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Nonlinear Analysis
journal homepage:
www.elsevier.com/locate/na
Generalized synchronization of chaotic systems by pure error dynamics
and elaborate Lyapunov function
Zheng-Ming Ge
∗
, Ching-Ming Chang
Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 4 December 2007 Accepted 6 April 2009 MSC: 34C15 34C28 37D45 Keywords: Chaos Generalized synchronization Double Mathieu system Lyapunov function Lyapunov direct methoda b s t r a c t
The generalized synchronization is studied by applying pure error dynamics and elaborate
Lyapunov function in this paper. Generalized synchronization can be obtained by pure error
dynamics without auxiliary numerical simulation, instead of current mixed error dynamics
in which master state variables and slave state variables are presented. The elaborate
Lyapunov function is applied rather than the current plain square sum Lyapunov function,
deeply weakening the power of Lyapunov direct method. The scheme is successfully
applied to both autonomous and nonautonomous double Mathieu systems with numerical
simulations.
© 2009 Elsevier Ltd. All rights reserved.
1. Introduction
Chaos synchronization has been applied in secure communication [
1
,
2
], biological systems [
3
,
4
], and many other
fields [
5–25
]. One of the intricate types of chaos synchronization is generalized synchronization, which has been extensively
investigated recently [
26–33
]. The generalized synchronization is studied by applying pure error dynamics and elaborate
Lyapunov function in this paper.
The pure error dynamics can be analyzed theoretically without auxiliary numerical simulation, whereas the aid of
additional numerical simulation is unavoidable for current mixed error dynamics in which master state variables and slave
state variables are presented, while their maximum values must be determined by simulation [
34–38
]. Besides, the elaborate
Lyapunov function is applied rather than current plain square sum Lyapunov function, V
(
e
) =
12
e
T
e, which is currently
used for convenience. However, the Lyapunov function can be chosen in a variety of forms for different systems. Restricting
Lyapunov function to square sum makes a long river brook-like, and greatly weakens the power of Lyapunov direct method.
Based on the Lyapunov direct method [
39
], generalized synchronization is achieved and a systematic method of designing
Lyapunov function is proposed. The technique is successfully applied to both autonomous and nonautonomous double
Mathieu systems. This paper is organized as follows. In Section
2
, the method of designing Lyapunov function is presented,
∗
Corresponding address: Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, ROC. Tel.: +886 3 5712121 55119; fax: +886 3 5720634.E-mail address:zmg@cc.nctu.edu.tw(Z.-M. Ge).
0362-546X/$ – see front matter©2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.04.020
and generalized synchronization is obtained. Section
3
contains the examples of autonomous and nonautonomous double
Mathieu systems, and numerical simulations show the feasibility of the proposed method. Finally, the conclusions are
drawn.
2. Design of Lyapunov function
Consider the master and slave nonlinear dynamic systems described by
˙
x
=
f
(
t
,
x
)
(2.1)
˙
y
=
f
(
t
,
y
) +
u
(
t
,
x
,
y
)
(2.2)
where x
,
y
∈
R
nare master and slave state vectors, f
:
R
+×R
n→
R
nis a nonlinear vector function, and u
:
R
+×R
n×R
n→
R
nis controller vector.
Generalized synchronization means that there is a functional relation y
=
g
(
t
,
x
)
between master and slave states as
time goes to infinity, where g
:
R
+×
R
n→
R
nis a continuously differentiable vector function. Define e
=
y
−
g
(
t
,
x
)
as the
generalized synchronization error vector, and the error dynamics can be obtained:
˙
e
= ˙
y
− ˙
g
(
t
,
x
)
= ˙
y
−
∂
g
(
t
,
x
)
∂
x
x
˙
−
∂
g
(
t
,
x
)
∂
t
=
f
(
t
,
y
) −
∂
g
(
t
,
x
)
∂
x
f
(
t
,
x
) −
∂
g
(
t
,
x
)
∂
t
+
u
(
t
,
x
,
y
).
(2.3)
Based on Lyapunov direct method [
39
], the scheme of generalized synchronization and the procedure of designing Lyapunov
function are described as follows:
Step 1. Construct a Lyapunov function
V
(
t
,
e
) =
1
2
e
T3
(
t
)
e
=
1
2
λ
11(
t
)
e
2 1+
1
2
λ
22(
t
)
e
2 2+ · · · +
1
2
λ
nn(
t
)
e
2 n(2.4)
where
3
(
t
) = [λ
ii(
t
)] ∈
R
n×nis an unknown continuously differentiable positive definite diagonal matrix to be designed.
Its derivative is
˙
V
(
t
,
e
) = ˙
e
T3
(
t
)
e
+
1
2
e
T3
˙
(
t
)
e
=
λ
11(
t
)
e
1˙
e
1+
λ
22(
t
)
e
2˙
e
2+ · · · +
λ
nn(
t
)
e
n˙
e
n+
1
2
˙
λ
11(
t
)
e
21+
1
2
˙
λ
22(
t
)
e
22+ · · · +
1
2
˙
λ
nn(
t
)
e
2n.
