PHYSICAL REVIEW E 76, 036212 共2007兲
Synchronization of time-delayed systems
1
Maoyin Chen1,2 and Jürgen Kurths2
Department of Automation, Tsinghua University, Beijing 100084, China
Institut für Physik, Potsdam Universität, Am Neuen Palais 10, D-14469, Germany
共Received 15 May 2007; revised manuscript received 4 July 2007; published 21 September 2007兲
2
In this paper we study synchronization in linearly coupled time-delayed systems. We first consider coupled
nonidentical Ikeda systems with a square wave coupling rate. Using the theory of the time-delayed equation,
we derive less restrictive synchronization conditions than those resulting from the Krasovskii-Lyapunov theory
关Yang Kuang, Delay Differential Equations 共Academic Press, New York, 1993兲兴. Then we consider a wide
class of nonlinear nonidentical time-delayed systems. We also propose less restrictive synchronization conditions in an approximative sense, even if the coefficients in the linear time-delayed equation on the synchronization error are time dependent. Theoretical analysis and numerical simulations fully verify our main results.
DOI: 10.1103/PhysRevE.76.036212
PACS number共s兲: 05.45.Xt, 05.45.Vx, 42.55.Px, 42.65.Sf
I. INTRODUCTION
Time-delayed systems are ubiquitous in nature, technology, and society because of finite signal transmission times,
switching speeds, and memory effects 关1兴. It is well known
that dissipative systems with a nonlinear time-delayed feedback or memory can produce chaotic dynamics 关2,3兴, and the
dimension of their chaotic attractors can be made arbitrarily
large by increasing their delay time sufficiently 关4兴. Recently,
the synchronization of the time-delayed systems has attracted
much attention 关5–12兴. Complete synchronization 关7兴, phase
synchronization 关8兴, anticipating synchronization 关9兴, and
generalized synchronization 关10兴 are considered in the timedelayed coupled systems.
From recent works 关1,6,9,11,12兴, the KrasovskiiLyapunov theory 关13兴 is useful for discussing the synchronization in the coupled time-delayed systems. According to
this theory, some sufficient conditions are given to ensure
synchronization in the coupled time-delayed systems
关1,6,9,11兴. However, these conditions are not valid for the
general case where the linear time-delayed equation on the
synchronization error are time varying; especially the coefficients in this equation are time dependent. This has been
already shown in detail by Zhou et al. 关12兴. Therefore, it is
necessary and important to propose the synchronization conditions for the above case.
In this paper we also consider the synchronization in the
linearly coupled time-delayed systems. Here we study two
cases of the coupled time-delayed systems. The first case is
the coupled nonidentical Ikeda systems with a square wave
coupling rate. Using the stability theory of the time-delayed
equation, we can derive less restrictive synchronization conditions than those resulting from the Krasovskii-Lyapunov
theory. The second case is a wide class of nonlinear nonidentical time-delayed systems. In an approximative sense, we
also propose less restrictive synchronization conditions even
if the coefficients in the linear time-delayed equation on the
synchronization error are time dependent.
II. SYNCHRONIZATION IN THE COUPLED
IKEDA SYSTEMS
First we analyze two nonidentical linearly coupled Ikeda
systems
1539-3755/2007/76共3兲/036212共5兲
dx
= − ␣x + m1 sin x共t − ␶1兲,
dt
共1兲
dy
= − ␣y + m2 sin y共t − ␶2兲 + K共x − y兲,
dt
共2兲
where x is the phase lag of the electric field across the resonator, ␣ is the relaxation coefficient, m1,2 are the laser intensities injected into the systems, ␶1,2 are the round-trip times
of the light in the resonators or feedback delay times in the
coupled systems, and K is the coupling rate between the
drive and response systems 关10兴. The Ikeda model was introduced to describe the dynamics of an optical bistable resonator and is well known for delay-induced chaotic behavior
关9,11兴.
