Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 918943, 11 pages
http://dx.doi.org/10.1155/2013/918943
Research Article
Stability and Hopf Bifurcation Analysis of
Coupled Optoelectronic Feedback Loops
Gang Zhu1,2 and Junjie Wei1
1
2
Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China
School of Education Science, Harbin University, Harbin, Heilongjiang 150086, China
Correspondence should be addressed to Junjie Wei; weijj@hit.edu.cn
Received 7 January 2013; Accepted 28 January 2013
Academic Editor: Patricia J. Y. Wong
Copyright © 2013 G. Zhu and J. Wei. 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.
The dynamics of a coupled optoelectronic feedback loops are investigated. Depending on the coupling parameters and the feedback
strength, the system exhibits synchronized asymptotically stable equilibrium and Hopf bifurcation. Employing the center manifold
theorem and normal form method introduced by Hassard et al. (1981), we give an algorithm for determining the Hopf bifurcation
properties.
1. Introduction
In recent research [1–5], it is found that even if several
individual systems behave chaotically, in the case where the
systems are identical, by proper coupling, the systems can
be made to evolve toward a situation of exact isochronal
synchronism. Synchronization phenomena are common in
coupled semiconductor systems, and they are important
examples of oscillators in general, and many works are concerned with coupled semiconductor systems [6–15].
We consider a feedback loop comprises a semiconductor
laser that serves as the optical source, a Mach-Zehnder electrooptic modulator, a photoreceiver, an electronic filter, and
an amplifier. The dynamics of the feedback loop can be
modeled by the delay differential equations [14, 15]:
d𝑥1 (𝑡)
= − (𝛾1 + 𝛾2 ) 𝑥1 (𝑡) − 𝛾2 𝑦1 (𝑡)
d𝑡
− 𝛽𝛾2 cos2 [𝑥1 (𝑡 − 𝜏) + 𝜑0 ] ,
is the dimensionless feedback strength, they are all positive
constants, and 𝜑0 is the bias point of the modulator.
Depending on the value of the feedback strength 𝛽 and
delay 𝜏, the loop, which is modeled by system (1), is capable
of producing dynamics ranging from periodic oscillations to
high-dimensional chaos [1, 14, 15].
We couple two nominally identical optoelectronic feedback loops unidirectionally, that is, the transmitter affects
the dynamics of the receiver but not vice versa. Thus, the
equations of motion describing the coupled system are given
by (1) for the transmitter and
d𝑥2 (𝑡)
= − (𝛾1 + 𝛾2 ) 𝑥2 (𝑡) − 𝛾2 𝑦2 (𝑡)
d𝑡
− 𝛽𝛾2 cos2 [𝑘𝑥1 (𝑡 − 𝜏) + (1 − 𝑘) 𝑥2 (𝑡 − 𝜏) + 𝜑0 ] ,
d𝑦2 (𝑡)
= 𝛾1 𝑥2 (𝑡) ,
d𝑡
(1)
(2)
Here, 𝑥1 (𝑡) is the normalized voltage signal applied to the
electrooptic modulator, 𝜏 is the feedback time delay, 𝛾1 and
𝛾2 are the filter low-pass and high-pass corner frequencies, 𝛽
for the receiver. In (2), 𝑘 > 0 denotes the coupling strength.
We will find that with the variety of 𝑘, the dynamical behavior
of the coupled system can be different, while the feedback
strength 𝛽 keeps the same value.
The paper is organized as follows. In Section 2, using
the method presented in [16], we study the stability, and
d𝑦1 (𝑡)
= 𝛾1 𝑥1 (𝑡) .
d𝑡
2
Abstract and Applied Analysis
the local Hopf bifurcation of the equilibrium of the coupled
system (1) and (2) by analyzing the distribution of the roots
of the associated characteristic equation. In Section 3, we
use the normal form method and the center manifold theory
introduced by Hassard et al. [17] to analyze the direction,
stability and the period of the bifurcating periodic solutions at
critical values of 𝛽. In Section 4, some numerical simulations
are carried out to illustrate the results obtained from the
analysis. In Section 5, we come to some conclusion about the
effect caused by the variety of parameters.
1
0
𝑓 −1
−2
−3
−4
−5
0
2. Stability Analysis
In this section, we consider the linear stability of the nonlinear coupled system
d𝑥1 (𝑡)
= − (𝛾1 + 𝛾2 ) 𝑥1 (𝑡) − 𝛾2 𝑦1 (𝑡)
d𝑡
2
4
d𝑦1 (𝑡)
= 𝛾1 𝑥1 (𝑡) ,
d𝑡
10
Figure 1: The points of intersection of 𝑓1 = tan 𝜔𝜏 and 𝑓2 = (−𝜔2 +
𝛾1 𝛾2 )/𝜔(𝛾1 + 𝛾2 ), when 𝛾1 = 0.1, 𝛾2 = 2.5, 𝜏 = 1.5.
𝜆2 + (𝛾1 + 𝛾2 ) 𝜆 + 𝛾1 𝛾2 − 𝛽𝛿𝛾2 𝜆𝑒−𝜆𝜏 = 0,
(6)
𝜆2 + (𝛾1 + 𝛾2 ) 𝜆 + 𝛾1 𝛾2 − (1 − 𝑘) 𝛽𝛿𝛾2 𝜆𝑒−𝜆𝜏 = 0.
(7)
2
− 𝛽𝛾2 cos [𝑘𝑥1 (𝑡 − 𝜏) + (1 − 𝑘) 𝑥2 (𝑡 − 𝜏) + 𝜑0 ] ,
d𝑥1 (𝑡)
= − (𝛾1 + 𝛾2 ) 𝑥1 (𝑡) − 𝛾2 𝑦1 (𝑡) + 𝛽𝛿𝛾2 𝑥1 (𝑡 − 𝜏) ,
d𝑡
𝜆 3,4 = −𝛾2 .
(9)
So, we have the following lemma.
Lemma 1. The equilibrium 𝐸0 (0, −𝛽cos2 𝜑0 , 0, −𝛽cos2 𝜑0 ) is
asymptotically stable when 𝛽 = 0.
Next, we regard 𝛽 as the bifurcation parameter to investigate the distribution of roots of (6) and (7).