(2.5)
Step 2. Eq.
(2.5)
can be rewritten in the following form:
˙
V
(
t
,
e
) =
G
1(λ
11, ˙λ
11)
e
21+
G
2(λ
22, ˙λ
22)
e
22+ · · · +
G
n(λ
nn, ˙λ
nn)
e
2n+ [H
1(λ
11, . . . , λ
nn,
x
1, . . . ,
x
n,
y
1, . . . ,
y
n,
t
) + λ
11u
1]e
1+ [H
2(λ
11, . . . , λ
nn,
x
1, . . . ,
x
n,
y
1, . . . ,
y
n,
t
) + λ
22u
2]e
2+ · · · + [H
n(λ
11, . . . , λ
nn,
x
1, . . . ,
x
n,
y
1, . . . ,
y
n,
t
) + λ
nnu
n]e
n(2.6)
where G
i(λ
ii, ˙λ
ii)
and H
i(λ
11, . . . , λ
nn,
x
1, . . . ,
x
n,
y
1, . . . ,
y
n,
t
) (
i
=
1
,
2
, . . . ,
n
)
are continuous differentiable functions,
u
i(
i
=
1
,
2
, . . . ,
n
)
are controllers to be determined.
Step 3. Eq.
(2.6)
may be classified as two general forms: (1) All G
i(λ
ii, ˙λ
ii)
depend on
λ
ii(
t
)
and
λ
˙
ii(
t
)
, (2) Some of G
j(λ
jj, ˙λ
jj)
depend on
λ
jj(
t
)
and
λ
˙
jj(
t
)
, the remaining G
k(λ
kk, ˙λ
kk)
depend only on
λ
˙
kk(
t
)
.
Form (1) All G
i(λ
ii, ˙λ
ii)
depend on
λ
ii(
t
)
and
λ
˙
ii(
t
)
.
Step 4. Design the controllers u
isuch that
H
i(λ
11, . . . , λ
nn,
x
1, . . . ,
x
n,
y
1, . . . ,
y
n,
t
) + λ
iiu
i=
0
(
i
=
1
,
2
, . . . ,
n
)
(2.7)
i.e., current mixed error dynamics has been replaced by pure error dynamics. By Eq.
(2.7)
, we design the controllers u
isuch
that
λ
ii(
i
=
1
,
2
, . . . ,
n
)
are linear function of each other with positive coefficients.
Step 5. Design
λ
11(
t
), λ
22(
t
), . . . , λ
nn(
t
)
such that
where
λ
m ii,
λ
M iiare positive constants, and
∀t
≥
0
,
G
i(λ
ii, ˙λ
ii) <
0
(
i
=
1
,
2
, . . . ,
n
)
(2.9)
then the Lyapunov function can be obtained and the generalized synchronization is achieved according to the Lyapunov
direct method.
Form (2) Some of G
j(λ
jj, ˙λ
jj)
depend on
λ
jj(
t
)
and
λ
˙
jj(
t
)
, and the remaining G
k(λ
kk, ˙λ
kk)
depend only on
λ
˙
kk(
t
)
.
Step 4. Assume
∀k
, λ
kk(
t
) =
1
(2.10)
∀k
,
H
k(λ
11, . . . , λ
nn,
x
1, . . . ,
x
n,
y
1, . . . ,
y
n,
t
) + λ
kk(
t
)
u
k= −e
k(2.11)
∀j
,
H
j(λ
11, . . . , λ
nn,
x
1, . . . ,
x
n,
y
1, . . . ,
y
n,
t
) + λ
jj(
t
)
u
j=
0
(2.12)
i.e., current mixed error dynamics has been replaced by pure error dynamics, and appropriately design the controllers
u
i(
i
=
1
,
2
, . . . ,
n
)
and
λ
jj(
t
)
such that
∀t
≥
0
,
0
< λ
mjj≤
λ
jj(
t
) ≤ λ
Mjj(2.13)
where
λ
m jj,
λ
M jjare positive constants, and
∀t
≥
0
,
G
j(λ
jj, ˙λ
jj) <
0
(2.14)
then the Lyapunov function can be obtained and the generalized synchronization is achieved according to the Lyapunov
direct method.
3. Generalized synchronization of double Mathieu systems
In this section, the functional relation between master and slave states is y
i=
g
i(
t
,
x
i) = α(
t
)
x
i+
β(
t
) (
i
=
1
,
2
, . . . ,
n
)
.
To demonstrate the use of the proposed method, two examples of autonomous and nonautonomous double Mathieu systems
are presented.
