Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 357382, 13 pages
doi:10.1155/2012/357382
Research Article
Stability and Bifurcation Analysis of
a Three-Dimensional Recurrent Neural
Network with Time Delay
Yingguo Li
School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, China
Correspondence should be addressed to Yingguo Li, yguoli@fjnu.edu.cn
Received 14 August 2011; Revised 4 October 2011; Accepted 5 October 2011
Academic Editor: Shiping Lu
Copyright q 2012 Yingguo Li. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We consider the nonlinear dynamical behavior of a three-dimensional recurrent neural network
with time delay. By choosing the time delay as a bifurcation parameter, we prove that Hopf
bifurcation occurs when the delay passes through a sequence of critical values. Applying the normal form method and center manifold theory, we obtain some local bifurcation results and derive
formulas for determining the bifurcation direction and the stability of the bifurcated periodic
solution. Some numerical examples are also presented to verify the theoretical analysis.
1. Introduction
Starting with the work of Hopfield 1 on neural networks, recurrent neural networks
including Hopfield neural networks, Cohen-Grossberg neural networks, and cellular
neural networks have been used extensively in different areas such as signal processing,
pattern recognition, optimization, and associative memories. Many researchers studied the
dynamical behavior of Recurrent neural network systems, and most of papers are devoted
to the stability of equilibrium, existence and stability of periodic solutions, bifurcation, and
chaos 2–5. In 5, Ruiz et al. considered a particular configuration of a recurrent neural
network, illustrated in Figure 1. In Figure 1, ut is the input and yt is the output of the
network. This recurrent neural network can be described by the following system:
ẋ1 t −x1 t fx2 t,
..
.
ẋn−1 t −xn−1 t ut,
2
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ẋn t −xn t w1 fx1 t · · · wn−1 fxn−1 t,
yt fxn t.
1.1
Here, xt ∈ Rn is the state, wi ∈ R, i 1, . . . , n − 1 are the network parameters or weights,
ut is a smooth input, and yt is the output. The transfer function of the neurons is taken
as f· tanh·. A three-node network of the form 1.1 in feedback configuration, with
ut yt, has been studied in 5; that is,
ẋ1 t −x1 t tanhx2 t,
ẋ2 t −x2 t tanhx3 t,
1.2
ẋ3 t −x3 t w1 tanhx1 t w2 tanhx2 t.
Ruiz et al. found and analyzed the Hopf bifurcation behavior in system 1.2. In 4, Maleki
2i−1
, where α2i−1 > 0
et al. considered system 1.1 with a transfer function fx ∞
i1 α2i−1 x
for i odd and α2i−1 < 0 for i even. The authors analyzed the Bogdanov-Takens bifurcation in
the system.
It is well known that there exist time delays in the information processing of neurons.
The delayed axonal signal transmissions in the neural network models make the dynamical
behaviors become more complicated and may destabilize the stable equilibria and admit
periodic oscillation, bifurcation, and chaos. Therefore, the delay is an important control
parameter in living nervous system: different ranges of delays correspond to different
patterns of neural activities see, e.g., 6–11.
In the present paper, we consider the following three-dimensional recurrent neural
network model with time delay
ẋ1 t −x1 t fx2 t − τ,
ẋ2 t −x2 t fx3 t − τ,
1.3
ẋ3 t −x3 t w1 fx1 t − τ w2 fx2 t − τ.
By choosing the time delay as a bifurcation parameter, we prove that Hopf bifurcation occurs
in the neuron and study the properties of periodic solutions of this model.
The organization of this paper is as follows. In Section 2, by analyzing the characteristic
equation of the linearized system of system 1.3 at the equilibrium, we discuss the stability
of the equilibrium and the existence of the Hopf bifurcation occurring at the equilibrium. In
Section 3, the formulae determining the direction of the Hopf bifurcations and the stability of
bifurcating periodic solutions on the center manifold are obtained by using the normal form
theory and the center manifold theorem due to Hassard et al. 12. We do some computer
observations to validate our theoretical results in Section 4.