Let 𝜆 = i𝜔 (𝜔 > 0) be a root of (6) and substituting 𝜆 = i𝜔
into (6), separating the real and imaginary parts yields
d𝑦1 (𝑡)
= 𝛾1 𝑥1 (𝑡) ,
d𝑡
−𝜔2 + 𝛾1 𝛾2 = 𝛽𝛿𝛾2 𝜔 sin 𝜔𝜏,
𝜔 (𝛾1 + 𝛾2 ) = 𝛽𝛿𝛾2 𝜔 cos 𝜔𝜏.
d𝑥2 (𝑡)
= − (𝛾1 + 𝛾2 ) 𝑥2 (𝑡) − 𝛾2 𝑦2 (𝑡)
d𝑡
(8)
whose roots are
𝜆 1,2 = −𝛾1 ,
(3)
It is easy to see that 𝐸(0, −𝛽cos2 𝜑0 , 0, −𝛽cos2 𝜑0 ) is the
only equilibrium of system (3). Linearizing system (3) around
𝐸 and denote 𝛿 = sin 2𝜑0 , we get the linearization system
(10)
Then, we can get
+ 𝑘𝛽𝛿𝛾2 𝑥1 (𝑡 − 𝜏) + (1 − 𝑘) 𝛽𝛿𝛾2 𝑥2 (𝑡 − 𝜏) ,
tan 𝜔𝜏 =
(4)
and the characteristic equation of system (4)
× [𝜆2 + (𝛾1 + 𝛾2 ) 𝜆 + 𝛾1 𝛾2 − 𝛽𝛿𝛾2 𝜆𝑒−𝜆𝜏 ] = 0,
8
[𝜆2 + (𝛾1 + 𝛾2 ) 𝜆 + 𝛾1 𝛾2 ] = 0,
2
[𝜆2 + (𝛾1 + 𝛾2 ) 𝜆 + 𝛾1 𝛾2 − (1 − 𝑘) 𝛽𝛿𝛾2 𝜆𝑒−𝜆𝜏 ]
6
Notice that when 𝛽 = 0, (5) becomes
d𝑥2 (𝑡)
= − (𝛾1 + 𝛾2 ) 𝑥2 (𝑡) − 𝛾2 𝑦2 (𝑡)
d𝑡
d𝑦2 (𝑡)
= 𝛾1 𝑥2 (𝑡) ,
d𝑡
𝜔
which is equivalent to
− 𝛽𝛾2 cos2 [𝑥1 (𝑡 − 𝜏) + 𝜑0 ] ,
d𝑦2 (𝑡)
= 𝛾1 𝑥2 (𝑡) .
d𝑡
2
(5)
−𝜔2 + 𝛾1 𝛾2
.
𝜔 (𝛾1 + 𝛾2 )
(11)
Hence, (11) has a sequence of roots {𝜔𝑗 }𝑗≥0 (see Figure 1), and
2𝑗𝜋 2𝑗𝜋 + 𝜋/2
{
,
),
𝜔𝑗2 < 𝛾1 𝛾2 ,
(
{
{
𝜏
{ 𝜏
𝜔𝑗 ∈ {
{
2 (𝑗 + 1) 𝜋
{
{( 2𝑗𝜋 + 3𝜋/2 ,
) , 𝜔𝑗2 > 𝛾1 𝛾2 ,
𝜏
𝜏
{
𝑗 = 0, 1, 2, . . . .
(12)
Abstract and Applied Analysis
3
Define
where
𝛽𝑗 =
𝛾1 + 𝛾2
.
𝛿𝛾2 cos 𝜔𝑗 𝜏
2
𝛾𝛾
1
2
𝛽2 = 2 2 [( 1 2 − 𝜔) + (𝛾1 + 𝛾2 ) ] ,
𝜔
𝛿 𝛾2
2
(14)
which gives that
d𝛽
1
=
(𝜔2 + 𝛾1 𝛾2 ) (𝜔2 − 𝛾1 𝛾2 ) .
d𝜔 𝛽𝛿2 𝛾22 𝜔3
Lemma 2. There exists a sequence values of 𝛽 denoted by
0 < 𝛽0 < 𝛽1 < ⋅ ⋅ ⋅ ,
(16)
such that (6) has a pair of imaginary roots ±𝑖𝜔𝑗 when 𝛽 =
𝛽𝑗 (𝑗 = 0, 1, 2, . . .), where 𝛽𝑗 is defined by (13), and 𝜔𝑗 is the
root of (11).
(21)
As to (7), it can be easily found that −𝛾1 , −𝛾2 are two
negative roots when 𝑘 = 1, so, next, we only focus on (7) with
𝑘 ≠ 1.
Let 𝜆 = 𝑖𝜛(𝜛) > 0 be a root of (7). Using the same method
above, we get
−𝜛2 + 𝛾1 𝛾2 = (1 − 𝑘) 𝛽𝛿𝛾2 𝜛 sin 𝜛𝜏,
(15)
From Figure 1, we know that 𝜔𝑗 → ∞ when 𝑗 → ∞,
which means that 𝜔𝑗2 > 𝛾1 𝛾2 ; furthermore, 𝛽 is increasing
with respect to 𝜔, when 𝑗 is sufficiently big.
Reorder the set {𝛽𝑗 } such that 𝛽0 = min{𝛽𝑗 } and 𝜔𝑗
is correspondent of 𝛽𝑗 (𝑗 = 0, 1, 2, . . .). Then, we have the
following lemma.
(𝛾1 + 𝛾2 ) 𝜛 = (1 − 𝑘) 𝛽𝛿𝛾2 𝜛 cos 𝜛𝜏,
tan 𝜛𝜏 =
−𝜛2 + 𝛾1 𝛾2
.
𝜛 (𝛾1 + 𝛾2 )
(22)
(23)
Then, when 0 < 𝑘 < 1, (23) has a sequence of roots
{𝜛𝑗 }𝑗≥0 , which are the same as those of (11).
When 𝑘 > 1, (23) has a sequence of roots {𝜛𝑗 }𝑗≥0 , and
(2𝑗 + 1) 𝜋 (2𝑗 + 1) 𝜋 + 𝜋/2
{
,
) , 𝜛𝑗2 < 𝛾1 𝛾2 ,
(
{
{
{
𝜏
𝜏
𝜛𝑗 ∈ {
{
{
{( 2𝑗𝜋 + 𝜋/2 , (2𝑗 + 1) 𝜋 ) ,
𝜛𝑗2 > 𝛾1 𝛾2 , (24)
𝜏
𝜏
{
𝑗 = 0, 1, 2, . . . .