Here we choose the coupling rate K as a square wave
coupling, denoted by the following sequence;
兵共t0,K1兲,共t1,K2兲,共t2,K3兲,共t3,K4兲, . . . 其,
共3兲
where t j = t0 + 共j − 1兲␶s is the switching instant, the coupling
rate K2j−1 = k1 and K2j = k2 for all j 艌 1; k1 and k2 are different
values. Within the interval 关t j−1 , t j兲, the coupled Ikeda systems 共1兲 and 共2兲 become
dx
= − ␣x + m1 sin x共t − ␶1兲,
dt
dy
= − ␣y + m2 sin y共t − ␶2兲 + K j共x − y兲.
dt
When the time delays ␶1 = ␶2, one necessary condition for
complete synchronization in the coupled Ikeda systems 共1兲
and 共2兲 is m1 = m2 关10兴. The dynamics of the error ⌬ = x − y is
given by
d⌬
= − 共␣ + K兲⌬ − m1 cos x共t − ␶1兲⌬共t − ␶1兲.
dt
共4兲
It is obvious that ⌬ = 0 is a solution of system 共4兲.
To study the stability of system 共4兲, one can design a
Krasovskii-Lyapunov function as V共t兲 = 21 ⌬2共t兲 + ␮兰−0 ␶ ⌬2共t
1
+ ␪兲d␪, where ␮ ⬎ 0 is an arbitrary positive parameter.
036212-1
©2007 The American Physical Society
PHYSICAL REVIEW E 76, 036212 共2007兲
MAOYIN CHEN AND JÜRGEN KURTHS
The main purpose of the Krasovskii-Lyapunov theory is to
find the condition for the negativeness of ddtV when the synchronization error ⌬ is not zero. Since r共t兲 = ␣ + K and s共t兲
= −m1 cos x共t − ␶1兲 in system 共4兲 are time varying, a sufficient
condition for complete synchronization is ␣ + K ⬎ 兩m1兩
艌 兩s共t兲兩 for all time 关1,6,9,11,12兴. But this condition cannot
be applied to the general case where r共t兲 and s共t兲 are time
varying, especially s共t兲 is not bounded for all time. Even for
the square wave coupling rate, the above condition may not
ensure complete synchronization. The main reason is that the
parameter ␮ in the Krasovskii-Lyapunov function V共t兲 is
supposed to be time invariant. However, when r共t兲 and s共t兲
in the general case are time dependent, the parameter ␮
should be also time varying 关12兴.
Here we propose a less restrictive synchronization condition using the stability theory of the time-delayed equation.
Defining a positive-defined function V共t兲 = ⌬2共t兲, we get
冐 再冕 冋
再冕 冋
再冕 冋
再冕 冋
t
储⌫1共t兲储 = exp
t j−1
t0
t
t j−1
冉
再冕
t
−2 ␣+K−
t0
冊
共5兲
册
冊
兩m1兩
V共s兲 + 兩m1兩V共s − ␶1兲 ds,
2
where V0 is an initial condition of V共t兲. From the comparison
theorem of the delayed equation 关14兴, the solution V共t兲 satisfies
V共t兲 艋 ⌫共t兲,
共6兲
where ⌫共t兲 is the maximal solution of the following integral
equation:
冕冋 冉
t
⌫共t兲 = V0 +
−2 ␣+K−
t0
册
冊
兩m1兩
⌫共s兲 + 兩m1兩⌫共s − ␶1兲 ds,
2
冋 冉
冊
册
兩m1兩
+ 兩m1兩 ds ¯
2
− 2 ␣ + K1 −
兩m1兩
+ 兩m1兩 ds V0
2
冎
Hence we obtain a sufficient condition for complete synchronization in the coupled Ikeda systems 共1兲 and 共2兲,
⬁
关− 共␣ + K − 兩m1兩兲兴dt = − ⬁.
共9兲
If this condition is satisfied, we get limt→⬁⌫1共t兲 = 0, which
also means limt→⬁⌫共t兲 = 0. However, this condition does not
require that the derivative of a positive-defined function V is
negative for all time when the synchronization error is not
zero 关1,6,9,11,12兴. It aims to make limt→⬁⌫共t兲 = 0, which also
leads to limt→⬁V共t兲 = 0. Further, condition 共9兲 is less restrictive than the condition of ␣ + K ⬎ 兩m1兩 for all time 关1,6,9,11兴.
For the square wave coupling 共3兲, the condition for complete
synchronization is −共␣ + k1 − 兩m1兩兲␶s − 共␣ + k2 − 兩m1兩兲␶s ⬍ 0 during one period. Complete synchronization in the coupled
Ikeda systems 共1兲 and 共2兲 can also be ensured even if the
coupling rate k1 共or k2兲 could not satisfy ␣ + k1 ⬎ 兩m1兩 共or ␣
+ k2 ⬎ 兩m1兩兲.
