Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 257635, 20 pages
doi:10.1155/2012/257635
Research Article
Synchronized Hopf Bifurcation Analysis in
a Delay-Coupled Semiconductor Lasers System
Gang Zhu1, 2 and Junjie Wei1
1
2
Department of Mathematics, Harbin Institute of Technology, Heilongjiang, Harbin 150001, China
School of Education Science, Harbin University, Heilongjiang, Harbin 150086, China
Correspondence should be addressed to Junjie Wei, weijj@hit.edu.cn
Received 30 May 2012; Revised 26 August 2012; Accepted 29 August 2012
Academic Editor: Nazim Idrisoglu Mahmudov
Copyright q 2012 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 system of two semiconductor lasers, which are delay coupled via a passive relay
within the synchronization manifold, are investigated. Depending on the coupling parameters,
the system exhibits synchronized Hopf bifurcation and the stability switches as the delay varies.
Employing the center manifold theorem and normal form method, an algorithm is derived for
determining the Hopf bifurcation properties. Some numerical simulations are carried out to
illustrate the analysis results.
1. Introduction
Systems of coupled semiconductor lasers SLs are receiving increasing interest, because
of their practical importance in more and more complex experimental devices, so coupled
system are studied by many researchers 1–3. Moreover, they are important examples of
delay-coupled oscillators in general 4, 5. The distance between the lasers always results
in a time delay in the coupling. In many situations, the time delay has been neglected.
However, for semiconductor lasers, this is not always justified due to their large bandwidth
and fast time scales of their dynamics. It is well known that delay effects can destabilize a
system, furthermore, delay may even result in chaotic dynamics as was shown in 6–8. On
the other hand, time delay in the coupling can also be used to stabilize a chaotic system
9–12. Synchronization phenomena are common in coupled semiconductor lasers systems.
Research shows 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 isochronal synchronism 13–18.
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In this paper, we consider the Lang-Kobayashi rate equation 19–22:
τ
1
,
1 iαnj Ej keiϕ EY t −
2
2
2
T ṅj p − nj − 1 nj Ej , j 1, 2,
Ėj 1.1
which has already been analyzed by many authors during these years from the physics point
of views. Here, Ej and EY are the complex electric field amplitudes of the jth system and the
relay, respectively, nj is the excess carrier density, α is the linewidth enhancement factor, k is
the coupling strength, p is the pump current, and the time scale parameter T τc /τp is the
ratio of the carrier and the photon lifetime, ϕ is the feedback phase, and all the parameters in
system 1.1 are constants.
In this paper, for the passive relay EY realized through a semitransparent mirror,
which only receives, reflects, and passes part of the laser from E1 and E2 , we consider the
algebraic equation:
EY t τ
τ 1 E1 t −
E2 t −
.
2
2
2
1.2
Noticing the coefficient of the relay EY , we found that since k, e, i, and ϕ are all positive
constants, so similar to 22, we choose the feedback phase ϕ 0 for simplicity, whose results
are different from those of the system with ϕ /
0 only by a constant multiple. Splitting the
complex electric field Ej xj iyj and using the vector Xj xj , yj , nj , j 1, 2, we consider
the dynamics within the synchronization manifold SM, that is, X1 t X2 t Xt, then
we have
τ
Et − τ,
EY t −
2
1.3
and 1.1 can be written in the following form:
ẋt nt xt − αyt kxt − τ,
2
nt αxt yt kyt − τ,
2
T ṅt p − nt − 1 nt x2 t y2 t .
ẏt 1.4
It is found that, under certain conditions, the equilibrium of system 1.4 is unstable
when the delay τ varies from zero, and as τ passes through a critical value, the equilibrium
becomes asymptotically stable, and after that when τ passes through another critical value,
the equilibrium becomes unstable again, which means that there are stability switches as τ is
increasing. Hence, a Hopf bifurcation occurs at the equilibrium when τ equals to each critical
value, which means system 1.4 has periodic solutions and 1.1 exhibits synchronized
periodic oscillation. Since the delay is caused by the distance between the lasers and the
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3
receiver, we know that the variety of the distance may result in amplitude death amplitude
tending to zero or periodic oscillation in the complex electric field.