3.1. Regular and chaotic dynamics of autonomous and nonautonomous double Mathieu systems
The nonlinear damped Mathieu system is [
40
,
41
]
˙
x
1=
x
2˙
x
2= −a
(
1
+
sin
ω
t
)
x
1−
(
1
+
sin
ω
t
)
x
31−
ax
2.
(3.1)
An autonomous double Mathieu system can be constructed by mutual linear coupling of two Mathieu systems:
˙
x
1=
x
2˙
x
2= −a
(
1
+
x
4)
x
1−
(
1
+
x
4)
x
31−
ax
2+
bx
3˙
x
3=
x
4˙
x
4= −
(
1
+
x
2)
x
3−
a
(
1
+
x
2)
x
33−
ax
4+
bx
1.
(3.2)
The parameters in simulation are a
=
0
.
5
,
b
=
1–1
.
254
,
and the initial condition is x
1(
0
) =
0
.
1
,
x
2(
0
) =
0
.
1
,
x
3(
0
) =
0
.
2
,
x
4(
0
) =
0
.
2. The phase portraits, Poincaré maps, bifurcation diagram, and Lyapunov exponents are shown in
Fig. 1
. It
can be observed that the motion is period 1 for b
=
1
.
1, period 4 for b
=
1
.
243, and period 8 for b
=
1
.
246. For b
=
1
.
24,
the motion is chaotic.
A nonautonomous double Mathieu system can also be constructed by mutual linear coupling of two Mathieu systems:
˙
x
1=
x
2˙
x
2= −a
(
1
+
sin
ω
t
)
x
1−
(
1
+
sin
ω
t
)
x
31−
ax
2+
bx
3˙
x
3=
x
4˙
x
4= −
(
1
+
sin
ω
t
)
x
3−
a
(
1
+
sin
ω
t
)
x
33−
ax
4+
bx
1.
(3.3)
The parameters in simulation are a
=
0
.
5
,
b
=
0
.
9–1
, ω =
1
,
and the initial condition is x
1(
0
) =
0
.
1
,
x
2(
0
) =
0
.
1
,
x
3(
0
) =
0
.
2
,
x
4(
0
) =
0
.
2. The phase portraits, Poincaré maps, bifurcation diagram, and Lyapunov exponents are shown in
Fig. 2
. It
can be observed that the motion is period 1 for b
=
0
.
9, period 2 for b
=
0
.
93, and period 4 for b
=
0
.
934. For b
=
1, the
motion is chaotic.
1 0.5 0 x2 -0.5 -1 x2 0 0.5 x1 x1 1 1 0.5 0 -0.5 -1.5 -1 -0.5 0 0.5 1 1.5 x2 x1 1 0.5 0 -0.5 -1.5 -1 x2 x1 x1 1 0.5 0 -0.5 -1.5 -1 -0.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0 0.5 1 1.5 1 1.05 1.1 1.15 b 1.2 1.254 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 1 1.05 1.1 b 1.15 1.2 1.254 0.1 -1 L y apuno v e xponents
a
b
c
-0.5 0 0.5 1 1.5Fig. 1. (a) Phase portraits and Poincaré maps (b) Bifurcation diagram (c) Lyapunov exponents for autonomous double Mathieu system.
3.2. Generalized synchronization of autonomous double Mathieu systems
The master and slave autonomous double Mathieu systems can be described by
˙
x
1=
x
2˙
x
2= −a
(
1
+
x
4)
x
1−
(
1
+
x
4)
x
31−
ax
2+
bx
3˙
x
3=
x
4˙
x
4= −
(
1
+
x
2)
x
3−
a
(
1
+
x
2)
x
33−
ax
4+
bx
1(3.4)
x1 x2 L y apuno v e xponents
a
b
c
0.1 0.05 0 -0.05 -0.1 x1 x2 x2 0.2 0.1 0 -0.1 -0.2 x1 x1 x1 x2 5 0 -5 -4 -2 0 2 4 0.9 0.91 0.92 0.93 0.94 0.95 b 0.96 0.97 0.98 0.99 1 0.9 0.91 0.92 0.93 0.94 0.95 b 0.96 0.97 0.98 0.99 1 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 0 0.1 0.2 0.3 0.4 -0.4 -0.3 -0.2 -0.1 0 0.2 0.1 0 -0.1 -0.2 -0.4 -0.3 -0.2 -0.1 0 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2Fig. 2. (a) Phase portraits and Poincaré maps (b) Bifurcation diagram (c) Lyapunov exponents for nonautonomous double Mathieu system.
˙
y
1=
y
2+
u
1˙
y
2= −a
(
1
+
y
4)
y
1−
(
1
+
y
4)
y
31−
ay
2+
by
3+
u
2˙
y
3=
y
4+
u
3˙
y
4= −
(
1
+
y
2)
y
3−
a
(
1
+
y
2)
y
33−
ay
4+
by
1+
u
4.
(3.5)
The parameters in simulation are a
=
0
.
5, b
=
1
.