When the time delays ␶1 ⫽ ␶2, complete synchronization is
not possible. But we can analyze the condition for generalized synchronization in the coupled Ikeda systems 共1兲 and
共2兲 based on the auxiliary approach 关16兴. Therefore, we
should consider complete synchronization between the following Ikeda systems:
共7兲
with the same initial condition V0. If we prove that
limt→⬁⌫共t兲 = 0, we get limt→⬁V共t兲 = 0, which further results in
the limit limt→⬁⌬共t兲 = 0. From Ref. 关15兴, the stability of system 共7兲 is equivalent to the stability of
兩m1兩
d⌫1
= −2 ␣+K−
+ ␽兩m1兩 ⌫1 ,
2
dt
兩m1兩
+ ␽兩m1兩 ds V0
2
关− 共␣ + K − 兩m1兩兲兴ds V0 .
t0
or equivalently,
兩m1兩
d⌫
= − 2共␣ + K − 兩m1兩兲⌫ +
⌫共t − ␶1兲,
2
dt
− 2 ␣ + K1 −
t0
兩m1兩
V + 兩m1兩V共t − ␶1兲.
2
冕冋 冉
t0
t
= exp 2
冕
Integrating the above inequality, we have
V共t兲 艋 V0 +
t1
⫻exp
冊 册冎
冊 册冎冐
冊 册冎
冊 册冎
兩m1兩
+ ␽兩m1兩 ds ¯
2
− 2 ␣ + Kj −
艋 exp
dV
艋 − 2共␣ + K兲V + 2兩m1兩兩⌬⌬共t − ␶1兲兩
dt
艋−2 ␣+K−
t1
⫻exp
冉
冉
冉
冉
− 2 ␣ + Kj −
dy
= − ␣y + m2 sin y共t − ␶2兲 + K共x − y兲,
dt
共10兲
dz
= − ␣z + m2 sin z共t − ␶2兲 + K共x − z兲.
dt
共11兲
Applying the above approach for the case of complete synchronization, we get a sufficient condition for generalized
synchronization in the coupled Ikeda systems 共1兲 and 共2兲 as
follows:
冕
共8兲
⬁
关− 共␣ + K − 兩m2兩兲兴dt = − ⬁,
共12兲
t0
where ␽ = exp共j␪兲 , ␪ 苸 关0 , 2␲兴 , j = 冑−1. From the square
wave coupling rate 共3兲, if time t belongs to the interval
关t j−1 , t j兲, we get
where the coupling rate K is a square wave signal 共3兲. This
condition is also less restrictive than the condition of ␣ + K
⬎ 兩m2兩 for all time 关1,6,9,11兴.
036212-2
PHYSICAL REVIEW E 76, 036212 共2007兲
SYNCHRONIZATION OF TIME-DELAYED SYSTEMS
III. SYNCHRONIZATION IN THE NONLINEAR
TIME-DELAYED SYSTEMS
Now we generalize the above approach to a wide class of
nonlinear nonidentical time-delayed systems. Consider a
general form of one-way coupled scalar time-delayed systems,
system 共15兲. Therefore, if ␶0 is sufficiently small, condition
共18兲 can be approximately regarded as a condition for complete synchronization of the coupled systems 共13兲 and 共14兲.
When the delays ␶1 ⫽ ␶2, we can also study generalized
synchronization in the coupled systems 共13兲 and 共14兲. From
the auxiliary approach 关16兴, we should consider complete
synchronization between the following systems:
dx
= F共p,x,x␶1兲,
dt
共13兲
dy
= F共q,y,y ␶2兲 + K共x − y兲,
dt
共19兲
dy
= F共q,y,y ␶2兲 + K共x − y兲,
dt
共14兲
dz
= F共q,z,z␶2兲 + K共x − z兲.
dt
共20兲
where K is the coupling rate, and p and q are the parameters
for the drive system and the response system, respectively.
When the delays ␶1 = ␶2 and the parameters p = q, we consider complete synchronization in the coupled systems 共13兲
and 共14兲. A small deviation ⌬ = x − y is governed by the linearized time-delayed equation
d⌬
= − r共t兲⌬ + s共t兲⌬共t − ␶1兲,
dt
共15兲
where r共t兲 = 共K − ⳵x兲F共p , x , x␶1兲 and s共t兲 = ⳵x␶ F共p , x , x␶1兲.