The paper is organized as follows. In Section 2, using the method presented in 23,
we study the stability and the existence of Hopf bifurcation of system 1.4 by analyzing the
distribution of the roots of the associated characteristic equation, which is a transcendental
equation. In Section 3, we use the normal form method and the center manifold theory
presented in Hassard et al. 24 to analyze the direction, stability, and period of the bifurcating
periodic solution at critical values of τ. In Section 4, some numerical simulations are carried
out to illustrate the analytical results.
2. Stability Analysis
For 1.4, it is straightforward to see that E0, 0, p is an equilibrium. Linearizing equation
1.4 around 0, 0, p, it follows that
p
αp
xt −
yt kxt − τ,
2
2
αp
p
ẏt xt yt kyt − τ,
2
2
ẋt ṅt −
2.1
nt
,
T
whose characteristic equation is given by
2 α2 p2
p
1
−λτ
λ − − ke
0,
λ
T
2
4
2.2
which is equivalent to the following two equations:
1
0,
T
2 α2 p2
p
λ − − ke−λτ 0.
2
4
λ
2.3
2.4
So λ −1/T is always a negative root. Next, we study 2.4.
Obviously, 2.4 is equivalent to
p
αp
− ke−λτ i ,
2
2
p
αp
λ − − ke−λτ −i .
2
2
λ−
2.5
2.6
It is easy to see that the roots of 2.5 are conjugatives of 2.6, so we study 2.5 only.
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Journal of Applied Mathematics
When τ 0, we get the root of 2.5 easily
λ
αp
p
ki ,
2
2
2.7
so we have
Lemma 2.1. The equilibrium E0, 0, p is unstable when τ 0.
Let λ iωω /
0 be a root of 2.5 and substitute λ iω into 2.5 yields
p
k cos ωτ 0,
2
αp
ω k sin ωτ ,
2
2.8
which leads to
αp 2
p2
k2 − .
2
4
2.9
1
αp ± 4k2 − p2 .
2
2.10
ω−
Then, one gets
ω± Hence,
cos ω τ cos ω− τ −p
,
2k
sin ω τ −p
,
2k
αp − 2ω
−
2k
sin ω− τ αp − 2ω−
2k
4k2 − p2
2k
4k2 − p2
2k
,
2.11
.
Consequently, for k > p/2, one has
⎛
1 ⎜
τj− ⎝π − arcsin
ω−
⎛
τj 1 ⎜
⎝π arcsin
ω
4k2 − p2
2k
4k2 − p2
2k
⎞
⎟
2jπ ⎠,
2.12
⎞
⎟
2jπ ⎠,
j 0, 1, 2, . . . .
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5
Let
λτ γτ iωτ
2.13
be the root of 2.5 satisfying γτj± 0, ωτj± ω± .
We have the following conclusion.
Lemma 2.2. It holds that
i γ τj > 0
ii γ τj−
⎧
⎪
⎪
⎪< 0,
⎨
⎪
⎪
⎪
⎩> 0,
√
p
1 α2
p
when < k <
2
2
√
1 α2
when k >
p.
2
2.14
Proof. Substituting λτ into 2.5 and taking the derivative with respect to τ, it follows that
dλ
dλ
λke−λτ kτe−λτ
0.
dτ
dτ
2.15
Therefore, noting that ke−λτ λ − p/2 − i/2αp, we have
−λ λ − p/2 − i/2αp
dλ
,
dτ 1 τ λ − p/2 − i/2αp
2.16
and by a straight computation, we get
γ τj±
αp ω± ω±
ω± −
±
Δ
2
Δ
k2 −
p2
,
4
2.17
√
where Δ 1 pτ/2√2 τ 2 ω −αp/22 . Notice that ω− > 0 when p/2 < k < 1 α2 /2p
and ω− < 0 when k > 1 α2 /2p, this completes the proof.
As to the order of the sequence {τj± }, we have
Lemma 2.3. Suppose p/2 < k < αp/2π. Then, τ0− < τ0 , and there exists an integer m ≥ 0 such
that
−
−
< τm
< τm1
< τm1
.
τ0− < τ0 < τ1− < · · · < τm
2.18
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Journal of Applied Mathematics
Proof. Condition p/2 < k < αp/2π implies that τj± are well defined. So it is sufficient to
verify that τ0− < τ0 . From
arcsin x x ∞
2n − 1!!
x2n1 : x A,
12n!!