24, and the initial condition is x
1(
0
) =
0
.
1, x
2(
0
) =
0
.
1, x
3(
0
) =
0
.
2,
x
4(
0
) =
0
.
2, y
1(
0
) =
0
.
3, y
2(
0
) =
0
.
3, y
3(
0
) =
0
.
4, y
4(
0
) =
0
.
4.
Let e
i=
y
i−
α(
t
)
x
i−
β(
t
) (
i
=
1
, . . . ,
4
)
and subtract Eq.
(3.4)
from Eq.
(3.5)
, then the error dynamics can be
obtained:
˙
e
1=
e
2− ˙
α(
t
)
x
1+
β(
t
) − ˙β(
t
) +
u
1˙
e
2= −ae
1−
ae
2+
be
3−
a
(
y
1y
4−
α(
t
)
x
1x
4) − [(
1
+
y
4)
y
31−
α(
t
)(
1
+
x
4)
x
31]
− ˙
α(
t
)
x
2+
(
b
−
2a
)β(
t
) − ˙β(
t
) +
u
2˙
e
3=
e
4− ˙
α(
t
)
x
3+
β(
t
) − ˙β(
t
) +
u
3˙
e
4= −e
3−
ae
4+
be
1−
(
y
2y
3−
α(
t
)
x
2x
3) −
a[
(
1
+
y
2)
y
33−
α(
t
)(
1
+
x
2)
x
33]
− ˙
α(
t
)
x
4+
(
b
−
a
−
1
)β(
t
) − ˙β(
t
) +
u
4.
(3.6)
Step 1. Construct a Lyapunov function
V
(
t
,
e
) =
1
2
e
T3
(
t
)
e
=
1
2
λ
11(
t
)
e
2 1+
1
2
λ
22(
t
)
e
2 2+
1
2
λ
33(
t
)
e
2 3+
1
2
λ
44(
t
)
e
2 4.
(3.7)
Its derivative is
˙
V
(
t
,
e
) =
1
2
˙
λ
11(
t
)
e
21+
λ
11(
t
)
e
1˙
e
1+
1
2
˙
λ
22(
t
)
e
22+
λ
22(
t
)
e
2˙
e
2+
1
2
˙
λ
33(
t
)
e
23+
λ
33(
t
)
e
3e
˙
3+
1
2
˙
λ
44(
t
)
e
24+
λ
44(
t
)
e
4e
˙
4.
(3.8)
Step 2. Eq.
(3.8)
can be rewritten in the following form
˙
V
(
t
,
e
) =
G
1(λ
11, ˙λ
11)
e
21+
G
2(λ
22, ˙λ
22)
e
22+
G
3(λ
33, ˙λ
33)
e
32+
G
4(λ
44, ˙λ
44)
e
24+ [H
1(λ
11, . . . , λ
44,
x
1, . . . ,
x
4,
y
1, . . . ,
y
4,
t
) + λ
11u
1]e
1+ [H
2(λ
11, . . . , λ
44,
x
1, . . . ,
x
4,
y
1, . . . ,
y
4,
t
) + λ
22u
2]e
2+ [H
3(λ
11, . . . , λ
44,
x
1, . . . ,
x
4,
y
1, . . . ,
y
4,
t
) + λ
33u
3]e
3+ [H
4(λ
11, . . . , λ
44,
x
1, . . . ,
x
4,
y
1, . . . ,
y
4,
t
) + λ
44u
4]e
4(3.9)
where
G
1(λ
11, ˙λ
11) =
1
2
˙
λ
11(
t
) − λ
11(
t
)
G
2(λ
22, ˙λ
22) =
1
2
˙
λ
22(
t
) −
a
λ
22(
t
)
G
3(λ
33, ˙λ
33) =
1
2
˙
λ
33(
t
) − λ
33(
t
)
G
4(λ
44, ˙λ
44) =
1
2
˙
λ
44(
t
) −
a
λ
44(
t
)
H
1(λ
11, . . . ,
t
) = λ
11(
t
)[− ˙α(
t
)
x
1+
β(
t
) − ˙β(
t
) +
e
1] +
b
λ
44(
t
)
e
4H
2(λ
11, . . . ,
t
) = λ
11(
t
)
e
1+
λ
22(
t
)[−
ae
1−
a
(
y
4y
1−
α(
t
)
x
4x
1) − ((
1
+
y
4)
y
13−
α(
t
)(
1
+
x
4)
x
31)
− ˙
α(
t
)
x
2+
(
b
−
2a
)β(
t
) − ˙β(
t
)]
H
3(λ
11, . . . ,
t
) =
b
λ
22(
t
)
e
2+
λ
33(
t
)[− ˙α(
t
)
x
3+
β(
t
) − ˙β(
t
) +
e
3]
H
4(λ
11, . . . ,
t
) = λ
33(
t
)
e
3+
λ
44(
t
)[−
e
3−
(
y
2y
3−
α(
t
)
x
2x
3) −
a
((
1
+
y
2)
y
33−
α(
t
)(
1
+
x
2)
x
33)
− ˙
α(
t
)
x
4+
(
b
−
a
−
1
)β(
t
) − ˙β(
t
)].