2. Stability and Existence of Hopf Bifurcation
For most of the models in the literature, including the ones in 5, 7, 8, the activation function
f is fu tanhcu. However, we only make the following assumption on function f:
H f ∈ C3 R, f0 0, and f 0 /
0.
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3
u(t)
wn−1
xn−1
1
wn−2
xn−2
1
1
y(t)
xn
.
.
.
1
w1
x1
Figure 1: Class of recurrent neural networks.
Clearly, x1 , x2 , x3 T 0, 0, 0T is equilibrium of system 1.3. Linearization of 1.3 at
the zero equilibrium yields
ẋ1 t −x1 t f 0x2 t − τ,
ẋ2 t −x2 t f 0x3 t − τ,
2.1
x˙3 t −x3 t w1 f 0x1 t − τ w2 f 0x2 t − τ,
whose characteristic equation is
⎛
⎜
det⎜
⎝
λ1
−f 0e−λτ
0
λ1
−w1 f 0e
−λτ
−w2 f 0e
0
⎞
⎟
−f 0e−λτ ⎟
⎠ 0,
−λτ
2.2
λ1
that is,
λ 13 − w2 f 2 0λ 1e−2λτ − w1 f 3 0e−3λτ 0.
2.3
The zero equilibrium is stable if all roots of 2.3 have negative real parts and unstable if at
least one root has positive real part. Therefore, in order to study the local stability of the zero
equilibrium of system 2.3, we need to investigate the distribution of the roots of 2.3.
When τ 0, characteristic equation 2.3 yields
λ 13 − w2 f 2 0λ 1 − w1 f 3 0 0.
2.4
Let y λ 1, 2.4 reduces to
y3 − w2 f 2 0y − w1 f 3 0 0.
2.5
4
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Denote
√
w12 f 6 0 w23 f 6 0
1
3
−
,
ε− i,
Δ
4
27
2
2
3
3
√
√
3 w1 f 0
3 w1 f 0
Δ,
− Δ.
β
α
2
2
2.6
From Cardano formula for the third-degree algebra equation, we have the following lemma.
Lemma 2.1. (1)√If Δ > 0, then 2.5 has a real root α β and a pair of conjugate complex roots
−α β/2 ± i 3/2α √
− β. Furthermore, the roots of 2.4 are given by λ1 −1 α β and
λ2,3 −1 − α β/2 ± i 3/2α − β.
(2) If Δ 0, then 2.5 has a simple root 2α and a multiple root −α with the multiplicity
of 2. Furthermore, the roots of 2.4 are given by λ1 −1 2α and λ2,3 −1 − α. Meanwhile, if
w1 w2 0, that is, α 0, then 2.4 has a multiple root −1 with the multiplicity of 3.
(3) If Δ < 0, then 2.5 has three real roots 2 Re{α}, 2 Re{αε}, and 2 Re{αε}. Furthermore,
the roots of 2.4 are given by λ1 −1 2 Re{α}, λ2 −1 2 Re{αε}, and λ3 −1 2 Re{αε}.
Applying Lemma 2.1, we have the following lemma.
Lemma 2.2. All roots of 2.4 have negative real parts if one of the following holds.
(1) Δ > 0 and −2 < α β < 1.
(2) Δ 0 and −1 < α < 1/2.
(3) Δ < 0 and max{Re{α}, Re{αε}, Re{αε}} < 1/2.
Let z λ 1eλτ , then 2.3 becomes
z3 − w2 f 2 0z − w1 f 3 0 0.
2.7
We notice that 2.7 and 2.5 have the same coefficients.
Denote the three roots of 2.7 by zn Rn iIn n 1, 2, 3. Hence, 2.3 is equivalent
to
λ 1eλτ zn
n 1, 2, 3.
2.8
Clearly, iω ω > 0 is a root of 2.3 if and only if ω satisfies
cos ωτ − ω sin ωτ Rn ,
2.9
sin ωτ ω cos ωτ In ,
which implies that
cos ωτ Rn ωIn
,
1 ω2
sin ωτ In − ωRn
1 ω2
n 1, 2, 3.