2n
n1
x ∈ −1, 1
2.19
and 2.12, we have
⎛
τ0− 1 ⎜
⎝π −
ω−
⎛
τ0 1 ⎜
⎝π ω
4k2 − p2
2k
4k2 − p2
2k
⎞
⎟
− A⎠,
2.20
⎞
⎟
A⎠,
and if τ0− < τ0 , then
⎛
⎜
ω ⎝π −
4k2 − p2
2k
⎞
⎛
⎟
⎜
− A⎠ < ω− ⎝π 4k2 − p2
2k
⎞
⎟
A⎠,
2.21
which is equivalent to
⎞
⎞
⎛
⎛
2 − p2
2 − p2
4k
4k
⎟
⎟
⎜
⎜
αp 4k2 − p2 ⎝π −
− A⎠ < αp − 4k2 − p2 ⎝π A⎠,
2k
2k
2.22
if and only if
⎞
⎛
2 − p2
4k
⎟
⎜
π 4k2 − p2 < αp⎝
A⎠.
2k
2.23
So, from the condition above, we have
αp
k<
⇒ π 4k2 − p2 < αp
2π
This implies that τ0− < τ0 .
4k2 − p2
2k
⎛
⎞
2 − p2
4k
⎜
⎟
< αp⎝
A⎠.
2k
2.24
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7
−
− τj 2π/ω , τj1
− τj− 2π/ω− , and ω > ω− , we have
From τj1
−
− τj < τj1
− τj− .
τj1
2.25
Hence, the conclusion follows.
For convenience, we make the following assumption:
√
H1 k ∈ 0, p/2 ∪ 1 α2 /2p, ∞,
H2 p/2 < k < αp/2π.
√
From lemmas 2.1–2.3 and the fact that αp/2π < 1 α2 /2p, then by the Hopf
bifurcation theorem for functional differential equations 25, we have the following results
on stability and bifurcation to system 1.4.
Theorem 2.4. For system 1.4, the following hold.
i If H1 is satisfied, then E is unstable for all τ ≥ 0.
ii If H2 is satisfied, then system 1.4 undergoes a Hopf bifurcation at E when τ τj± ,
j 0, 1, . . .. Particularly, there exists an integer m ≥ 0 such that E is unstable when
−
τ ∈ {0} ∪ ∪m
j0 τj−1 , τj ∪ τm , ∞ with τ−1 0, and asymptotically stable when τ ∈
− m
∪j0 τj , τj .
Remark 2.5. From Lemmas 2.1–2.3, we have that all other roots, except iω− Res. iω , of 2.5
with τ τj− resp., τ τj have negative real parts for j 0, 1, . . . , m when H2 holds.
3. The Direction and Stability of Hopf Bifurcation
In Section 2, we obtained that, under the assumption H2 , system 1.4 undergoes a Hopf
bifurcation at some critical values of τ. In this section, we study the direction, stability, and
the period of the bifurcating periodic solutions. The method we used is based on the normalform method and the center manifold theory presented by Hassard et al. 24.
Transform E0, 0, p to the origin O0, 0, 0 and rescale the time by t → t/τ to
normalize the delay so that system 1.4 can be written as the form
p
nt xt − αyt kxt − 1 xt − αyt ,
2
2
p
nt ẏt τ
αxt yt kyt − 1 αxt yt ,
2
2
τ
−nt − 1 p nt x2 t y2 t .
ṅt T
ẋt τ
3.1
Clearly, the phase space is C C−1, 0, R3 . For convenience, let τ τ0 μ, μ ∈ R
and τ0 be taken in {τj } ∪ {τj− }. From the analysis in Section 2, we know that μ 0 is the
Hopf bifurcation value for system 3.1, and iω0 τ0 is the root of the characteristic equation
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Journal of Applied Mathematics
associated with the linearization of system 3.1 when τ τ0 , where either ω0 ω or ω0 ω− . For φ φ1 , φ2 , φ3 ∈ C, let
Lμ φ τ0 μ Bφ0 τ0 μ Cφ−1,
3.2
where
⎞
⎛ p
αp
−
0
2
⎟
⎜ 2
⎟
⎜
⎟
⎜ αp p
⎟,
0
B⎜
⎟
⎜ 2
2
⎟
⎜
⎠
⎝
1
0
0 −
T
⎛
⎞
k 0 0
C ⎝ 0 k 0⎠.