(3.10)
Step 3. Since all G
i(λ
ii, ˙λ
ii)
depend on
λ
ii(
t
)
and
λ
˙
ii(
t
) (
i
=
1
, . . . ,
4
)
, Eq.
(3.9)
can be classified as form (1).
Step 4. Design the controllers
u
1= −y
1−
by
4+
(α(
t
) + ˙α(
t
))
x
1+
b
α(
t
)
x
4+
b
β(
t
) + ˙β(
t
)
u
2=
a
(
y
1y
4−
α(
t
)
x
1x
4) + (
1
+
y
4)
y
31−
α(
t
)(
1
+
x
4)
x
31+ ˙
α(
t
)
x
2−
(
b
−
2a
)β(
t
) + ˙β(
t
)
u
3= −by
2−
y
3+
(α(
t
) + ˙α(
t
))
x
3+
b
α(
t
)
x
2+
b
β(
t
) + ˙β(
t
)
u
4=
y
2y
3−
α(
t
)
x
2x
3+
a
(
1
+
y
2)
y
33−
α(
t
)(
1
+
x
2)
x
33+
1
−
1
a
y
3−
1
−
1
a
α(
t
)
x
3+ ˙
α(
t
)
x
4−
b
−
a
−
1
a
β(
t
) + ˙β(
t
)
(3.11)
such that
H
i(λ
11, . . . , λ
44,
x
1, . . . ,
x
4,
y
1, . . . ,
y
4,
t
) + λ
ii(
t
)
u
i=
0
(
i
=
1
, . . . ,
4
)
(3.12)
and
λ
ii(
i
=
1
, . . . ,
4
)
are linear function of each other with positive coefficients:
λ
11(
t
) = λ
44(
t
)
λ
22(
t
) =
1
a
λ
11(
t
)
λ
33(
t
) =
1
a
λ
11(
t
).
(3.13)
Now, the mixed error dynamics is replaced by pure error dynamics:
˙
V
(
t
,
e
) =
G
1(λ
11, ˙λ
11)
e
21+
G
2(λ
22, ˙λ
22)
e
22+
G
3(λ
33, ˙λ
33)
e
23+
G
4(λ
44, ˙λ
44)
e
24.
(3.14)
Step 5. Design
λ
11(
t
) =
1
1
+
e
−tλ
22(
t
) =
1
a
(
1
+
e
−t)
λ
33(
t
) =
1
a
(
1
+
e
−t)
λ
44(
t
) =
1
1
+
e
−t(3.15)
such that
∀t
≥
0
,
0
< λ
m11(
t
) =
1
2
≤
λ
11(
t
) ≤ λ
M11(
t
) =
1
∀t
≥
0
,
0
< λ
m22(
t
) =
1
2a
≤
λ
22(
t
) ≤ λ
M22(
t
) =
1
a
∀t
≥
0
,
0
< λ
m33(
t
) =
1
2a
≤
λ
33(
t
) ≤ λ
M33(
t
) =
1
a
∀t
≥
0
,
0
< λ
m44(
t
) =
1
2
≤
λ
44(
t
) ≤ λ
M44(
t
) =
1
(3.16)
∀t
≥
0
,
G
1(λ
11, ˙λ
11) =
1
2
˙
λ
11(
t
) − λ
11(
t
)
=
−
2
−
e
−t2
(
1
+
e
−t)
2<
0
∀t
≥
0
,
G
2(λ
22, ˙λ
22) =
1
2
˙
λ
22(
t
) −
a
λ
22(
t
)
=
−
2a
+
(
1
−
2a
)
e
−t2a
(
1
+
e
−t)
2=
−
1
(
1
+
e
−t)
2<
0
(∵
a
=
0
.
5 in simulation
)
∀t
≥
0
,
G
3(λ
33, ˙λ
33) =
1
2
˙
λ
33(
t
) − λ
33(
t
)
=
−
2
−
e
−t2a
(
1
+
e
−t)
2=
−
2
−
e
−t(
1
+
e
−t)
2<
0
(∵
a
=
0
.
5 in simulation
)
∀t
≥
0
,
G
4(λ
44, ˙λ
44) =
1
2
˙
λ
44(
t
) − λ
44(
t
)
=
−
2a
+
(
1
−
2a
)
e
−t2
(
1
+
e
−t)
2=
−
1
2
(
1
+
e
−t)
2<
0
(
∵
a
=
0
.