2.10
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5
This yields
ω2 R2n In2 − 1
n 1, 2, 3.
2.11
Obviously, we have the following lemma.
Lemma 2.3. Equation 2.11 is meaningless when |zn | R2n In2 ≤ 1 n 1, 2, 3. If 2.7 has a
root, denoted by zj , satisfying |zn | > 1, then 2.11 has a positive root given by
ωj R2n In2 − 1.
2.12
Summarizing the discussion above, we have the following.
Lemma 2.4. If Δ ≥ 0, then 2.11 has at most two positive roots. If Δ < 0, then 2.11 has at most
three positive roots.
Without loss of generality, one assumes that 2.11 has three positive roots ωn n 1, 2, 3.
Define
n
τj
1
Rn ωn In
arc cos
2jπ
,
ωn
1 ωn2
n 1, 2, 3, j 0, 1, 2, . . . .
2.13
n
Then, ±iωn is a pair of purely imaginary roots of 2.3 with τ τj .
Let λτ στ iωτ be the root of 2.3 near τ τjn satisfying
σ τjn 0,
ω τjn ωn
n 1, 2, 3, j 0, 1, 2, . . . .
2.14
We have the following.
Lemma 2.5. One has
dστ > 0,
dτ ττjn
n 1, 2, 3, j 0, 1, 2, 3, . . . .
2.15
Proof. Differentiating both sides of 2.8 with respect to τ, we have
dλ
dτ
−1
−
τ
1
− .
λλ 1 λ
2.16
6
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n
Note that λτj iωn , therefore
sign
dRe λ
dτ
n
sign Re
ττj
dλ
dτ
−1 sign Re
λiω0
−1
iωn iωn 1
1
sign
.
ω2 1
2.17
We have
dστ dRe λ
> 0.
n
dτ ττjn
dτ
ττj
2.18
This completes the proof.
n ∞
For convenience, we let ∪3n1 {τj }
j0
{τi }∞
i0 , such that
τ0 < τ1 < τ2 < · · · < τi < · · · ,
2.19
0 0 0
τ0 min τ1 , τ2 , τ3
2.20
where
n
and τj
is defined in 2.13.
Applying Lemmas 2.1–2.5 and Corollary 2.4 of Ruan and Wei 13, we have the
following results.
Lemma 2.6. All roots of 2.3 have negative real parts if one of the following holds:
H1 Δ > 0, −1 < α β < 1 and 1/4α β2 3/4α − β2 < 1,
H2 Δ 0 and −1 < α < 1/2,
H3 Δ < 0 and max{Re{α}, Re{αε}, Re{αε}} < 1/2.
Lemma 2.7. Suppose that one of the following hypothesis is satisfied:
H4 Δ > 0, −2 < α β < −1,
H5 Δ > 0, −1 < α β < 1 and 1/4α β2 3/4α − β2 > 1.
Then, there exists a sequence values of τ defined by 2.19 such that all roots of 2.3 have negative
real parts for all τ ∈ 0, τ0 , and 2.3 has at least one root with positive real part when τ > τ0 , and
n
2.3 exactly has a pair of purely imaginary roots ±iωn n 1, 2, 3 when τ τj n 1, 2, 3; j n
0, 1, 2, 3, . . ., where ωn and τ τj
are defined by 2.12 and 2.13, respectively.
From Lemmas 2.5–2.7 and the Hopf bifurcation theorem for functional differential
equations in 14, we have the theorem.
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7
Theorem 2.8. (1) If one of the hypothesis H1 , H2 , H3 is satisfied, then the zero solution of
system 1.3 is asymptotically stable for all τ ≥ 0.
(2) If H4 or H5 is satisfied, then the zero solution of system 1.3 is asymptotically stable
for τ ∈ 0, τ0 and unstable for τ > τ0 , and system 1.3 undergoes a Hopf bifurcation at the origin
when τ τj j 0, 1, 2, 3, . . ..
3. Direction of Hopf Bifurcations and Stability of the Bifurcating
Periodic Orbits
In this section, we will study the direction of the Hopf bifurcation and stability of bifurcating
periodic solutions by using the normal theory and the center manifold theorem due to
Hassard et al. 12.
