, c12 = d12 = p12 − q12 = 0.01, c13 = d13 = p13 − q13 = 0.84;
m14 = n14 = p11 + q11 = 0.47, m12 = n12 = p12 + q12 = 0.49, m13 = n13 = p13 + q13 = 0.12;
c25 = d25 = p22 − q22 = −0.31, c21 = d21 = p21 − q21 = −0.005, c23 = d23 = p23 − q23 = 1;
m25 = n25 = p22 + q22 = −0.01, m21 = n21 = p21 + q21 = 0.355, m23 = n23 = p23 + q23 = 0.3;
c36 = d36 = p33 − q33 = −0.04, c31 = d31 = p31 − q31 = −0.13, c32 = d32 = p32 − q32 = −0.01;
m36 = n36 = p33 + q33 = 0.76, m31 = n31 = p31 + q31 = 0.43, m32 = n32 = p32 + q32 = −0.49. We see
that kBk = 1, μ(A) = 1.85. Therefore, kBk + μ(A) = 2.85 > 0. Based on Theorem 2, there exists a
periodic oscillatory solution (see Fig. 4).
5 Conclusion
In this paper, we have discussed the oscillatory behavior of the solutions for a complex-valued neural
network model with discrete and distributed delays. By means of the mathematical analysis method,
two criteria to guarantee the existence of periodic oscillations, which is easy to check, as compared
11
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Chunhua Feng; Book Review: Periodic oscillation for a complex-valued neural network model with
discrete and distributed delays. European Journal of Applied Sciences, Volume 8 No 5, Oct 2020; pp:1-17
URL: http://dx.doi.org/10.14738/aivp.85.8873 12
to find the regions of bifurcation have been proposed. In this network, we decomposed the activation
functions and connection weights into their real and imaginary parts, so as to discuss an equivalent
real-valued system. The activation function affects the oscillatory amplitude. When these time delay
systems generate a periodic oscillatory solution, the delays affect oscillatory frequencies.
References
[1] R. Samidurai, R. Sriraman, S. Zhu, Leakage delay-dependent stability analysis for complex-valued neural
networks with discrete and distributed time-varying delays, Neurocomputing, 338 (2019) 262-273.
[2] R. Sriraman, Y. Cao, R. Samidurai, Global asymptotic stability of stochastic complex-valued neural net- works with probabilistic time-varying delays, Math. Comput. Simul. 171 (2020) 103-118.
[3] P.F. Wang, X.L. Wang, H. Su, Stability analysis for complex-valued stochastic delayed networks with
Markovian switching and impulsive effects, Commun. Nonlinear Sci. Numer. Simul. 731 (2019) 35-51.
[4] R. Guo, Z. Zhang, X. Liu, C. Lin, Existence, uniqueness, and exponential stability analysis for complex- valued memristor-based BAM neural networks with time delays, Appl. Math. Comput. 311 (2017) 100-117.
[5] C.A. Popa, Global μ-stability of neutral-type impulsive complex-valued BAM neural networks with leakage
delay and unbounded time-varying delays, Neurocomputing, 376 (2020) 73-94.
[6] W. Zhou, J.M. Zurada, Discrete-time recurrent neural networks with complex-valued linear threshold
neurons, IEEE Trans. Circuits Syst. II Express Briefs 56 (8) (2009) 669-673.
[7] P.F. Wang, W.Q. Zou, H. Su, J.Q. Feng, Exponential synchronization of complex-valued delayed coupled
systems on networks with aperiodically on-off coupling, Neurocomputing, 369 (2019) 155-165.
[8] Y. Kan, J.Q. Lu, J.L. Qiu, J. Kurths, Exponential synchronization of time-varying delayed complex-valued
neural networks under hybrid impulsive controllers, Neural Networks, 114 (2019) 157-163.
[9] L. Li, X.H. Shi, J.L. Liang, Synchronization of impulsive coupled complex-valued neural networks with
delay: The matrix measure method, Neural Networks, 117 (2019) 285-294.
[10] D. Ding, Z. Tang, Y. Wang, Z.C. Ji, Synchronization of nonlinearly coupled complex networks: Distributed
impulsive method, Chaos, Solitons and Fractals, 133 (2020) 109620.
[11] C. Hu, H.B. He, H.J. Jiang, Synchronization of complex-valued dynamic networks with intermittently
adaptive coupling: A direct error method, Automatica, 112 (2020) 108675.
[12] J. Yu, C. Hu, H.J. Jiang, L.M. Wang, Exponential and adaptive synchronization of inertial complex-valued
neural networks: A non-reduced order and non-separation approach, Neural Networks, 124 (2020) 50-59.
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S e r v i c e s f o r S c i e n c e a nd E d u c a t i o n , U n i te d K i ng d om 13
[13] S. Zheng, L.G. Yuan, Nonperiodically intermittent pinning synchronization of complex-valued complex
networks with non-derivative and derivative coupling, Physica A, 525 (2019) 587-605.
[14] W. Xu, S. Zhu, X.Y. Fang, W. Wang, Adaptive anti-synchronization of memristor-based complex-valued
neural networks with time delays, Physica A, 535 (2019) 122427.
[15] Z.Y. Wang, J.D. Cao, Z.W. Cai, L.H. Huang, Periodicity and finite-time periodic synchronization of
discontinuous complex-valued neural networks, Neural Networks, 119 (2019) 249-260.
[16] R. Trabelsi, I. Jabri, F. Melgani, F. Smach, A. Bouallegue, Indoor object recognition in RGBD images
with complex-valued neural networks for visually-impaired people, Neurocomputing, 330 (2019) 94-103.