Define
Let
𝜆 (𝜏) = 𝛼 (𝛽) + 𝑖𝜔 (𝛽)
Lemma 3. 𝛼󸀠 (𝛽𝑗 ) > 0.
Proof. Substituting 𝜆(𝛽) into (6) and taking the derivative
with respect to 𝛽, it follows that
d𝜆
d𝜆
d𝜆
+ (𝛾1 + 𝛾2 )
− 𝛿𝛾2 𝜆𝑒−𝜆𝜏 − 𝛽𝛿𝛾2 𝑒−𝜆𝜏
d𝛽
d𝛽
d𝛽
−𝜆𝜏 d𝜆
+ 𝜏𝛽𝛿𝛾2 𝜆𝑒
d𝛽
(18)
(19)
𝛼 (𝛽𝑗 ) =
𝛽𝑗 Δ
Then, (𝜛𝑗 , 𝛽𝑗 ) is the solution of (22).
Repeat the previous process, we have
d𝛽
1
(𝜛2 + 𝛾1 𝛾2 ) (𝜛2 − 𝛾1 𝛾2 ) .
=
2
d𝜛 (1 − 𝑘) 𝛽𝛿2 𝛾22 𝜛3
Lemma 4. There exists a sequence values of 𝛽 denoted by
0 < 𝛽0 < 𝛽1 < ⋅ ⋅ ⋅ ,
+
𝛾22 )
+
(𝜔𝑗2
(27)
such that (7) has a pair of imaginary roots ±𝑖𝜛𝑗 when 𝛽 =
𝛽𝑗 (𝑗 = 0, 1, 2, . . .), where 𝛽𝑗 is defined by (25), and 𝜛𝑗 is the
root of (23).
𝜆 (𝜏) = 𝛼 (𝛽) + i𝜛 (𝛽)
(𝛾12
(26)
Let
and by a straight computation, we get
[𝜏𝜔𝑗2
(25)
Reorder the set {𝛽𝑗 } such that 𝛽0 = min{𝛽𝑗 } and 𝜛𝑗 is
Therefore, noting that 𝛽𝛿𝛾2 𝜆𝑒−𝜆𝜏 = 𝜆2 + (𝛾1 + 𝛾2 )𝜆 + 𝛾1 𝛾2 , we
have
𝜔𝑗2
𝛾1 + 𝛾2
.
(1 − 𝑘) 𝛿𝛾2 cos 𝜛𝜏
correspondent of 𝛽𝑗 (𝑗 = 0, 1, 2, . . .).
= 0.
𝜆3 + (𝛾1 + 𝛾2 ) 𝜆2 + 𝛾1 𝛾2 𝜆
d𝜆 1
,
=
2
d𝛽 𝛽 𝜆 − 𝛾1 𝛾2 + 𝜏 [𝜆3 + (𝛾1 + 𝛾2 ) 𝜆2 + 𝛾1 𝛾2 𝜆]
𝛽𝑗 =
(17)
be the root of (6) satisfying 𝛼(𝛽𝑗 ) = 0 and 𝜔(𝛽𝑗 ) = 𝜔𝑗 . We
have the following conclusion.
󸀠
2
+ [𝜏𝜔𝑗 (𝜔𝑗2 − 𝛾1 𝛾2 )] .
Then, (𝜔𝑗 , 𝛽𝑗 ) is the solution of (10).
From (10), we know that
2𝜆
Δ = [(𝜔𝑗2 + 𝛾1 𝛾2 ) + 𝜏𝜔𝑗2 (𝛾1 + 𝛾2 )]
(13)
+ 𝛾1 𝛾2 ) (𝛾1 + 𝛾2 )
+𝜏𝜔𝑗4 + 𝜏𝛾12 𝛾22 ] > 0,
(20)
(28)
be the root of (7) satisfying 𝛼(𝛽𝑗 ) = 0, 𝜛(𝛽𝑗 ) = 𝜛𝑗 . Then,
similar to the proof of Lemma 3, we have the following
conclusion.
4
Abstract and Applied Analysis
Lemma 5. 𝛼󸀠 (𝛽𝑗 ) > 0.
Clearly, the phase space is C = C([−𝜏, 0], R4 ). For
convenience, let
Compare 𝛽𝑗 , 𝛽𝑗 and reorder the set {𝛽𝑗 } and {𝛽𝑗 } and
𝛽∗ ∈ {𝛽𝑗 } ∪ {𝛽𝑗 } ,
remove the “−” of 𝛽𝑗 , such that
0 < 𝛽0 < 𝛽1 < ⋅ ⋅ ⋅ ,
(29)
then from previous lemmas and the Hopf bifurcation theorem for functional differential equations [18], we have the
following results on stability and bifurcation to system (3).
Theorem 6. For system(3), the equilibrium 𝐸 is asymptotically
stable when 𝛽 ∈ [0, 𝛽0 ) and unstable when 𝛽 ∈ (𝛽0 , +∞);
system (3) undergoes a Hopf bifurcation at 𝐸 when 𝛽 = 𝛽𝑗 ,
𝑗 = 0, 1, 2, . . ., where 𝛽𝑗 are defined by (13) or (25).
and 𝛽 = 𝛽∗ + 𝜇, 𝜇 ∈ R. From the analysis above we know
that 𝜇 = 0 is the Hopf bifurcation value for system(30).
Let i𝜔∗ be the root of the characteristic equation associate
with the linearization of system (30) when 𝛽 = 𝛽∗ . For
𝜙 = (𝜙1 , 𝜙2 , 𝜙3 , 𝜙4 ) ∈ C, let
𝐿 𝜇 (𝜙) = 𝐵𝜙 (0) + 𝐶𝜙 (−𝜏) ,
d𝑥1 (𝑡)
= − (𝛾1 + 𝛾2 ) 𝑥1 (𝑡) − 𝛾2 𝑦1 (𝑡) + 𝛽𝛿𝛾2 𝑥1 (𝑡 − 𝜏)
d𝑡
− (𝛾1 + 𝛾2 ) −𝛾2
0
0
0
0
0
𝛾1
𝐵=(
),
0
0 − (𝛾1 + 𝛾2 ) −𝛾2
0
0
𝛾1
0
𝛽𝛿𝛾2
0
𝐶=(
𝑘𝛽𝛿𝛾2
0
0
0
0
0
0 (1 − 𝑘) 𝛽𝛿𝛾2
0
0
0
0
).