Let u1 t x1 τt, u2 t x2 τt, u3 t x3 τt, then system 1.3 becomes functional
differential equation in C C−1, 0, R3 as
⎛
u̇1 t
⎞
⎜
⎟
⎜u̇2 t⎟
⎝
⎠
u̇3 t
⎛
⎛
⎞
⎞
u1 t
u1 t − 1
⎜
⎜
⎟
⎟
⎜
⎟
⎟
τB1 ⎜
⎝u2 t⎠ τB2 ⎝u2 t − 1⎠
u3 t
u3 t − 1
⎛
⎞
f 0 2
f 0 3
u
u
−
1
−
1
·
·
·
t
t
2
2
⎜
⎟
2!
3!
⎜
⎟
⎜
⎟
f 0 2
f 0 3
⎜
⎟
τ⎜
⎟,
u
u
−
1
−
1
·
·
·
t
t
3
3
⎜
⎟
2!
3!
⎜
⎟
⎝
⎠
w1 f 0 3
w2 f 0 2
w2 f 0 3
w1 f 0 2
u1 t − 1
u1 t − 1
u2 t − 1
u2 t − 1 · · ·
2!
3!
2!
3!
3.1
where
⎞
⎛
−1 0 0
⎟
⎜
⎟
B1 ⎜
⎝ 0 −1 0 ⎠,
0 0 −1
⎛
⎜
B2 ⎜
⎝
0
f 0
0
0
w1 f 0 w2 f 0
0
⎞
⎟
f 0⎟
⎠.
3.2
0
Setting τ ν τj , we know that ν 0 is Hopf bifurcation value of system 3.1.
For φ φ1 , φ2 , φ3 T ∈ C−1, 0, R3 , let
Lν φ τj ν B1 φ0 B2 φ−1 .
3.3
8
Journal of Applied Mathematics
By the Riesz representation theorem, there exists a function ηθ, ν of bounded variation for
θ ∈ −1, 0, such that
Lν φ 0
−1
dηθ, νφθ
for φ ∈ C −1, 0, R3 .
3.4
In fact, we can choose
ηθ, ν τj ν B1 δθ − τj ν B2 δθ 1,
3.5
where δ denotes the Dirac delta function. For φ ∈ C−1, 0, R3 , define
⎧
dφθ
⎪
⎪
⎪
⎨ dθ ,
Aνφ 0
⎪
⎪
⎪
dηs, νφs,
⎩
θ ∈ −1, 0,
θ 0,
−1
Rνφ 0,
f ν, φ ,
3.6
θ ∈ −1, 0,
θ 0,
where
f ν, φ
τj ν
⎛
⎞
f 0 2
f 0 3
u
u
−
1
−
1
·
·
·
t
⎜
⎟
2 t
2! 2
3!
⎜
⎟
⎜
⎟
f 0 2
f 0 3
⎜
⎟
×⎜
⎟.
u
u
−
1
−
1
·
·
·
t
t
3
3
⎜
⎟
2!
3!
⎜
⎟
⎝ w1 f 0
⎠
w1 f 0 3
w2 f 0 2
w2 f 0 3
2
u1 t − 1
u1 t − 1
u2 t − 1
u2 t − 1 · · ·
2!
3!
2!
3!
3.7
Then system 3.1 is equivalent to
u̇t Aνut Rνut ,
3.8
where ut u1 t, u2 t, u3 tT , ut ut θ for θ ∈ −1, 0.
For ψ ∈ C1 0, 1, R3 ∗ , define
⎧
dψs
⎪
⎪
,
−
⎪
⎪
⎨
ds
∗
A ψs ⎪ 0
⎪
⎪
⎪
⎩
dηT t, 0ψ−t,
−1
s ∈ −1, 0,
3.9
s 0,
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9
and a bilinear inner product
ψs, φθ ψ0φ0 −
0 θ
−1
ψξ − θdηθψξdξ,
3.10
ξ0
where ηθ ηθ, 0. Then, A0 and A∗ are adjoint operators. By the discussion in Section 2,
we know that ±iω0 τj are eigenvalues of A0 and A∗ corresponding to iω0 τj and −iω0 τj ,
respectively.