5 in simulation
)
(3.17)
x2 x3 y1 y3 e1 x4 x4 1 0.5 0 -0.5 -1 -1.5 x1 -0.5 0 0.5 1 1.5 x1 x2 x3 -0.5 0 0.5 1 1.5 x1 1 0.5 0 -0.5 -1.5 -1 1 0.5 0 -0.5 -1.5 -1 x2 x4 -0.5 0 0.5 1 1.5 x3 -0.5 0 0.5 1 1.5 2 1 0 -1 -2 y2 2 1 0 -1 -2 y4 2 1 0 -1 -2 2 1 0 -1 -2 0 -0.5 -1 0 5 10 15 20 25 30 35 40 45 50 e2 0 -0.5 -1 0 5 10 15 20 25 30 35 40 45 50 e3 0 -0.5 -1 0 5 10 15 20 25 30 35 40 45 50 e4 0 -0.5 -1 0 5 10 15 20 25 t 30 35 40 45 50
a
b
c
1 1.5 0.5 0 -0.5 -2 -1 0 1 -0.5 0 0.5 1 1.5 -2 -1 0 1 -2 -1 0 1Fig. 3. (a) Phase portraits of master system (b) Phase portraits of xito yi
(
i=
1, . . . ,
4)
when generalized synchronization is obtained (c) Time history oferrors.
then the Lyapunov function can be obtained
V
(
t
,
e
) =
1
2
(
1
+
e
−t)
e
2 1+
1
2a
(
1
+
e
−t)
e
2 2+
1
2a
(
1
+
e
−t)
e
2 3+
1
2
(
1
+
e
−t)
e
2 4(3.18)
and
˙
V
(
t
,
e
) = −
2
+
e
−t2
(
1
+
e
−t)
2e
2 1−
1
(
1
+
e
−t)
2e
2 2−
2
+
e
−t(
1
+
e
−t)
2e
2 3−
1
2
(
1
+
e
−t)
2e
2 4.
(3.19)
x2 x4 x1 5 10 5 0 -5 -10 x4 x3 10 5 0 -5 -10 0 -5 e1 0 -0.5 -1 0 5 10 15 20 25 30 35 40 45 50 e2 0 -0.5 -1 0 5 10 15 20 25 30 35 40 45 50 e3 0 -0.5 -1 0 5 10 15 20 25 30 35 40 45 50 e4 0 -0.5 -1 t
a
b
c
-4 -2 0 2 4 x1 x2 x2 -4 -2 0 2 4 y1 x1 -4 -2 0 2 4 x3 -4 -5 0 5 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 y3 x3 x4 -4 -2 0 2 4 y4 -4 -10 -5 0 5 10 -2 0 2 4 y2 -4 -5 0 5 -2 0 2 4 -4 -2 0 2 4 0 5 10 15 20 25 30 35 40 45 50Fig. 4. (a) Phase portraits of master system (b) Phase portraits of xito yi
(
i=
1, . . . ,
4)
when generalized synchronization is obtained (c) Time historyof errors.
Since Lyapunov global asymptotical stability theorem is satisfied, the global generalized synchronization is achieved.
α(
t
) =
sin
ω
t,
β(
t
) =
cos
ω
t are chosen in simulation, and the results are shown in
Fig. 3
.
3.3. Generalized synchronization of nonautonomous double Mathieu systems
˙
x
1=
x
2˙
x
2= −a
(
1
+
sin
ω
t
)
x
1−
(
1
+
sin
ω
t
)
x
31−
ax
2+
bx
3˙
x
3=
x
4˙
x
4= −
(
1
+
sin
ω
t
)
x
3−
a
(
1
+
sin
ω
t
)
x
33−
ax
4+
bx
1(3.20)
˙
y
1=
y
2+
u
1˙
y
2= −a
(
1
+
sin
ω
t
)
y
1−
(
1
+
sin
ω
t
)
y
31−
ay
2+
by
3+
u
2˙
y
3=
y
4+
u
3˙
y
4= −
(
1
+
sin
ω
t
)
y
3−
a
(
1
+
sin
ω
t
)
y
33−
ay
4+
by
1+
u
4.
(3.21)
The parameters in simulation are a
=
0
.
5, b
=
1,
ω =
1, and the initial condition is x
1(
0
) =
0
.
1, x
2(
0
) =
0
.
1, x
3(
0
) =
0
.
2,
x
4(
0
) =
0
.
2, y
1(
0
) =
0
.
3, y
2(
0
) =
0
.
3, y
3(
0
) =
0
.
4, y
4(
0
) =
0
.
4.
Let e
i=
y
i−
α(
t
)
x
i−
β(
t
) (
i
=
1
, . . . ,
4
)
and subtract Eq.
(3.20)
from Eq.