0
0
(33)
By the Rieze representation theorem, there exists a 4 × 4
matrix, 𝜂(𝜃, 𝜇) (−𝜏 ≤ 𝜃 ≤ 0), whose elements are of bounded
variation functions such that
0
𝐿 𝜇 (𝜙) = ∫ d𝜂 (𝜃, 𝜇) 𝜙 (𝜃) ,
2
+ 𝛽𝛾2 𝜌𝜑0 𝑥12 (𝑡 − 𝜏) − 𝛽𝛿𝛾2 𝑥13 (𝑡 − 𝜏) + 𝑂 (4) ,
3
d𝑦1 (𝑡)
= 𝛾1 𝑥1 (𝑡) ,
d𝑡
(32)
where
3. The Direction and Stability of
the Hopf Bifurcation
In Section 2 we obtained some conditions under which
system (3) undergoes the Hopf bifurcation at some critical
values of 𝛽. In this section, we study the direction, stability,
and the period of the bifurcating periodic solutions. The
method we used is based on the normal form method and
the center manifold theory introduced by Hassard et al. [17].
Move 𝐸(0, −𝛽cos2 𝜑0 , 0, −𝛽cos2 𝜑0 ) to the origin 𝑂(0, 0,
0, 0) and denote 𝛿 = sin 2𝜑0 , 𝜌 = cos 2𝜑0 , then system (3)
can be written as the form
(31)
𝜙 ∈ C.
−𝜏
(34)
In fact, we can choose
𝐵,
𝜃 = 0,
{
{
𝜂 (𝜃, 𝜇) = {0,
𝜃 ∈ (−𝜏, 0)
{
{−𝐶, 𝜃 = −𝜏.
d𝑥2 (𝑡)
= − (𝛾1 + 𝛾2 ) 𝑥2 (𝑡) − 𝛾2 𝑦2 (𝑡)
d𝑡
+ 𝛽𝛾2 [𝑘𝛿𝑥1 (𝑡 − 𝜏) + (1 − 𝑘) 𝛿𝑥2 (𝑡 − 𝜏)
Then, (30) is satisfied.
For 𝜙 ∈ C, define the operator 𝐴(𝜇) as
+ 𝑘2 𝜌𝑥12 (𝑡 − 𝜏)
+ 2𝑘 (1 − 𝑘) 𝜌𝑥1 (𝑡 − 𝜏) 𝑥2 (𝑡 − 𝜏)
2
+ (1 − 𝑘)2 𝜌𝑥22 (𝑡 − 𝜏) − 𝑘3 𝛿𝑥13 (𝑡 − 𝜏)
3
2
− 2𝑘 (1 −
𝑘) 𝛿𝑥12
(35)
(𝑡 − 𝜏) 𝑥2 (𝑡 − 𝜏)
− 2𝑘(1 − 𝑘)2 𝛿𝑥1 (𝑡 − 𝜏) 𝑥22 (𝑡 − 𝜏)
d𝜙 (𝜃)
{
,
{
{
{ d𝜃
𝐴 (𝜇) 𝜙 (𝜃) = { 0
{
{
{∫ d𝜂 (𝑡, 𝜇) 𝜙 (𝑡) ,
{ −𝜏
𝜃 ∈ [−𝜏, 0) ,
(36)
𝜃 = 0,
and 𝑅(𝜇)𝜙 as
2
− (1 − 𝑘)3 𝛿𝑥23 (𝑡 − 𝜏) ] + 𝑂 (4) ,
3
0,
𝜃 ∈ [−𝜏, 0) ,
𝑅 (𝜇) 𝜙 (𝜃) = {
𝑓 (𝜇, 𝜙) , 𝜃 = 0,
d𝑦2 (𝑡)
= 𝛾1 𝑥2 (𝑡) .
d𝑡
(30)
where
(37)
Abstract and Applied Analysis
5
2
𝜌𝜙12 (−𝜏) − 𝛿𝜙13 (−𝜏) + 𝑂 (4)
3
(0
)
(
)
(
)
( 𝑘2 𝜌𝜙2 (−𝜏) + 𝑘 (1 − 𝑘) 𝜌𝜙 (−𝜏) 𝜙 (−𝜏)
)
(
)
1
3
1
)
∗ (
𝑓 (𝜇, 𝜙) = 𝛽 𝛾2 (
).
(
)
2
( +(1 − 𝑘)2 𝜌𝜙32 (−𝜏) − 𝑘3 𝛿𝜙13 (−𝜏) − 2𝑘2 (1 − 𝑘) 𝛿𝜙12 (−𝜏) 𝜙3 (𝑡 − 𝜏) )
(
)
3
(
)
(
)
2
−2𝑘(1 − 𝑘)2 𝛿𝜙1 (−𝜏) 𝜙32 (−𝜏) − (1 − 𝑘)3 𝛿𝜙33 (−𝜏) + 𝑂 (4)
3
(0
)
Then, system (30) is equivalent to the following operator
equation:
𝑢̇ 𝑡 = 𝐴 (𝜇) 𝑢𝑡 + 𝑅 (𝜇) 𝑢𝑡 ,
d𝜓 (𝑠)
{
,
𝑠 ∈ (0, 𝜏] ,
−
{
{
{ d𝑠
∗
𝐴 𝜓 (𝑠) = { 0
{
{
{∫ 𝜓 (−𝜉) d𝜂 (𝜉, 0) , 𝑠 = 0.
{ −𝜏
(40)
For 𝜙 ∈ C[−𝜏, 0] and 𝜓 ∈ C[0, 𝜏], define the bilinear
form
⟨𝜓 (𝑠) , 𝜙 (𝜃)⟩ = 𝜓 (0) 𝜙 (0)
0
𝜃
− ∫ ∫ 𝜓 (𝜉 − 𝜃) d𝜂 (𝜃) 𝜙 (𝜉) d𝜉,
On the center manifold C0 , we have
𝑊 (𝑡, 𝜃) = 𝑊 (𝑧 (𝑡) , 𝑧 (𝑡) , 𝜃) ,
(39)
where 𝑢(𝑡) = (𝑥1 (𝑡), 𝑦1 (𝑡), 𝑥2 (𝑡), 𝑦2 (𝑡))𝑇 , 𝑢𝑡 = 𝑢(𝑡 + 𝜃), for
𝜃 ∈ [−𝜏, 0].