0 0 0
3.3
By the Rieze representation theorem, there exists a 3 × 3 matrix, ηθ, μ−1 ≤ θ ≤ 0, whose
elements are of bounded variation functions such that
Lμ φ 0
−1
dη θ, μ φθ,
φ ∈ C.
3.4
In fact, we can choose
⎧
⎪
τ0 μ B,
⎪
⎨
η θ, μ 0,
⎪
⎪
⎩−τ μC,
0
θ 0,
θ ∈ −1, 0,
θ −1.
3.5
Then, 3.1 is satisfied.
For φ ∈ C, define the operator Aμ as
⎧
dφθ
⎪
⎪
,
⎪
⎨
dθ
A μ φθ 0
⎪
⎪
⎪
⎩
dη t, μ φt,
−1
θ ∈ −1, 0,
3.6
θ 0,
and Rμφ as
R μ φθ 0,
f μ, φ ,
θ ∈ −1, 0,
θ 0,
3.7
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9
where
⎛
φ1 0φ3 0 αφ2 0φ3 0
−
⎜
2
2
⎜
⎜
⎜ αφ1 0φ3 0 φ2 0φ3 0
f μ, φ τ0 μ ⎜
⎜
2
2
⎜
⎝ −1 2
1 p φ3 0 φ1 0 φ22 0
T
⎞
⎟
⎟
⎟
⎟
⎟.
⎟
⎟
⎠
3.8
Then, system 3.1 is equivalent to the following operator equation:
u̇t A μ ut R μ ut ,
3.9
where ut xt, yt, ntT , ut ut θ, for θ ∈ −1, 0.
For ψ ∈ C1 0, 1, R3 , define
⎧
dψs
⎪
⎪
⎪
⎨− ds ,
A∗ ψs 0
⎪
⎪
⎪
⎩
ψ−ξdηξ, 0,
−1
s ∈ 0, 1,
3.10
s 0.
For φ ∈ C−1, 0 and ψ ∈ C0, 1, define the bilinear form
!
ψs, φθ ψ0φ0 −
0 θ
−1
ψξ − θdηθφξdξ,
3.11
0
where ηθ ηθ, 0. Then, A0 and A∗ are adjoint operators.
Let qθ, q∗ s be the eigenvectors of A0 and A∗ associated with iω0 τ0 and −iω0 τ0 ,
respectively. It is not difficult to verify that
T
qθ 1, β, 0 eiω0 τ0 θ ,
q∗ s 1
D
1, ν, 0eiω0 τ0 s ,
3.12
where
p
−2 iω0 − − ke−iω0 τ0 ,
αp
2
p
−2 iω0 keiω0 τ0 ,
ν
αp
2
D 1 βν 1 kτ0 e−iω0 τ0 ,
β
and q∗ s, qθ 1, q∗ s, qθ 0.
3.13
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Journal of Applied Mathematics
Let ut be the solution of 3.9, and define
"
#
Wt, θ ut θ − 2 Re ztqθ .
!
zt q∗ , ut ,
3.14
On the center manifold C0 , we have
Wt, θ Wzt, zt, θ,
3.15
where
Wz, z, θ W20
z2
z2
W11 zz W02 · · · ,
2
2
3.16
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
"
#!
żt iω0 τ0 z q∗ θ, f 0, W 2 Re ztqθ
"
#
iω0 τ0 z q∗ 0, f 0, Wz, z, 0 2 Re ztq0
3.17
iω0 τ0 z q∗ 0f0 z, z.
We rewrite this equation as
żt iω0 τ0 z gz, z,
3.18
where
gz, z g20
z2
z2
z2 z
g11 zz g02 g21
··· .
2
2
2
3.19
By 3.9 and 3.18, we have
Ẇ u̇t − żq − ż q
%
$
⎧
⎪
⎨AW − 2 Re q∗ 0f0 qθ ,
%
$
⎪
⎩AW − 2 Re q∗ 0f q0 f ,
0
0
θ ∈ −1, 0,
θ 0,
3.20
AW Hz, z, θ,
where
Hz, z, θ H20 θ
z2
z2
H11 θzz H02 θ · · · .