(3.21)
, then the error dynamics can be obtained:
˙
e
1=
e
2− ˙
α(
t
)
x
1+
β(
t
) − ˙β(
t
) +
u
1˙
e
2= −a
(
1
+
sin
ω
t
)
e
1−
ae
2+
be
3−
(
1
+
sin
ω
t
)(
y
31−
α(
t
)
x
3 1) − ˙α(
t
)
x
2+
(−
a
(
1
+
sin
ω
t
) −
a
+
b
)β(
t
) − ˙β(
t
) +
u
2˙
e
3=
e
4− ˙
α(
t
)
x
3+
β(
t
) − ˙β(
t
) +
u
3˙
e
4=
(
1
+
sin
ω
t
)
e
3−
ae
4+
be
1−
a
(
1
+
sin
ω
t
)(
y
33−
α(
t
)
x
3 3) − ˙α(
t
)
x
4+
(−(
1
+
sin
ω
t
) −
a
+
b
)β(
t
) − ˙β(
t
) +
u
4.
(3.22)
Step 1. Construct a Lyapunov function
V
(
t
,
e
) =
1
2
e
T3
(
t
)
e
=
1
2
λ
11(
t
)
e
2 1+
1
2
λ
22(
t
)
e
2 2+
1
2
λ
33(
t
)
e
2 3+
1
2
λ
44(
t
)
e
2 4.
(3.23)
Its derivative is
˙
V
(
t
,
e
) =
1
2
˙
λ
11(
t
)
e
21+
λ
11(
t
)
e
1˙
e
1+
1
2
˙
λ
22(
t
)
e
22+
λ
22(
t
)
e
2˙
e
2+
1
2
˙
λ
33(
t
)
e
23+
λ
33(
t
)
e
3e
˙
3+
1
2
˙
λ
44(
t
)
e
24+
λ
44(
t
)
e
4e
˙
4.
(3.24)
Step 2. Eq.
(3.24)
can be rewritten in the following form
˙
V
(
t
,
e
) =
G
1(λ
11, ˙λ
11)
e
21+
G
2(λ
22, ˙λ
22)
e
22+
G
3(λ
33, ˙λ
33)
e
32+
G
4(λ
44, ˙λ
44)
e
24+ [H
1(λ
11, . . . , λ
44,
x
1, . . . ,
x
4,
y
1, . . . ,
y
4,
t
) + λ
11u
1]e
1+ [H
2(λ
11, . . . , λ
44,
x
1, . . . ,
x
4,
y
1, . . . ,
y
4,
t
) + λ
22u
2]e
2+ [H
3(λ
11, . . . , λ
44,
x
1, . . . ,
x
4,
y
1, . . . ,
y
4,
t
) + λ
33u
3]e
3+ [H
4(λ
11, . . . , λ
44,
x
1, . . . ,
x
4,
y
1, . . . ,
y
4,
t
) + λ
44u
4]e
4(3.25)
where
G
1(λ
11, ˙λ
11) =
1
2
˙
λ
11(
t
)
G
2(λ
22, ˙λ
22) =
1
2
˙
λ
22(
t
) −
a
λ
22(
t
)
G
3(λ
33, ˙λ
33) =
1
2
˙
λ
33(
t
)
G
4(λ
44, ˙λ
44) =
1
2
˙
λ
44(
t
) −
a
λ
44(
t
)
H
1(λ
11, . . . ,
t
) = λ
11(
t
)[− ˙α(
t
)
x
1+
β(
t
) − ˙β(
t
)] +
b
λ
44(
t
)
e
4H
2(λ
11, . . . ,
t
) = λ
11(
t
)
e
1+
λ
22(
t
)[−
a
(
1
+
sin
ω
t
)
e
1−
(
1
+
sin
ω
t
)(
y
31−
α(
t
)
x
3 1) − ˙α(
t
)
x
2+
(−
a
(
1
+
sin
ω
t
) −
a
+
b
)β(
t
) − ˙β(
t
)]
H
3(λ
11, . . . ,
t
) =
b
λ
22(
t
)
e
2+
λ
33(
t
)[− ˙α(
t
)
x
3+
β(
t
) − ˙β(
t
)]
H
4(λ
11, . . . ,
t
) = λ
33(
t
)
e
3+
λ
44(
t
)[−(
1
+
sin
ω
t
)
e
3−
a
(
1
+
sin
ω
t
)(
y
33−
α(
t
)
x
3 3) − ˙α(
t
)
x
4+
(−(
1
+
sin
ω
t
) −
a
+
b
)β(
t
) − ˙β(
t
)].
(3.26)
Step 3. Since some of G
j(λ
jj, ˙λ
jj)
depend on
λ
jj(
t
)
and
λ
˙
jj(
t
) (
j
=
2
,
4
)
, the remaining G
k(λ
kk, ˙λ
kk)
depend only on
λ
˙
kk(
t
) (
k
=
1
,
3
)
, Eq.
(3.26)
can be classified as form (2).