1
Similarly, one can also design a Krasovskii-Lyapunov
function as V共t兲 = 21 ⌬2共t兲 + ␮兰−0 ␶ ⌬2共t + ␪兲d␪ to study the sta1
bility of system 共15兲. Further, a sufficient condition for complete synchronization is r共t兲 ⬎ 兩s共t兲兩 for all time 关1,6,9,11,12兴.
However, this condition cannot be applied to the general case
where r共t兲 and s共t兲 are time varying 关12兴. Here we give one
sufficient synchronization condition in an approximative
sense.
We first segment the time interval 关t0 , ⬁兲 into 关t0 , ⬁兲
= 艛 j艌1关t j−1 , t j兲, where ␶0 ⬎ 0 is sufficiently small, t j = t0 + j␶0,
and ␶1 is a multiple of ␶0. If ␶0 is sufficiently small, the
solution x共t兲 of the drive system 共13兲 can be approximated by
x共t j兲 within the interval 关t j−1 , t j兲, which further results in the
approximation of r共t兲 , s共t兲 by r共t j兲 , s共t j兲, respectively. Therefore, within the interval 关t j−1 , t j兲, system 共15兲 can be approximated by
d⌬
= − r⬘共t兲⌬ + s⬘共t兲⌬共t − ␶1兲,
dt
共16兲
The small deviation ⌬⬘ = y − z is governed by
d⌬⬘
= − r共t兲⌬⬘ + s共t兲⌬⬘共t − ␶2兲,
dt
where r共t兲 = 共K − ⳵x兲F共q , x , x␶2兲 and s共t兲 = ⳵x␶ F共q , x , x␶2兲.
2
Similarly, we segment the time interval 关t0 , ⬁兲 into 关t0 , ⬁兲
= 艛 j艌1关t j−1 , t j兲, where ␶0 is sufficiently small, and ␶2 is the
multiple of ␶0. Therefore, system 共21兲 can be approximated
sufficiently by
d⌬⬘
= − r⬘共t兲⌬ + s⬘共t兲⌬⬘共t − ␶2兲,
dt
„t2,兵s⬘共t3兲,r⬘共t3兲其…, . . . 其.
共17兲
Applying the approach for the case of the coupled Ikeda
systems, we can derive the stability condition for the approximation system 共16兲 as follows:
冕
⬁
兵− 关r⬘共t兲 − 兩s⬘共t兲兩兴其dt = − ⬁,
冕
where r⬘共t兲 , s⬘共t兲 satisfy the sequence 共17兲. As ␶0 tends to
zero, the stability of system 共16兲 becomes the stability of
⬁
兵− 关r⬘共t兲 − 兩s⬘共t兲兩兴其dt = − ⬁,
共23兲
t0
where r⬘共t兲 , s⬘共t兲 satisfy the sequence 共17兲. If ␶0 is sufficiently small, condition 共23兲 can be also approximately regarded as a condition for generalized synchronization of the
coupled systems 共13兲 and 共14兲.
From the above analysis, if ␶0 approaches zero,
x共t j兲 , r⬘共t兲 , s⬘共t兲 approach x共t兲 , r共t兲 , s共t兲, respectively, and
system 共16兲 关or system 共22兲兴 can also approximate system
共15兲 关or system 共21兲兴. If ␶0 tends to zero, conditions 共18兲 and
共23兲 become
冕
⬁
兵− 关r共t兲 − 兩s共t兲兩兴其dt = − ⬁,
共24兲
t0
where r共t兲 = 共K − ⳵x兲F共p , x , x␶1兲 and s共t兲 = ⳵x␶ F共p , x , x␶1兲 for
1
complete synchronization, and r共t兲 = 共K − ⳵x兲F共q , x , x␶2兲 and
s共t兲 = ⳵x␶ F共q , x , x␶2兲 for generalized synchronization. Hence
2
condition 共24兲 can be considered as a sufficient synchronization condition in an approximative sense.