2
2
3.21
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11
Expanding the above series and comparing the coefficients, we obtain
A − 2iω0 τ0 IW20 θ −H20 θ,
AW11 θ −H11 θ, . . . .
3.22
Noticing that
T
qθ 1, β, 0 eiω0 τ0 θ ,
3.23
ut θ zqθ z qθ Wz, z, θ,
we have
xt z z W 1 z, z, 0,
yt zβ zβ W 2 z, z, 0,
xt − 1 ze−iω0 τ0 zeiω0 τ0 W 1 z, z, −1,
3.24
yt − 1 zβe−iω0 τ0 zβeiω0 τ0 W 2 z, z, −1,
nt W 3 z, z, 0,
where
1
W 1 z, z, θ W20 θ
z2
z2
1
1
W11 θzz W02 θ · · · ,
2
2
3.25
z2
z2
2
2
2
W 2 z, z, θ W20 θ W11 θzz W02 θ · · · ,
2
2
and recalling that
⎛
xtnt αytnt
−
⎜
2
2
⎜
⎜
⎜ αxtnt ytnt
f0 τ0 ⎜
⎜
2
2
⎜
⎝ −1 1 p nt x2 t y2 t
T
⎞
⎟
⎟
⎟
⎟
⎟,
⎟
⎟
⎠
3.26
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Journal of Applied Mathematics
we have
#
τ0 "
xtnt − αytnt μ αxtnt ytnt
2D
τ0
z2
z2
1
1
1
1 αν z z W20 0 W11 0zz W02 0 · · ·
2D
2
2
gz, z q∗ 0f0 ×
z2
3
W20 0
2
3
W11 0zz
ν − α zβ zβ 2
×
2
3
W11 0zz
2
2
z2
2
W20 0
z2
3
W20 0
z
3
W02 0
···
2
W11 0zz
z2
3
W02 0
2
3.27
z2
2
W02 0
2
···
&
···
.
We can obtain the coefficients which will be used in determining the important quantities:
g20 g11 g02 0,
g21
3
τ0 W20 0 3
1 − αβ αν βν W11 0 1 − αβ αν βν .
D
2
3.28
We still need to compute W20 θ and W11 θ for θ ∈ −1, 0. We have
Hz, z, θ −q∗ 0f0 qθ − q∗ 0f0 qθ −gz, zqθ − gz, zqθ.
3.29
Comparing the coefficients about Hz, z, θ gives that
H20 θ −g20 qθ − g 02 qθ 0,
H11 −g11 qθ − g 11 qθ 0.
3.30
Then, from 3.22, we get
Ẇ20 θ 2iω0 τ0 W20 θ,
Ẇ11 θ 0,
3.31
which implies that
W20 θ Ee2iω0 τ0 θ ,
W11 θ F,
3.32
Journal of Applied Mathematics
13
where E, F are both three-dimensional vectors and can be determined by setting θ 0 in
Hz, z, θ. In fact, from
"
#
Hz, z, 0 − 2 Re q∗ 0f0 q0 f0
⎛
⎞
xtnt − αytnt
⎟
⎜
⎟
τ0 ⎜
⎟
⎜αxtnt ytnt
− gz, zq0 − gz, zq0 ⎜
⎟
⎟
2⎜
⎠
⎝ −2 2
2
1 p nt x t y t
T
3.33
− gz, zq0 − gz, zq0
W 1 z z W 3 − α W 2 zβ zβ W 3
⎜
⎜ τ0 ⎜
α W 1 z z W 3 W 2 zβ zβ W 3
⎜
2⎜
⎜
2 ⎝ −2 2 1
W z z W 2 zβ zβ
1 p W 3
T
⎛
⎞
⎟
⎟
⎟
⎟,
⎟
⎟
⎠
we have
T
2
H20 0 −g20 q0 − g 02 q0 τ0 0, 0, − 1 p 1 β2
T
τ0
T
2
0, 0, − 1 p 1 β2
T
,
H11 0 −g11 q0 − g 11 q0 τ0 0, 0, −
2
1p
T
T
2 1 β
3.34
2 T
2
τ0 0, 0, − 1 p 1 β
.
T
It follows from 3.22 and the definition of A that
τ0 BW20 0 τ0 CW20 −1 2iω0 τ0 W20 0 − H20 0,
τ0 BW11 0 τ0 CW11 −1 − H11 0.