Suppose qθ 1, q1 , q2 T eiω0 τj θ is the eigenvectors of A0 corresponding to iω0 τj ,
then A0qθ iω0 τj qθ. Then from the definition of A0 and 3.3–3.5, we have
τj B1 q0 τj B2 q−1 iω0 τj q0.
3.11
For q−1 q0e−iω0 τj , then we obtain
iω0 1eiω0 τj
,
f 0
q1 3.12
iω0 12 e2iω0 τj
q2 .
f 2 0
Similarly, we can obtain the eigenvector q∗ s D1, q1∗ , q2∗ eω0 τj s of A∗ corresponding to
−iω0 τj , where
q1∗ q2∗
iω0 − 12 e−2iω0 τj
,
f 2 0w1
3.13
1 − iω0 e−iω0 τj
.
f 0w1
In order to assure q∗ s, qθ 1, we need to determine the value of D. By 3.10, we
have
∗
q s, qθ
D
T
1, q∗1 , q∗2 1, q1 , q2
!
D 1
q1 q∗1
q2 q∗2
−
−
0 θ
0
−1
−1
ξ0
T
D 1, q∗1 , q∗2 e−iω0 τj ξ−θ dηθ 1, q1 , q2 eiω0 τj ξ dξ
T
1, q∗1 , q∗2 θeiω0 τj ξ dηθ 1, q1 , q2
"
#
$
D 1 q1 q∗1 q2 q∗2 τj f 0e−iω0 τj w1 q∗2 q1 w2 q1 q∗2 q2 q∗1
1.
3.14
Journal of Applied Mathematics
1.2
1.2
1
1
0.8
0.8
0.6
0.6
x2 (t)
x1 (t)
10
0.4
0.4
0.2
0.2
0
0
−0.2
0
200
400
600
800
−0.2
1000
0
200
400
600
800
1000
t
t
a
b
1.2
0.8
1
0.6
0.5
x3 (t)
x3 (t)
1
0.4
0
0.2
−0.5
1
0
−0.2
0
200
400
600
t
800
1000
0.5
x2 (
t)
c
0
−0.5 −0.5
0
0.5
1
)
x 1(t
d
Figure 2: When τ 1.7 < τ0 1.8434, the zero equilibrium is asymptotically stable. Here initial value is 1,
1, 1.
Therefore, we can choose D as
D
1
.
1 q1 q1∗ q2 q2∗ τj f 0eiω0 τj w1 q2∗ q1 w2 q1 q2∗ q2 q1∗
3.15
Following the algorithms given in 12 and using similar computation process in 7,
we can get that the coefficients which will be used to determine the important quantities:
#
$
g20 τj f 0De−2iω0 τj w1 q∗2 q12 w2 q∗2 q12 q∗1 q22 ,
%
&
g11 τj f 0D w1 q∗2 1 w2 q∗2 q1 q1 q∗1 q2 q2 ,
#
$
g02 τj f 0De2iω0 τj w1 q∗2 1 w2 q∗2 q21 q∗1 q22 ,
#
1
1
g21 τj f 0D w1 q∗2 W20 −1eiω0 τj 2W11 −1e−iω0 τj
2
2
1 w2 q∗2 W20 −1q1 eiω0 τj 2W11 −1q1 e−iω0 τj
$
3
3
q∗1 W20 −1q2 eiω0 τj 2W11 −1q2 e−iω0 τj ,
3.16
11
1.2
1.2
1
1
0.8
0.8
0.6
0.6
x2 (t)
x1 (t)
Journal of Applied Mathematics
0.4
0.4
0.2
0.2
0
0
−0.2
0
200
400
600
800
−0.2
1000
0
200
400
600
800
1000
t
t
a
b
1
0.6
1
0.4
0.5
x3 (t)
x3 (t)
0.8
0.2
0
0
−0.5
1
−0.2
−0.4
0
200
400
600
800
0.5
x2
(t)
1000
t
c
0
0
−0.5 −0.5
x1
0.5
(t)
1
d
Figure 3: When τ 1.9 > τ0 1.8434, a periodic orbit bifurcates from the zero equilibrium. Here, initial
value is 1, 1, 1.