For 𝜓 ∈ C1 ([0, 𝜏], R4 ), define
𝑊 (𝑧, 𝑧, 𝜃) = 𝑊20
𝑞∗ (𝑠) =
∗
𝛾
𝛾
1
(1, 2∗ , 1, 2∗ ) 𝑒𝑖𝜔 𝑠 ,
𝑖𝜔
𝑖𝜔
𝐷
𝑧2
𝑧2
+ 𝑊11 𝑧𝑧 + 𝑊02 + ⋅ ⋅ ⋅ ,
2
2
(46)
𝑧 and 𝑧 are local coordinates for center manifold C0 in the
direction of 𝑞∗ and 𝑞∗ . Note that 𝑊 is real if 𝑢𝑡 is real. We
only consider real solutions.
For solution 𝑢𝑡 in C0 , since 𝜇 = 0, we have
𝑧̇ (𝑡) = 𝑖𝜔∗ 𝑧 + ⟨𝑞∗ (𝜃) , 𝑓 (0, 𝑊 + 2 Re {𝑧 (𝑡) 𝑞 (𝜃)})⟩
= 𝑖𝜔∗ 𝑧 + 𝑞∗ (0) , 𝑓 (0, 𝑊 (𝑧, 𝑧, 0) + 2 Re {𝑧 (𝑡) 𝑞 (0)})
(41)
= 𝑖𝜔∗ 𝑧 + 𝑞∗ (0) 𝑓0 (𝑧, 𝑧) .
where 𝜂(𝜃) = 𝜂(𝜃, 0). Then, 𝐴(0) and 𝐴∗ are adjoint
operators.
Let 𝑞(𝜃) and 𝑞∗ (𝑠) be eigenvectors of 𝐴(0) and 𝐴∗
associated to 𝑖𝜔∗ and −𝑖𝜔∗ , respectively. It is not difficult with
verify that
𝛾1 𝑇 𝑖𝜔∗ 𝜃
𝛾1
,
1,
) 𝑒 ,
𝑖𝜔∗
𝑖𝜔∗
(45)
where
−𝜏 0
𝑞 (𝜃) = (1,
(38)
(47)
We rewrite this equation as
𝑧̇ (𝑡) = 𝑖𝜔∗ 𝑧 + 𝑔 (𝑧, 𝑧) ,
(48)
where
(42)
𝑔 (𝑧, 𝑧) = 𝑔20
𝑧2
𝑧2
𝑧2 𝑧
+ 𝑔11 𝑧𝑧 + 𝑔02 + 𝑔21
⋅⋅⋅.
2
2
2
(49)
where
𝐷=2+
∗
2𝛾1 𝛾2
+ 2𝛽∗ 𝛿𝛾2 𝜏𝑒−i𝜔 𝜏 .
∗2
𝜔
By (39) and (48), we have
(43)
Then, ⟨𝑞∗ (𝑠), 𝑞(𝜃)⟩ = 1, ⟨𝑞∗ (𝑠), 𝑞(𝜃)⟩ = 0.
Let 𝑢𝑡 be the solution of (39) and define
𝑧 (𝑡) = ⟨𝑞∗ , 𝑢𝑡 ⟩ ,
𝑊 (𝑡, 𝜃) = 𝑢𝑡 (𝜃) − 2 Re {𝑧 (𝑡) 𝑞 (𝜃)} .
(44)
̇
̇ − 𝑧𝑞
𝑊̇ = 𝑢̇ 𝑡 − 𝑧𝑞
𝐴𝑊 − 2 Re {𝑞∗ (0) 𝑓0 𝑞 (𝜃)} ,
= {
𝐴𝑊 − 2 Re {𝑞∗ (0) 𝑓0 𝑞 (0)} + 𝑓0 ,
= 𝐴𝑊 + 𝐻 (𝑧, 𝑧, 𝜃) ,
𝜃 ∈ [−𝜏, 0) ,
(50)
𝜃 = 0,
6
Abstract and Applied Analysis
−0.32
0.3
−0.34
0.25
−0.36
−0.38
0.15
𝑦1 (𝑡)
𝑥1 (𝑡)
0.2
0.1
−0.44
0.05
−0.46
0
−0.05
−0.4
−0.42
−0.48
0
50
100
150
𝑡
200
250
−0.5
300
0
50
100
(a)
−0.32
0.25
−0.34
250
300
150
𝑡
200
250
300
−0.36
0.2
−0.38
0.15
𝑦2 (𝑡)
𝑥2 (𝑡)
200
(b)
0.3
0.1
−0.4
−0.42
−0.44
0.05
−0.46
0
−0.05
150
𝑡
−0.48
0
50
100
150
𝑡
200
250
300
−0.5
0
50
100
(c)
(d)
Figure 2: 𝛾1 = 0.1, 𝛾2 = 2.5, 𝜏 = 1.5, 𝑘 = 1.9, which means that condition (𝐻1 ) holds, and 𝛽 = 0.7 < 𝛽0 . The initial value is (0.1, −0.5, 0.1, −0.5).
where
𝐻 (𝑧, 𝑧, 𝜃) = 𝐻20 (𝜃)
𝑧2
𝑧2
+ 𝐻11 (𝜃) 𝑧𝑧 + 𝐻02 (𝜃)
+ ⋅⋅⋅.
2
2
(51)
Expanding the above series and comparing the coefficients,
we obtain
(𝐴 − 2𝑖𝜔∗ 𝐼) 𝑊20 (𝜃) = −𝐻20 (𝜃) ,
𝐴𝑊11 (𝜃) = −𝐻11 (𝜃) , . . . .