3.35
Combining the conditions above, we have
T
2
B e−2iω0 τ0 C − 2iω0 I E 0, 0,
,
1 p 1 β2
T
2 T
2
,
1p 1 β
B CF 0, 0,
T
3.36
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Journal of Applied Mathematics
which implies
E
0, 0,
T
−2
,
1 p 1 β2
1 2iT ω0
2 .
F 0, 0, −2 1 p 1 β
3.37
Consequently, the above g21 can be expressed by the parameters and delay in system 3.1.
Thus, we can compute the following quantities:
2 1 2
g21
i
g20 g11 − 2 g11 − g02
c1 0 2ω0 τ0
3
2
−τ0 1 p
1
1 β2 1 − αβ αν βν
2D
1 2iω0 T
2 2 1 β 1 − αβ αν βν ,
μ2 −
3.38
Re c1 0
,
Re λ τ0 β2 2 Re c1 0,
T2 −
Im c1 0 μ2 Im λ τ0 ,
ω0 τ0
which determine the properties of bifurcating periodic solutions at the critical value τ0 . The
direction and stability of 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; 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
i Re λ τj > 0,
ii Re λ τj−
⎧
⎪
⎪< 0,
⎨
⎪
⎪
⎪
⎪
⎩> 0,
√
p
1 α2
p,
when < k <
2
2
√
1 α2
p.
when k >
2
From Remark 2.5, we have the following results.
3.39
Journal of Applied Mathematics
15
Theorem 3.1. For system 1.4, suppose that condition H2 is satisfied. Then, when 0 ≤ j ≤ m, one
has
i the Hopf bifurcation at E0, 0, p when τ τj− is backward (forward) and the bifurcation
periodic solutions are stable (unstable) if Rec1 0 < 0> 0;
ii the Hopf bifurcation at E0, 0, p when τ τj is forward (backward) and the bifurcation
periodic solutions are stable (unstable) if Rec1 0 < 0> 0.
Here τj± and m are given in ii of Theorem 2.4.
4. Numerical Simulations
In this section, we will carry out numerical simulation on system 1.4. Set
A α 9.42, k 1, p 1, T 20.
.
.
Clearly, H2 is satisfied. We have ω− 3.845, ω 5.575 and
.
τ0− 0.5448,
.
τ0 0.7507,
.
τ1 1.8771,
.
τ1− 2.1782 · · · .
4.1
Thus, the equilibrium 0, 0, 1 is stable when τ ∈ τ0− , τ0 , and unstable when τ ∈ 0, τ0− ∪
τ0 , ∞. Furthermore, we get
.
Re λ τ0− −1.8411,
.
Re λ τ0 2.3133.
4.2
By the algorithm derived in Section 3, we can obtain
.
Re c1 0 −0.7414,
.
μ2 −0.1027,
.
β2 −1.4828
4.3
at τ τ0− , and
.
Re c1 0 −4.7053,
.
μ2 2.0340,
.
β2 −9.4106
4.4
at τ τ0 , respectively. These imply that the direction of Hopf bifurcations is backward when
τ τ0− , and forward when τ τ0 , respectively, and the bifurcating periodic solutions are
orbitally asymptotically stable. These are shown in Figures 1, 2, and 3.
5. Conclusion
Flunkert et al. 22 explored an experimental system of two semiconductor lasers coupled
via a passive relay within the synchronization manifold. They calculated the maximum
transversal Lyapunov exponential and got blow-out bifurcations when the coupling strength
k passed through critical values.
In this paper, we also study the coupled system realized by a passive relay within
the synchronization manifold. By analyzing the distribution of eigenvalues, we study the
16
Journal of Applied Mathematics
6
6
5
4
3
2
1
0
−1
−2
−3
−4
4
2
0
−2
−4
−6
0
500
1000
1500
2000
2500
0
500
1000
a
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
500
1000
1500
2000
2500
b
1500
2000
2500
6
5
4
3
2
1
0
−1
−2
−3
−4
−6
−4
0
−2
c
2
4
6
d
1
0.5
0
−0.5
−1
10
5
0
−5
−10
−5
0
5
e
Figure 1: For system 1.4 with the data A, the Hopf bifurcation is backward at the first critical value
.