IV. NUMERICAL SIMULATIONS
共18兲
t0
共22兲
in which r⬘共t兲 = r共t j兲 and s⬘共t兲 = s共t j兲 within the interval
关t j−1 , t j兲. We also derive the stability condition for system
共22兲
where r⬘共t兲 = r共t j兲 and s⬘共t兲 = s共t j兲. In fact, the above approximation of system 共15兲 can be regarded as the following sequence:
兵„t0,兵s⬘共t1兲,r⬘共t1兲其…,„t1,兵s⬘共t2兲,r⬘共t2兲其…,
共21兲
We choose two typical examples to confirm our main results. One is the coupled nonidentical Ikeda systems; the
other is the coupled nonidentical Mackey-Glass systems 关2兴.
036212-3
PHYSICAL REVIEW E 76, 036212 共2007兲
MAOYIN CHEN AND JÜRGEN KURTHS
1.5
x
y
y
5
x,y
1
x,y
y,z
5
z
0
0
−5
0
−5
0
0.5
x
5
(a)
10
t
5
(b)
t
0
0
10
20
y
40
60
80
(a)
100
t
1.5
5
x
y
5
y
z
y,z
x,y
y,z
1
0
0
−5
0
−5
0
0.5
y
5
(c)
t
10
5
(d)
10
t
Note that all the results are simulated using the DDE23 program 关17兴 in MATLAB 7.
Example 1. The coupled nonidentical Ikeda systems (1)
and (2). We first choose a square wave coupling rate K as the
sequence 兵共0, 0兲,共2, 40兲,共4, 0兲,共6, 40兲,…其. Obviously, within
the intervals 关0, 2兲,关4, 6兲,…, there exists no coupling
共namely, the coupling rate K = 0兲, which means that ␣
⬎ 兩m1兩 for complete synchronization and ␣ ⬎ 兩m2兩 for generalized synchronization are not satisfied within these intervals. Hence conditions resulting from the KrasovskiiLyapunov theory cannot ensure the synchronization.
However, after a simple computation, condition 共9兲 for complete synchronization and condition 共12兲 for generalized synchronization are satisfied. Figure 1共a兲 shows complete synchronization for ␶1 = ␶2 = 1, ␣ = 5, and m1 = m2 = 20. Figure
1共b兲 shows generalized synchronization in the coupled Ikeda
systems 共1兲 and 共2兲 for ␶1 = 1, ␶2 = 1.2.
For the general case of the coupling rate K共t兲, we still
derive the approximated synchronization condition 共24兲
where r共t兲 = ␣ + K共t兲 and s共t兲 = −m1 cos x共t − ␶1兲 for complete
synchronization, and r共t兲 = ␣ + K共t兲 and s共t兲 = −m2 cos x共t
− ␶2兲 for generalized synchronization. For complete synchronization, we choose K共t兲 = −␣ + 2m1兩cos x共t − ␶1兲兩. Hence r共t兲
= 2m1兩cos x共t − ␶1兲兩 艌 兩s共t兲兩 = m1兩cos x共t − ␶1兲兩. From Refs.
关1,6,9,11,12兴, complete synchronization cannot be ensured
by the condition of r共t兲 ⬎ m1 艌 兩s共t兲兩 共or r共t兲 ⬎ 兩s共t兲兩兲 for all
time. However, we get 兰t⬁ 兵−关r共t兲 − 兩s共t兲兩兴其dt = 兰t⬁ 关−m1兩cos x共t
0
20
z
40
60
80
(b)
FIG. 1. Complete synchronization and generalized synchronization in the coupled Ikeda systems 共1兲 and 共2兲. 共a兲 Complete synchronization for ␶1 = ␶2 = 1, ␣ = 5, m1 = m2 = 20, and the square wave
coupling rate 兵共0, 0兲,共2, 40兲,共4, 0兲,共6, 40兲,…其. 共b兲 Generalized synchronization for ␶1 = 1, ␶2 = 1.2 关or complete synchronization in systems 共10兲 and 共11兲兴. All the other parameters have the same values
as those in Fig. 1共a兲. 共c兲 Complete synchronization in the coupled
Ikeda systems 共1兲 and 共2兲 where the coupling rate K共t兲 = −␣
+ 2m1兩cos x共t − ␶1兲兩. All the other parameters have the same values
as those in Fig. 1共a兲. 共d兲 Generalized synchronization in the coupled
Ikeda systems 共1兲 and 共2兲 where the coupling rate K共t兲 = −␣
+ 2m2兩cos x共t − ␶2兲兩 关or complete synchronization in the systems 共10兲
and 共11兲兴. All the other parameters have the same values as those in
Fig. 1共b兲.