(52)
Notice that
𝑞 (𝜃) = (1,
𝛾1
𝛾1 𝑇 𝑖𝜔∗ 𝜃
,
1,
) 𝑒 ,
𝑖𝜔∗
𝑖𝜔∗
(53)
2𝛽∗ 𝛾2 𝜌 −2𝑖𝜔∗ 𝜏 2
(𝑘 − 𝑘 + 2) ,
𝑒
𝐷
𝑔11 =
2𝛽∗ 𝛾2 𝜌 2
(𝑘 − 𝑘 + 2) ,
𝐷
𝑔02 =
2𝛽∗ 𝛾2 𝜌 2𝑖𝜔∗ 𝜏 2
(𝑘 − 𝑘 + 2) ,
𝑒
𝐷
{
{
{
{
{
∗
∗
(1)
(1)
𝜌 (𝑒𝑖𝜔 𝜏 𝑊20
(−𝜏) + 2𝑒−𝑖𝜔 𝜏 𝑊11
(−𝜏))
{
{
{
{
{
{
∗
∗
(1)
(1)
+ 𝑘2 𝜌 (𝑒𝑖𝜔 𝜏 𝑊20
(−𝜏) + 2𝑒−𝑖𝜔 𝜏 𝑊11
(−𝜏))
+ 𝑘 (1 − 𝑘) 𝜌
where
𝑊 (𝑧, 𝑧, 𝜃) =
𝑔20 =
2𝛽∗ 𝛾2
𝑔21 =
𝐷
𝑢𝑡 (𝜃) = 𝑧𝑞 (𝜃) + 𝑧𝑞 (𝜃) + 𝑊 (𝑧, 𝑧, 𝜃) ,
(𝑖)
Combing (38) and by straightforward computation, we can
obtain the coefficients which will be used in determining the
important quantities:
(𝑖)
𝑊20
𝑧2
(𝑖)
+ 𝑊11
(𝜃)
(𝜃) 𝑧𝑧
2
𝑧2
(𝑖)
+ 𝑊02
+ ⋅⋅⋅,
(𝜃)
2
𝑖 = 1, 2, 3, 4.
∗
∗
(3)
× (𝑒−𝑖𝜔 𝜏 𝑊11
(−𝜏) + 𝑒𝑖𝜔
(54)
∗
+𝑒𝑖𝜔
(1)
𝜏 𝑊20
(3)
𝜏 𝑊20
(−𝜏)
2
∗
(−𝜏)
(1)
+ 𝑒−𝑖𝜔 𝜏 𝑊11
(−𝜏))
2
Abstract and Applied Analysis
7
−0.4
0.5
0.4
−0.45
0.3
−0.5
0.2
0.1
0
−0.1
−0.2
𝑦1 (𝑡)
𝑥1 (𝑡)
−0.55
−0.65
−0.7
−0.3
−0.4
−0.5
−0.6
−0.75
0
100
200
300
𝑡
400
500
−0.8
600
0
100
200
(a)
500
600
300
𝑡
400
500
600
(b)
−0.45
0.3
0.2
0.1
−0.5
𝑦2 (𝑡)
𝑥2 (𝑡)
400
−0.4
0.5
0.4
0
−0.1
−0.55
−0.6
−0.65
−0.2
−0.7
−0.3
−0.4
−0.5
300
𝑡
−0.75
0
100
200
300
𝑡
400
500
600
−0.8
0
100
(c)
200
(d)
Figure 3: 𝛾1 = 0.1, 𝛾2 = 2.5, 𝜏 = 1.5, 𝑘 = 1.9, which means that condition (𝐻1 ) holds, and 𝛽 = 1.2 > 𝛽0 . The initial value is (0.1, −0.5, 0.1, −0.5).
∗
(3)
+ (1 − 𝑘)2 𝜌 (2𝑒−𝑖𝜔 𝜏 𝑊11
(−𝜏)
which implies that
}
}
}
}
∗
∗ }
𝑖𝜔 𝜏 (3)
−𝑖𝜔 𝜏
+𝑒 𝑊20 (−𝜏) ) − 4𝛿𝑒
.
}
}
}
}
}
}
(55)
We still need to compute 𝑊20 (𝜃) and 𝑊11 (𝜃), for 𝜃 ∈ [−𝜏, 0).
We have
𝐻 (𝑧, 𝑧, 𝜃) = −𝑞∗ (0) 𝑓0 𝑞 (𝜃) − 𝑞∗ (0) 𝑓0 𝑞 (𝜃)
= −𝑔 (𝑧, 𝑧) 𝑞 (𝜃) − 𝑔 (𝑧, 𝑧) 𝑞 (𝜃) .
𝐻11 = −𝑔11 𝑞 (𝜃) − 𝑔11 𝑞 (𝜃) .
(59)
Here, 𝐸 and 𝐹 are both four-dimensional vectors and can be
determined by setting 𝜃 = 0 in 𝐻(𝑧, 𝑧, 𝜃). In fact, from (38)
and
𝐻 (𝑧, 𝑧, 0) = −2 Re {𝑞∗ (0) 𝑓0 𝑞 (0)} + 𝑓0 ,
(60)
𝐻20 (0) = − 𝑔20 𝑞 (0) − 𝑔02 𝑞 (0)
𝑇
∗
(57)
𝐻11 (0) = − 𝑔11 𝑞 (0) − 𝑔11 𝑞 (0)
(61)
𝑇
+ 2𝛽∗ 𝛾2 𝜌(1, 0, 𝑘2 − 𝑘 + 1, 0) .
It follows from (52) and the definition of 𝐴 that
∗
𝑊̇ 11 (𝜃) = 𝑔11 𝑞 (𝜃) + 𝑔11 𝑞 (𝜃) ,
∗
𝑔 𝑞 (0)
𝑔 𝑞 (0) ∗
𝑊11 (𝜃) = 11 ∗ 𝑒𝑖𝜔 𝜃 + 11 ∗ 𝑒−𝑖𝜔 𝜃 + 𝐹.