where
W20 θ ig 02
ig20
q0eiω0 τj θ q0e−iω0 τj θ E1 e2iω0 τj θ ,
ω0 τj
3ω0 τj
3.17
ig
ig11
W11 θ −
q0eiω0 τj θ 11 q0e−iω0 τj θ E2 ,
ω0 τj
ω0 τj
moreover E1 , E2 satisfy the following equations, respectively,
⎛
⎜
⎜
⎝
2iω0 1
−f 0e−2iω0 τj
0
2iω0 1
0
⎞
⎛
⎜
⎟
−2iω0 τj ⎜
−f 0e−2iω0 τj ⎟
⎝
⎠E1 f 0e
q12
q22
−w1 f 0e−2iω0 τj −w2 f 0e−2iω0 τj
2iω0 1
w1 w2 q12
⎞
⎛
⎛
⎞
1
−f 0
q1 q1
0
⎟
⎜
⎜
⎟
⎟.
⎜
⎜
0
1
−f 0⎟
q2 q2
⎠
⎝
⎠E2 f 0⎝
−w1 f 0 −w2 f 0
w1 w2 q1 q1
1
⎞
⎟
⎟,
⎠
3.18
12
Journal of Applied Mathematics
Therefore, all gij in 3.16 can be expressed in terms of parameters. And we can compute the
following values:
c1 0 2 g21
i ,
g20 g11 − 2g11 −
2ω0 τj
2
μ2 −
Re{c1 0}
' ( ,
Re λ τj
3.19
β2 2 Re{c1 0},
' (
im{c1 0} μ2 im λ τj
,
T2 −
ω0 τj
j 0, 1, 2, . . . ,
which determine the qualities of bifurcating periodic solution in the center manifold at the
critical values τj , that is, μ2 determines the directions of the Hopf bifurcation: if μ2 > 0μ2 < 0,
then the Hopf bifurcation is supercritical subcritical and the bifurcating periodic solutions
exist for τ > τj τ < τj ; β2 determines the stability of the bifurcating periodic solutions: the
bifurcating periodic solutions are stable unstable if β2 < 0β2 > 0; T2 determines the period
of the bifurcating periodic solutions: the period increases decreases if T2 > 0T2 < 0.
4. Computer Simulation
In this section, we will confirm our theoretical analysis by numerical simulation. We give
an example of system 3.1 with w1 1, w2 −1, and f· tanh·. Then, f0 0 and
f 0 1.
Equation 1.3 becomes
ẋ1 t −x1 t tanhx2 t − τ,
ẋ2 t −x2 t tanhx3 t − τ,
4.1
ẋ3 t −x3 t tanhx1 t − τ − tanhx2 t − τ.
From 2.6, we have Δ 0.2870, α 1.0118, β −0.3295.
By Lemma 2.1, we know 2.7 has
√
roots z1 α β 0.6823 and z2,3 −α β/2 ± i 3/2α − β −0.3412
± 1.1615i. Clearly,
|z1 | 0.6823 < 1, |z2,3 | 1.4655 > 1. From 2.12, it follows that ω |z2,3 |2 − 1 0.6823,
and, from 2.13, we get τ0 1.8434. Thus, the zero equilibrium is asymptotically stable when
τ ∈ 0, τ0 as is illustrated by the computer simulations see Figures 2a–2d. When τ
passes through the critical value τ0 , zero equilibrium loses its stability and a Hopf bifurcation
occurs, that is, a family of periodic solutions bifurcates from the origins 0, 0, 0, which are
depicted in Figures 3a–3d.
Acknowledgment
The author is very grateful to the referees for their valuable suggestions, which help to improve the paper significantly.
Journal of Applied Mathematics
13
References
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