τ0− 0.5448, and the bifurcating periodic solutions are asymptotically stable, where τ 0.54 < 0.5448 and
the initial value is taken as 0.01, 0.01, 0.98.
stability of the equilibrium and the existence of periodic solutions. We find that as the
coupling strength increases, under the condition H2 , the stability switch for τ occurs, which
means that there exists a sequence values of τj± and an integer m satisfying
−
0 < τ0− < τ0 < τ1− < · · · < τm
< τm
< τm1
,
5.1
Journal of Applied Mathematics
17
0.02
0.02
0.015
0.015
0.01
0.01
0.005
0.005
0
0
−0.005
−0.005
−0.01
−0.01
−0.015
−0.015
−0.02
−0.02
−0.025
−0.025
0
500
1000
1500
2000
2500
3000
0
500
1000
a
1500
2000
2500
3000
b
0.02
1.005
0.015
1
0.01
0.005
0.995
0
−0.005
0.99
−0.01
−0.015
0.985
c
0.02
0.015
0.01
3000
0.005
2500
0
2000
−0.005
1500
−0.01
1000
−0.015
500
−0.02
0
−0.025
0.98
−0.02
−0.025
d
1.005
1
0.995
0.99
0.985
0.98
0.02
0
−0.02
−0.04
−0.01
−0.02
−0.03
0
0.01
0.02
e
Figure 2: For system 1.4 with the data A, the equilibrium E0, 0, 1 is asymptotically stable when τ ∈
τ0− , τ0 , where τ 0.6 and the initial value is taken as 0.01, 0.01, 0.98.
− such that the equilibrium E is asymptotically stable when τ ∈ ∪m
j0 τj , τj , and unstable when
−
m
τ ∈ {0} ∪ ∪j0 τj−1 , τj ∪ τm , ∞, and the system undergoes a Hopf bifurcation at τ τj± ,
where j 1, 2, . . ..
As a result, the modulation of the coupling strength k and the delay τ which is caused
by the distance between the lasers and the relay would be an efficient method to control the
18
Journal of Applied Mathematics
0.15
8000
7800
7600
7400
7200
7000
6800
6600
6000
6400
−0.2
8000
−0.2
7800
−0.15
7600
−0.15
7400
−0.1
7200
−0.1
7000
−0.05
6800
−0.05
6600
0
6400
0
6200
0.05
6000
0.1
0.05
6200
0.15
0.1
6000–8000
b
a
0.1
0.2
0.995
0.15
0.99
0.1
0.05
0.985
0
0.98
−0.05
0.975
−0.1
0.97
−0.2
−0.2 −0.15 −0.1 −0.05
8000
7000
6000
5000
4000
3000
2000
1000
0
0.965
−0.15
c
0
0.05
0.1
0.15
0.2
d
1
0.99
0.98
0.97
0.96
0.2
0.1
0
−0.1
−0.2
−0.2
−0.1
0
0.1
0.2
e
.
Figure 3: For system 1.4 with the data A, the Hopf bifurcation is forward at τ0 0.7507, and the
bifurcating periodic solutions are asymptotically stable, where τ 0.77 > τ0 and the initial value is taken
as 0.01, 0.01, 0.98.
system in the complex electric field; the amplitude either vanishes or presents a periodic
oscillation.
As per the coupled system which is realized by an active relay and the systems without
synchronization, we will study in the future.
Journal of Applied Mathematics
19
Acknowledgments
The authors are grateful to the anonymous referees for their helpful comments and valuable
suggestions. This research is supported by the National Natural Science Foundation of China
no. 11031002.
References
1 M. Möhrle, B. Sartorius, C. Bornholdt et al., “Detuned grating multisection-RW-DFB lasers for highspeed optical signal processing,” IEEE Journal on Selected Topics in Quantum Electronics, vol. 7, no. 2,
pp. 217–223, 2001.
2 M. T. Hill, H. De Waardt, G. D. Khoe, and H. J. S. Dorren, “All-optical flip-flop based on coupled laser
diodes,” IEEE Journal of Quantum Electronics, vol. 37, no. 3, pp. 405–413, 2001.