𝑖𝜔
−𝑖𝜔
+ 2𝛽∗ 𝛾2 𝜌𝑒−2𝑖𝜔 𝜏 (1, 0, 𝑘2 − 𝑘 + 1, 0) ,
Then, from (52), we get
𝑊̇ 20 (𝜃) = 2𝑖𝜔 𝑊20 (𝜃) + 𝑔20 𝑞 (𝜃) + 𝑔02 𝑞 (𝜃) ,
∗
𝑔20 𝑞 (0) 𝑖𝜔∗ 𝜃 𝑔02 𝑞 (0) −𝑖𝜔𝑗∗ 𝜃
𝑒
+
𝑒
+ 𝐸𝑒2i𝜔𝑗 𝜃 ,
−𝑖𝜔∗
−3𝑖𝜔∗
we have
(56)
Comparing the coefficients about 𝐻(𝑧, 𝑧, 𝜃) gives that
𝐻20 (𝜃) = −𝑔20 𝑞 (𝜃) − 𝑔02 𝑞 (𝜃) ,
𝑊20 (𝜃) =
(58)
𝛽∗ 𝐵𝑊20 (0) + 𝛽∗ 𝐶𝑊20 (−𝜏) = 2𝑖𝜔∗ 𝑊20 (0) − 𝐻20 (0) ,
𝛽∗ 𝐵𝑊11 (0) + 𝛽∗ 𝐶𝑊11 (−𝜏) = −𝐻11 (0) ,
(62)
8
Abstract and Applied Analysis
0.3
−0.25
0.25
−0.3
0.15
𝑦1 (𝑡)
𝑥1 (𝑡)
0.2
0.1
−0.35
−0.4
0.05
−0.45
0
−0.05
0
50
100
150
𝑡
200
250
−0.5
300
0
50
100
(a)
150
𝑡
200
250
300
150
𝑡
200
250
300
(b)
−0.25
0.3
0.25
−0.3
0.15
𝑦2 (𝑡)
𝑥2 (𝑡)
0.2
0.1
0.05
−0.4
−0.45
0
−0.05
−0.35
0
50
100
150
𝑡
200
250
300
−0.5
0
50
100
(c)
(d)
Figure 4: 𝛾1 = 0.1, 𝛾2 = 2.5, 𝜏 = 1.5, 𝑘 = 3, which means that condition (𝐻2 ) holds, and 𝛽 = 0.6 < 𝛽0 . The initial value is
(0.1, −0.5, 0.1, −0.5).
which implies that
+ 𝐶(
−1
∗
𝐸 = (𝐵 + 𝑒−2𝑖𝜔 𝜏 𝐶 − 2𝑖𝜔∗ I)
+
𝑇
(63)
[
𝑔20 𝑞 (0) −𝑖𝜔∗ 𝜏 𝑔02 𝑞 (0) 𝑖𝜔∗ 𝜏
𝑒
+
𝑒 )
𝑖𝜔∗
3𝑖𝜔∗
𝑖
󵄨 󵄨2
(𝑔 𝑔 − 2 󵄨󵄨󵄨𝑔11 󵄨󵄨󵄨 −
2𝜔∗ 20 11
𝜇2 = −
]
∗
𝑇]
×2𝛽∗ 𝛾2 𝜌𝑒−2𝑖𝜔 𝜏 (1, 0, 𝑘2 − 𝑘 + 1, 0) ]
],
]
]
𝑔11 𝑞 (0) 𝑔11 𝑞 (0)
+
)
−𝑖𝜔∗
𝑖𝜔∗
Consequently, the above 𝑔21 can be expressed by the parameters and delay in system (30). Thus, we can compute the
following quantities:
𝑐1 (0) =
1
+ ∗ (𝑔20 𝑞 (0) + 𝑔02 𝑞 (0))
𝛽
𝐹 = (𝐵 + 𝐶)−1 [𝐵 (
1
(𝑔 𝑞 (0) + 𝑔11 𝑞 (0))
𝛽∗ 11
−2𝛽∗ 𝛾2 𝜌(1, 0, 𝑘2 − 𝑘 + 1, 0) ] .
[
[
𝑔20 𝑞 (0) 𝑔02 𝑞 (0)
×[
[𝐵 ( 𝑖𝜔∗ + 3𝑖𝜔∗ )
[
+ 𝐶(
𝑔11 𝑞 (0) −𝑖𝜔∗ 𝜏 𝑔11 𝑞 (0) 𝑖𝜔∗ 𝜏
𝑒
+
𝑒 )
−𝑖𝜔∗
𝑖𝜔∗
𝑔
1 󵄨󵄨 󵄨󵄨2
󵄨󵄨𝑔20 󵄨󵄨 ) + 21 ,
3
2
Re 𝑐1 (0)
,
Re 𝜆󸀠 (𝛽∗ )
(64)
𝛽2 = 2 Re 𝑐1 (0) ,
𝑇2 = −
Im 𝑐1 (0) + 𝜇2 Im 𝜆󸀠 (𝛽∗ )
,
𝜔∗
which determine the properties of bifurcating periodic solutions at the critical value 𝜏0 . The direction and stability of the
Abstract and Applied Analysis
9
−0.32
0.3
−0.34
0.25
−0.36
−0.38
0.15
−0.4
𝑦1 (𝑡)
𝑥1 (𝑡)
0.2
0.1
−0.42
−0.44
0.05
−0.46
0
−0.05
− 0.48
0
50
100
150
𝑡
200
250
−0.5
300
0
50
100
150
𝑡
0.3
−0.33
0.2
−0.335
0.1
−0.34
0
−0.1
300
−0.345
−0.35
−0.355
−0.2
−0.36
−0.3
−0.4
1000
250
(b)
𝑦2 (𝑡)
𝑥2 (𝑡)
(a)
200
−0.365
1100
1200
𝑡
1300
1400
1500
−0.37
1000
1100
1200
(c)
𝑡
1300
1400
1500
(d)
Figure 5: 𝛾1 = 0.1, 𝛾2 = 2.5, 𝜏 = 1.5, 𝑘 = 3, which means that condition (𝐻2 ) holds, and 𝛽0 < 𝛽 = 0.7 < 𝛽0 . The initial value is
(0.1, −0.5, 0.1, −0.5).
Hopf bifurcation in the center manifold can be determined
by 𝜇2 and 𝛽2 , respectively. In fact, if 𝜇2 > 0 (𝜇2 < 0), then
the bifurcating periodic solutions are forward (backward);
the bifurcating periodic solutions on the center manifold are
stable (unstable) if 𝛽2 < 0 (𝛽2 > 0); and 𝑇2 determines
the period of the bifurcating periodic solutions: the period
increases (decreases) if 𝑇2 > 0 (𝑇2 < 0).
From the discussion in Section 2, we have known that
Re 𝜆󸀠 (𝛽𝑗 ) > 0; therefore; we have the following result.