0
0
0
100
t
FIG. 2. Complete synchronization and generalized synchronization in the coupled Mackey-Glass systems 共25兲 and 共26兲. 共a兲 Complete synchronization for a1 = a2 = 2, b1 = b2 = 10, c1 = c2 = 1, ␶1 = ␶2
= 10. 共b兲 Generalized synchronization for a1 = a2 = 2, b1 = b2 = 10,
c1 = c2 = 1, ␶1 = 10, ␶2 = 12 关or complete synchronization in the sysdy
dz
tems dt = −c2y + a2y ␶2 / 共1 + y b␶ 2兲 + K共x − y兲 and dt = −c2z + a2z␶2 / 共1
2
b2
+ z␶ 兲 + K共x − z兲兴.
2
− ␶1兲兩兴dt = −⬁ due to the chaotic nature of state x. Figure 1共c兲
shows the simulation result for complete synchronization.
For generalized synchronization, we choose K共t兲 = −␣
+ 2m2兩cos x共t − ␶2兲兩. Hence r共t兲 = 2m2兩cos x共t − ␶2兲兩 艌 兩s共t兲兩
= m2兩cos x共t − ␶2兲兩. Since the condition of r共t兲 ⬎ m2 艌 兩s共t兲兩 关or
r共t兲 ⬎ 兩s共t兲兩兴 for all time is not satisfied, generalized synchronization cannot be ensured by synchronization conditions
关1,6,9,11,12兴. However, we still have 兰t⬁ 兵−关r共t兲 − 兩s共t兲兩兴其dt
0
= 兰t⬁ 关−m2兩cos x共t − ␶2兲兩兴dt = −⬁. Figure 1共d兲 shows the simu0
lation result for generalized synchronization.
Example 2. The coupled Mackey-Glass systems. The
coupled nonidentical Mackey-Glass systems 关2,6兴 are
dx
= − c1x + a1x␶1/共1 + x␶b1兲,
1
dt
共25兲
dy
= − c2y + a2y ␶2/共1 + y ␶b2兲 + K共x − y兲.
2
dt
共26兲
Initially the Mackey-Glass system has been introduced as a
model of blood generation for patients with leukemia. Later
this model became popular in chaos theory as a model for
producing high dimensional chaos to test various methods of
chaotic time series analysis, controlling chaos, etc.
We consider the synchronization in the coupled systems
共25兲 and 共26兲, where the square wave coupling rate K is
denoted
by
the
sequence
兵共0 , −6兲 , 共0.2, 30兲 , 共0.4,
−6兲 , 共0.6, 30兲 , . . . 其. In this case we set a1 = a2 = 2, b1 = b2 = 10,
c1 = c2 = 1, ␶1 = ␶2 = 10 for complete synchronization, and ␶1
= 10, ␶2 = 12 for generalized synchronization. Here r共t兲 = K
+ c1 is a special kind of time-varying coupling rates, s共t兲
= ⳵x⳵␶ 关a1x␶1 / 共1 + x␶b1兲兴 for complete synchronization and s共t兲
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PHYSICAL REVIEW E 76, 036212 共2007兲
SYNCHRONIZATION OF TIME-DELAYED SYSTEMS
= ⳵x⳵␶ 关a2x␶2 / 共1 + x␶b2兲兴 for generalized synchronization are also
2
2
time varying. After a simple computation, we get
a b
兩 ⳵⳵x␶ 关a1x␶ / 共1 + x␶b1兲兴兩 艋 a1 + 12 1 = 12 for any ␶ ⬎ 0. Since the
coupling rate K within the intervals 关0, 0.2兲,关0.4, 0.6兲,… is
negative, the condition of r共t兲 ⬎ m2 艌 兩s共t兲兩 共or r共t兲 ⬎ 兩s共t兲兩兲
for all time is not satisfied 关1,6,9,11,12兴. Therefore we should
verify the satisfaction of condition 共24兲 for complete synchronization and generalized synchronization. It is easy to
verify that 兰t⬁ 兵−关K + c1 − 兩 ⳵x⳵␶ f共x␶1兲兩 兴其dt = −⬁ and 兰t⬁ 兵−关K
0
0
1
+ c1 − 兩 ⳵x⳵␶ f共x␶2兲兩 兴其dt = −⬁. Hence complete synchronization
2
共24兲 for ␶1 = ␶2 and generalized synchronization 共24兲 for ␶1
⫽ ␶2 in the coupled Mackey-Glass systems 共25兲 and 共26兲 are
ensured.