3 E. F. Manffra, I. L. Caldas, R. L. Viana, and H. J. Kalinowski, “Type-I intermittency and crisis-induced
intermittency in a semiconductor laser under injection current modulation,” Nonlinear Dynamics, vol.
27, no. 2, pp. 185–195, 2002.
4 H. Erzgräber, B. Krauskopf, and D. Lenstra, “Compound laser modes of mutually delay-coupled
lasers,” SIAM Journal on Applied Dynamical Systems, vol. 5, no. 1, pp. 30–65, 2006.
5 H. Erzgräber, D. Lenstra, B. Krauskopf et al., “Mutually delay-coupled semiconductor lasers: mode
bifurcation scenarios,” Optics Communications, vol. 255, no. 4–6, pp. 286–296, 2005.
6 G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions, vol. 210, Longman
Scientific & Technical, London, UK, 1989.
7 T. Heil, I. Fischer, W. Elsäßer, and A. Gavrielides, “Dynamics of semiconductor lasers subject to
delayed optical feedback: the short cavity regime,” Physical Review Letters, vol. 87, no. 24, Article ID
243901, 2001.
8 T. Heil, I. Fischer, W. Elsäßer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous
symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Physical Review Letters,
vol. 86, no. 5, pp. 795–798, 2001.
9 J. Wei and C. Yu, “Stability and bifurcation analysis in the cross-coupled laser model with delay,”
Nonlinear Dynamics, vol. 66, no. 1, pp. 29–38, 2011.
10 D. V. Ramana Reddy, A. Sen, and G. L. Johnston, “Time delay induced death in coupled limit cycle
oscillators,” Physical Review Letters, vol. 80, no. 23, pp. 5109–5112, 1998.
11 I. Fischer, Y. Liu, and P. Davis, “Synchronization of chaotic semiconductor laser dynamics on
subnanosecond time scales and its potential for chaos communication,” Physical Review A, vol. 62,
no. 1, Article ID 011801, 2000.
12 B. Ravoori, A. B. Cohen, A. V. Setty et al., “Adaptive synchronization of coupled chaotic oscillators,”
Physical Review E, vol. 80, no. 5, Article ID 056205, 2009.
13 A. Arenas, A. Dı́az-Guilera, J. Kurths, Y. Moreno, and C. Zhou, “Synchronization in complex
networks,” Physics Reports, vol. 469, no. 3, pp. 93–153, 2008.
14 M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster
formation in coupled laser networks,” Physical Review Letters, vol. 106, no. 22, Article ID 223901, 2011.
15 X. Liu, Q. Gao, and L. Niu, “A revisit to synchronization of Lurie systems with time-delay feedback
control,” Nonlinear Dynamics, vol. 59, no. 1-2, pp. 297–307, 2010.
16 C. F. Feng, “Projective synchronization between two different time-delayed chaotic systems using
active control approach,” Nonlinear Dynamics, vol. 62, pp. 453–459, 2010.
17 H. Huang, G. Feng, and J. Cao, “Exponential synchronization of chaotic Lur’e systems with delayed
feedback control,” Nonlinear Dynamics, vol. 57, no. 3, pp. 441–453, 2009.
18 J. Lu and J. Cao, “Adaptive synchronization of uncertain dynamical networks with delayed
coupling,” Nonlinear Dynamics, vol. 53, no. 1-2, pp. 107–115, 2008.
19 R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser
properties,” IEEE Journal of Quantum Electronics, vol. 16, no. 3, pp. 347–355, 1980.
20 P. M. Alsing, V. Kovanis, A. Gavrielides, and T. Erneux, “Lang and Kobayashi phase equation,”
Physical Review A, vol. 53, no. 6, pp. 4429–4434, 1996.
21 J. D. Farmer, “Chaotic attractors of an infinite-dimensional dynamical system,” Physica D, vol. 4, no.
3, pp. 366–393, 1982.
20
Journal of Applied Mathematics
22 V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,”
Physical Review E, vol. 79, no. 6, Article ID 065201, 2009.
23 S. Ruan and J. Wei, “On the zeros of transcendental functions with applications to stability of delay
differential equations with two delays,” Dynamics of Continuous, Discrete & Impulsive Systems Series A,
vol. 10, no. 6, pp. 863–874, 2003.
24 B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41,
Cambridge University Press, Cambridge, UK, 1981.
25 J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 1977.
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