Theorem 7. The direction of the Hopf bifurcation for system
(3) at the equilibrium 𝐸(0, −𝛽cos2 𝜑0 , 0, −𝛽cos2 𝜑0 ) when 𝛽 =
𝛽∗ is forward (backward), and the bifurcating periodic solutions on the center manifold are stable (unstable) if Re(𝑐1 (0)) <
0 (> 0). Particularly, the stability of the bifurcation periodic
solutions of system (3) and the reduced equations on the center
manifold are coincident at the first bifurcation value 𝛽 = 𝛽0 .
4. Numerical Simulations
In this section, we will carry out numerical simulations on
system (3) at special values of 𝛽. We choose a set of data as
follows:
𝜋
𝛾1 = 0.1, 𝛾2 = 2.5, 𝜑0 = , 𝜏 = 1.5,
(65)
4
which are the same as those in [1]. Then, 𝛿 = 1, 𝜌 = 0.
Then, we can obtain
⋅
𝜔1 = 3.5677, . . . ,
⋅
⋅
𝜛1 = 5.5531, . . . ,
⋅
𝛽1 = 1.7434, . . . ,
𝜔0 = 0.2225,
⋅
𝜛0 = 1.7294,
⋅
𝛽0 = 1.1008,
⋅
𝛽0 = 1.3534,
⋅
𝛽0 = 0.6093,
⋅
𝛽1 = 2.5235, . . . ,
⋅
𝛽1 = 1.1356, . . . ,
(66)
𝑘 = 1.9,
𝑘 = 3.
From the analysis in Section 2, we know that 𝛽(𝛽) is
increasing with respect to 𝜔(𝜛) when 𝜔(𝜛) > 𝛾1 𝛾2 , which
means that
𝛽0 = min {𝛽𝑗 } ,
𝛽0 = min {𝛽𝑗 } ,
𝑗 = 0, 1, 2, . . . ,
(67)
that is, 𝛽0 (𝛽0 ) is the first critical value at which system (3)
undergoes a Hopf bifurcation.
When 𝑘 = 1.9, by the previous results, it follows that
⋅
𝜆󸀠 (𝛽0 ) = 0.2440 − 0.0491i,
⋅
𝜇2 = 2.2020,
⋅
⋅
𝑐1 (0) = −0.5373 + 0.1082i,
𝛽2 = −1.0746,
𝑇2 = −0.0018.
(68)
Abstract and Applied Analysis
0.5
−0.4
0.4
0.3
−0.45
−0.5
0.2
0.1
−0.55
0
−0.1
𝑦1 (𝑡)
𝑥1 (𝑡)
10
−0.2
−0.3
−0.65
−0.7
−0.75
−0.4
−0.5
−0.6
0
500
1000
1500
2000
𝑡
2500
3000
3500
−0.8
4000
0
500
1000
0.8
−0.4
0.6
−0.45
0.4
−0.5
0.2
−0.55
0
−0.6
−0.2
−0.7
−0.6
−0.75
4500
2500
3000
3500
4000
−0.65
−0.4
−0.8
4000
2000
𝑡
(b)
𝑦2 (𝑡)
𝑥2 (𝑡)
(a)
1500
5000
𝑡
5500
6000
(c)
−0.8
4000
4500
5000
𝑡
5500
6000
(d)
Figure 6: 𝛾1 = 0.1, 𝛾2 = 2.5, 𝜏 = 1.5, 𝑘 = 3, which means that condition (𝐻2 ) holds, and 𝛽 = 1.2 > 𝛽0 > 𝛽0 . The initial value is
(0.1, −0.5, 0.1, −0.5).
Hence, we arrive at the following conclusion: the equilibrium 𝐸 is asymptotically stable when 𝛽 ∈ [0, 1.1008) and
unstable when 𝛽 ∈ (1.1008, +∞), and, at the first critical
value, the bifurcating periodic solutions are asymptotically
stable, and the direction of the bifurcation is forward (see
Figures 2 and 3).
When 𝑘 = 3, we can get
⋅
𝜆󸀠 (𝛽0 ) = 0.9004 + 0.0924i,
⋅
𝜇2 = 2.1231,
⋅
⋅
𝑐1 (0) = −1.9116 + 0.7930i,
𝛽2 = −3.8232,
𝑇2 = −0.5720.
(69)
Then, we have the following: the equilibrium 𝐸 is asymptotically stable when 𝛽 ∈ [0, 0.6093), and unstable when
𝛽 ∈ (0.6093, +∞), and, at the first critical value, the
bifurcating periodic solutions are asymptotically stable, and
the direction of the bifurcation is forward (see Figures 4, 5,
and 6).
5. Conclusion
Ravoori et al. [1] explored an experimental system of two
nominally identical optoelectronic feedback loops coupled
unidirectionally, which are described by system (3). In the
experiment, they found that depending on the value of the
feedback strength 𝛽 and delay 𝜏, system (1) is capable of
producing dynamics ranging from periodic oscillations to
high-dimensional chaos [14, 15].
This paper investigates the stability and the existence
of periodic solutions. We find that with the variety of the
coupling strength 𝑘, even if all other parameters keep the
same, the dynamical behavior can change greatly. In fact,
it is clear that the first two equations, 𝑥1 (𝑡) and 𝑦1 (𝑡) are
uncoupled with equations 𝑥2 (𝑡) and 𝑦2 (𝑡), so system (1) are
independent of (2), which means that coupling strength 𝑘
does not appear in (1). The characteristic equation of (1)
has the same form as (6), so the first critical value 𝛽0 is
independent of 𝑘. The analysis of characteristic equation (7)
shows that the value of 𝑘 can affect the first critical value
𝛽0 definitely. And we draw a conclusion that when 𝑘 is in
an interval, in which 𝛽0 < 𝛽0 holds, solutions of system (1)
and (2) keep synchronous; when 𝑘 belongs to the interval,
in which 𝛽0 < 𝛽0 holds, solutions of system (1) and (2) can
also keep synchronous with 𝛽 < 𝛽0 , while they lose their
synchronization when 𝛽 > 𝛽0 , no matter whether 𝛽 < 𝛽0
or not.
As a result, the modulation of the coupling strengths 𝑘
together with the feedback strength 𝛽 would be an efficient
and an easily implementable method to control the behavior
of the coupled chaotic oscillators.
Abstract and Applied Analysis
Acknowledgment
This paper is supported by the National Natural Science
Foundation of China (no. 11031002) and the Research Fund
for the Doctoral Program of Higher Education of China (no.
20122302110044).
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