[17] Z. Wang, X. Wang, Y. Li, X. Huang, Stability and Hopf bifurcation of fractional-order complex-valued
single neuron model with time delay, Int. J. Bifurc. Chaos 27 (13) (2017) 1750209.
[18] T. Dong, X. Liao, A. Wang, Stability and Hopf bifurcation of a complex-valued neural network with two
time delays, Nonlinear Dyn. 82 (1) (2015) 1-12.
[19] C.H. Ji, Y.H. Qiao, J. Miao, L.J. Duan, Stability and Hopf bifurcation analysis of a complex-valued
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[20] C.D. Huang, J.D. Cao, M. Xiao, A. Alsaedi, T. Hayat, Bifurcations in a delayed fractional complex-valued
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[21] L. Li, Z. Wang, Y.X. Li, H. Shen, J.W. Lu, Hopf bifurcation analysis of a complex-valued neural network
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Chunhua Feng; Book Review: Periodic oscillation for a complex-valued neural network model with
discrete and distributed delays. European Journal of Applied Sciences, Volume 8 No 5, Oct 2020; pp:1-17
URL: http://dx.doi.org/10.14738/aivp.85.8873 14
0 200 400 600 800 1000 1200
−200
0
200
(a) Solid line: x1
(t), dotted line: y1
(t).
Fig. 1 Oscillation of the solutions. delay: 0.8, alpha=0.24.
0 200 400 600 800 1000 1200
−50
0
50
(b) Solid line: x2
(t), dotted line: y2
(t).
0 200 400 600 800 1000 1200
−200
0
200
(c) Solid line: x3
(t), dotted line: y3
(t).
0 200 400 600 800 1000 1200
−200
0
200
(d) Solid line: x4
(t), dotted line: y4
(t).
0 200 400 600 800 1000 1200
−50
0
50
(e) Solid line: x5
(t), dotted line: y5
(t).
0 200 400 600 800 1000 1200
−200
0
200
(f) Solid line: x6
(t), dotted line: y6
(t), delay: 0.8, alpha: 0.24.
14
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E u r o p e a n J o u r na l o f Ap p l i e d S c i e n c e s , Vo l ume 8 No . 5, O c t 2 0 2 0
S e r v i c e s f o r S c i e n c e a nd E d u c a t i o n , U n i te d K i ng d om 15
0 200 400 600 800 1000 1200
−200
0
200
Fig. 2 Oscillation of the solutions, delay: 0.8, alpha=0.86.
(a) Solid line: x1
(t), dotted line: y1
(t).
0 200 400 600 800 1000 1200
−50
0
50
(b) Solid line: x2
(t), dotted line: y2
(t).
0 200 400 600 800 1000 1200
−200
0
200
(c) Solid line: x3
(t), dotted line: y3
(t).
0 200 400 600 800 1000 1200
−200
0
200
(d) Solid line: x4
(t), dotted line: y4
(t).
0 200 400 600 800 1000 1200
−50
0
50
(e) Solid line: x5
(t), dotted line: y5
(t).
0 200 400 600 800 1000 1200
−200
0
200
(f) Solid line: x6
(t), dotted line: y6
(t), delay: 0.8, alpha:0.86.
15
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Chunhua Feng; Book Review: Periodic oscillation for a complex-valued neural network model with
discrete and distributed delays. European Journal of Applied Sciences, Volume 8 No 5, Oct 2020; pp:1-17
URL: http://dx.doi.org/10.14738/aivp.85.8873 16
0 200 400 600 800 1000 1200
−400
−200
0
200
400
Fig. 3 Oscillation of the solutions, activation function: arctan(x)+ arctan(y).
(a) Solid line: x1
(t), dotted line: y1
(t).
0 200 400 600 800 1000 1200
−50
0
50
(b) Solid line: x2
(t), dotted line: y2
(t).
0 200 400 600 800 1000 1200
−500
0
500
(c) Solid line: x3
(t), dotted line: y3
(t).
0 200 400 600 800 1000 1200
−400
−200
0
200
400
(d) Solid line: x4
(t), dotted line: y4
(t).
0 200 400 600 800 1000 1200
−50
0
50
(e) Solid line: x5
(t), dotted line: y5
(t).
0 200 400 600 800 1000 1200
−500
0
500
(f) Solid line: x6
(t), dotted line: y6
(t), activation function: arctan(x)+arctan(y).
16
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E u r o p e a n J o u r na l o f Ap p l i e d S c i e n c e s , Vo l ume 8 No . 5, O c t 2 0 2 0
S e r v i c e s f o r S c i e n c e a nd E d u c a t i o n , U n i te d K i ng d om 17
0 20 40 60 80 100
−10
0
10
Fig. 4 Oscillation of the solutions, delay: 0.5, alpha: 1.6.
(a) Solid line: x1
(t), dotted line: y1
(t).
0 20 40 60 80 100
−5
0
5
(b) Solid line: x2
(t), dotted line: y2
(t).
0 20 40 60 80 100
−1
0
1
(c) Solid line: x3
(t), dotted line: y3
(t).
0 20 40 60 80 100
−10
0
10
(d) Solid line: x4
(t), dotted line: y4
(t).
0 20 40 60 80 100
−4
−2
0
2
4
(e) Solid line: x5
(t), dotted line: y5
(t).
0 20 40 60 80 100
−1
0
1
(f) Solid line: x6
(t), dotted line: y6
(t), delay: 0.5, alpha: 1.6.
17