Step 4. Assume
λ
11(
t
) =
1
λ
33(
t
) =
1
(3.27)
H
1(λ
11, . . . , λ
44,
x
1, . . . ,
x
4,
y
1, . . . ,
y
4,
t
) + λ
11(
t
)
u
1= −
e
1H
3(λ
11, . . . , λ
44,
x
1, . . . ,
x
4,
y
1, . . . ,
y
4,
t
) + λ
33(
t
)
u
3= −e
3(3.28)
H
2(λ
11, . . . , λ
44,
x
1, . . . ,
x
4,
y
1, . . . ,
y
4,
t
) + λ
22(
t
)
u
2=
0
H
4(λ
11, . . . , λ
44,
x
1, . . . ,
x
4,
y
1, . . . ,
y
4,
t
) + λ
44(
t
)
u
4=
0
(3.29)
and appropriately design the controllers u
i(
i
=
1
, . . . ,
4
)
and
λ
22(
t
)
,
λ
44(
t
)
u
1= −y
1−
b
2
+
sin
ω
t
y
4+
(α(
t
) + ˙α(
t
))
x
1+
b
α(
t
)
2
+
sin
ω
t
x
4+
b
β(
t
)
2
+
sin
ω
t
+ ˙
β(
t
)
u
2= −ay
1+
a
α(
t
)
x
1+ ˙
α(
t
)
x
2+
(
1
+
sin
ω
t
)(
y
31−
α(
t
)
x
3 1) + (
a sin
ω
t
+
3a
−
b
)β(
t
) + ˙β(
t
)
u
3= −y
3−
b
2
+
sin
ω
t
y
2+
(α(
t
) + ˙α(
t
))
x
3+
b
α(
t
)
2
+
sin
ω
t
x
2+
b
β(
t
)
2
+
sin
ω
t
+ ˙
β(
t
)
u
4= −y
3+
α(
t
)
x
3+ ˙
α(
t
)
x
4+
a
(
1
+
sin
ω
t
)(
y
33−
α(
t
)
x
3 3) + (
sin
ω
t
+
a
−
b
+
2
)β(
t
) + ˙β(
t
)
(3.30)
λ
22l
(
t
) =
1
a
(
2
+
sin
ω
t
)
λ
44(
t
) =
1
2
+
sin
ω
t
(3.31)
such that
∀t
≥
0
,
0
< λ
m22=
1
3a
≤
λ
22(
t
) ≤ λ
M22=
1
a
∀t
≥
0
,
0
< λ
m44=
1
3
≤
λ
44(
t
) ≤ λ
M44=
1
(3.32)
∀t
≥
0
,
G
2(λ
22, ˙λ
22) =
1
2
˙
λ
22(
t
) −
a
λ
22(
t
)
=
−
(
4a
+
2a sin
ω
t
+
ω
cos
ω
t
)
2a
(
2
+
sin
ω
t
)
2=
−
(
2
+
sin t
+
cos t
)
(
2
+
sin t
)
2<
0
(
∵
a
=
0
.
5
, ω =
1 in simulation
)
∀t
≥
0
,
G
4(λ
44, ˙λ
44) =
1
2
˙
λ
44(
t
) −
a
λ
44(
t
)
=
−
(
4a
+
2a sin
ω
t
+
ω
cos
ω
t
)
2
(
2
+
sin
ω
t
)
2=
−
(
2
+
sin t
+
cos t
)
2
(
2
+
sin t
)
2<
0
(
∵
a
=
0
.
5
, ω =
1 in simulation
).
(3.33)
Now, the mixed error dynamics is replaced by pure error dynamics:
˙
V
(
t
,
e
) = [
G
1(λ
11, ˙λ
11) − λ
11]e
21+
G
2(λ
22, ˙λ
22)
e
22+ [G
3(λ
33, ˙λ
33) − λ
33]e
23+
G
4(λ
44, ˙λ
44)
e
24.
(3.34)
Then the Lyapunov function can be obtained
V
(
t
,
e
) =
1
2
e
2 1+
1
2a
(
2
+
sin
ω
t
)
e
2 2+
1
2
e
2 3+
1
2
(
2
+
sin
ω
t
)
e
2 4(3.35)
and
˙
V
(
t
,
e
) = −
e
21−
2
+
sin t
+
cos t
(
2
+
sin t
)
2e
2 2−
e
2 3−
2
+
sin t
+
cos t
2
(
2
+
sin t
)
2e
2 4.
(3.36)
Since Lyapunov global asymptotical stability theorem is satisfied, the global generalized synchronization is achieved.
4. Conclusions
The generalized synchronization is studied by applying pure error dynamics and elaborate Lyapunov function in this
paper. By classification of the forms of
V
˙
(
t
,
e
)
, the complexity of designing suitable Lyapunov function is reduced greatly.
The proposed method is effectively applied to both autonomous and nonautonomous double Mathieu systems.
Acknowledgment
This research was supported by the National Science Council, Republic of China, under Grant Number NSC
94-2212-E-009-013.
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