The simulation results are plotted in Fig. 2. Figure 2共a兲
shows complete synchronization in the coupled nonidentical
Mackey-Glass systems 共25兲 and 共26兲 when a1 = a2 = 2, b1
= b2 = 10, c1 = c2 = 1, and ␶1 = ␶2 = 10. Figure 2共b兲 shows generalized synchronization in the coupled nonidentical
Mackey-Glass systems 共25兲 and 共26兲 for a1 = a2 = 2, b1 = b2
= 10, c1 = c2 = 1, ␶1 = 10, and ␶2 = 12.
关1兴 J. K. Hale and S. M. V. Lunel, Introduction to Functional
Differential Equations 共Springer, New York, 1993兲.
关2兴 M. C. Mackey and L. Glass, Science 197, 287 共1977兲.
关3兴 J. D. Farmer, Physica D 4, 366 共1982兲.
关4兴 B. Dorizzi, B. Grammaticos, M. Le Berre, Y. Pomeau, E. Ressayre, and A. Tallet, Phys. Rev. A 35, 328 共1987兲; K. Ikeda
and K. Matsumoto, Physica D 29, 223 共1987兲.
关5兴 S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, and C.
Zhou, Phys. Rep. 366, 1 共2002兲; R. He and P. G. Vaidya, Phys.
Rev. E 59, 4048 共1999兲; S. Sano, A. Uchida, S. Yoshimori, and
R. Roy, ibid. 75, 016207 共2007兲.
关6兴 K. Pyragas, Phys. Rev. E 58, 3067 共1998兲.
关7兴 A. S. Landsman and I. B. Schwartz, Phys. Rev. E 75, 026201
共2007兲.
关8兴 D. V. Senthilkumar, M. Lakshmanan, and J. Kurths, Phys. Rev.
E 74, 035205共R兲 共2006兲.
关9兴 H. U. Voss, Phys. Rev. E 61, 5115 共2000兲.
关10兴 E. M. Shahverdiev and K. A. Shore, Phys. Rev. E 71, 016201
V. CONCLUSION
We have studied synchronization in linearly coupled timedelayed systems. We first consider the coupled nonidentical
Ikeda systems with a square wave coupling rate. We derive
less restrictive synchronization conditions than those resulting from the Krasovskii-Lyapunov theory. Then we generalize the above approach to a wide class of nonlinear nonidentical time-delayed systems. Even if the coefficients in the
linear time-delayed equation on the synchronization error are
time dependent, we also propose less restrictive synchronization conditions in an approximative sense. Numerical
simulations of the coupled Ikeda systems and the coupled
Mackey-Glass systems fully verify our main results.
ACKNOWLEDGMENT
This work is supported by the Alexander von Humboldt
Foundation.
共2005兲.
关11兴 E. M. Shahverdiev, S. Sivaprakasam, and K. A. Shore, Phys.
Rev. E 66, 017204 共2002兲; E. M. Shahverdiev, ibid. 70,
067202 共2004兲; D. V. Senthilkumar and M. Lakshmanan, ibid.
71, 016211 共2005兲.
关12兴 S. B. Zhou, H. Li, and Z. F. Wu, Phys. Rev. E 75, 037203
共2007兲.
关13兴 Yang Kuang, Delay Differential Equations 共Academic Press,
London, 1993兲.
关14兴 V. Lakshmikantham and S. Leela, Differential and Integral
Inequalities 共Academic Press, New York, 1969兲.
关15兴 A. Hmamed, Int. J. Syst. Sci. 22, 605 共1991兲; J.-S. Chiou, IEE
Proc.: Control Theory Appl. 153, 684 共2006兲.
关16兴 N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, and H. D. I.
Abarbanel, Phys. Rev. E 51, 980 共1995兲; M. Zhan, X. Wang,
X. Gong, G. W. Wei, and C.-H. Lai, ibid. 68, 036208 共2003兲.
关17兴 L. F. Shampine and S. Thompson, Appl. Numer. Math. 37,
441 共2001兲.
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