# Finite-Time Synchronization of Chaotic Systems with Different Dimension and Secure Communication

Authors: Shouquan Pang; Yu Feng; Yongjian Liu
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

Finite-time synchronization of chaotic systems with different dimension and secure communication is investigated. It is rigorously proven that global finite-time synchronization can be achieved between three-dimension Lorenz chaotic system and four-dimension Lorenz hyperchaotic system which have certain parameters or uncertain parameters. The electronic circuits of finite-time synchronization using Multisim 12 are designed to verify our conclusion. And the application to the secure communications is also analyzed and discussed.


---

## Body

## 1. Introduction

Synchronization of chaotic systems recently has become one of the most interesting topics because of the seminal work of Pecora and Carroll [1]. These phenomena have arisen in various fields of science, like engineering technology [2], ecological system [3], biological neurons [4], and others [5–8] various types of synchronization. They have been deeply investigated in the previous literature. A large number of results have been reported in the past two decades [9].From a practical point of view, it will be more reasonable to realize synchronization in a given time [10–13]. This means optimality in convergence time [14]. This problem is important in all fields where synchronization finds or will find practical interest [15–17]. If we consider, for example, the application of synchronization in secure communications, the range of time during which the chaotic oscillators are not synchronized corresponds to the range of time during which the encoded message can unfortunately not be recovered or sent. More than a grave and irreversible loss of information, this is a catastrophe in digital telecommunications, since the first bits of standardized bit strings always contain signalization data, that is, the “identity card” of the message. Hence, it clearly appears that the synchronization time has to be minimized, so that the chaotic oscillators synchronize as fast as possible. In this context, it is fair to say that there is a need to study finite-time chaos synchronization and optimization problems of nonlinear systems with uncertainties.This paper mainly addresses the problem of finite-time master-slave synchronization of the three-dimensional (3D) Lorenz system and four-dimensional (4D) hyperchaotic Lorenz system. A family of feedbacks has been proposed to achieve the finite-time synchronization of the two systems with different dimension. It is rigorously proven that global finite-time synchronization can be achieved between three-dimension Lorenz chaotic system and four-dimension Lorenz hyperchaotic system which have certain parameters or uncertain parameters. Finally, the experimental setup for the coupling process has also been designed via Multisim 12 simulations. The numerical simulation and circuit experiment show that the proposed control technique is feasible and reliable.


## 2. Preliminary Definitions and Lemma

Finite-time synchronization means that the state of the slave system can track the state of the master system after the finite time. The definition of finite-time synchronization is given as follows:(1)
x
˙
t
=
f
t
,
x
,
y
˙
t
=
g
t
,
y
+
u
t
,
x
,
y
,where x
t and y
t are twoN-dimensional state vectors. x
,
y
∈
R
n, f, and g are differentiable functions in R
×
R
n
→
R
n. If there exists a constant t
s
>
0, such that lim
t
→
t
s
⁡
e
=
lim
t
→
t
s
⁡
y
(
t
)
-
x
(
t
)
=
0, one says that when t
>
t
s, synchronization of the x
˙
t
=
f
t
,
x and y
˙
t
=
g
t
,
y
+
u
t
,
x
,
y is achieved in a finite time.Lemma 1.
Let the nonlinear system(2)
z
˙
t
=
-
k
·
s
g
n
z
z
m
,
z
0
=
z
0
,where k
>
0, 0
<
m
<
1. For any initial value z
0, system (2) is stable in finite time t
s
=
z
0
1
-
m
/
k
1
-
m.


## 3. Finite-Time Synchronization of Different Dimension Systems

Lorenz system is considered a paradigm, since it captures many of the feathers of chaotic dynamics. The Lorenz system is described by the following nonlinear equations:(3)
x
˙
1
t
=
a
x
2
t
-
x
1
t
,
x
˙
2
t
=
c
x
1
t
-
x
2
t
-
x
1
t
x
3
t
,
x
˙
3
t
=
x
1
t
x
2
t
-
b
x
3
t
,where a
,
b, and c are the constants. With a
=
10, b
=
8
/
3, and c
=
28, the Lorenz system shows chaotic behavior.By introducing a nonlinear feedback controller to the second equation of the system (3), the follow hyperchaotic system is obtained [18]:(4)
y
˙
1
t
=
α
y
2
t
-
y
1
t
,
y
˙
2
t
=
γ
y
1
t
-
y
2
t
-
y
2
t
y
3
t
+
y
4
t
,
y
˙
3
t
=
-
β
y
3
t
+
y
1
t
y
2
t
,
y
˙
4
t
=
-
k
y
1
t
-
0.3
y
2
t
y
3
t
.When k
=
2, α
=
10, β
=
8
/
3, and γ
=
28, the four Lyapunov exponents are λ
1
=
0.2230, λ
2
=
0.1389, λ
3
=
0, and λ
4
=
-
14.0380. The Lyapunov dimension is D
L
=
3.0258, and numerical simulations have verified that system (4) indeed has a hyperchaotic attractor.


### 3.1. Finite-Time Synchronization of Systems with the Certain Parameters

This subsection deals with finite-time synchronization of Lorenz system and hyperchaotic Lorenz system. Considering any initial conditions, system (3) is regarded as the master and system (4) as the slaver.The master system(5)
X
˙
=
-
a
1
0
c
-
1
0
0
0
-
b
X
+
0
-
x
1
x
3
x
1
x
2could be described by(6)
X
˙
=
A
X
+
F
x
,where X
=
x
1
x
2
x
3, A
=
-
a
1
0
c
-
1
0
0
0
-
b, F
(
x
)
=
0
-
x
1
x
3
x
1
x
2, and x
1, x
2, and x
3 are the state constants and a
,
b, and c are state parameters of system.Hyperchaotic Lorenz system (4) is regarded as the slaver system, and u
(
e
,
y
) is controllers, which is determined for achieving synchronization between master system (3) and slaver system (4). Described by (7)
Y
˙
=
B
Y
+
g
y
+
u
e
,
y
,where Y
=
y
1
,
y
2
,
y
3
,
y
4
T is the state vector of the responder system, g
y
=
0
,
-
y
1
y
3
,
y
1
y
2
,
-
0.3
y
2
y
3
T is a continuous smooth function and u
=
u
1
,
u
2
,
u
3
,
u
4
T is a control function. To make the same structures between (6) and (7), a new artificial variable x
4 is introduced into system (6) and set x
˙
4
t
=
0. Subtract (6) from (7) and define the state errors as(8)
e
1
=
y
1
-
x
1
,
e
2
=
y
2
-
x
2
,
e
3
=
y
3
-
x
3
,
e
4
=
y
4
-
x
4
.Then the error system between the Lorenz system and the controlled hyperchaotic Lorenz system could be written as(9)
e
˙
1
=
a
e
2
-
e
1
+
u
1
,
e
˙
2
=
c
e
1
-
e
2
-
e
1
e
3
-
e
1
x
3
-
e
3
x
1
+
u
2
,
e
˙
3
=
b
e
3
+
e
1
e
2
+
e
1
x
2
+
e
2
x
1
+
u
3
,
e
˙
4
=
-
2
e
1
-
2
x
1
-
e
2
e
3
-
e
2
x
3
-
e
3
x
2
-
x
2
x
3
+
u
4
.The above four equations merge into a unified form:(10)
e
˙
=
B
e
+
f
e
,
x
+
u
e
,
x
,where(11)
e
=
e
1
e
2
e
3
e
4
,
B
=
-
a
a
0
0
c
-
1
0
0
0
0
b
0
-
2
0
0
0
,
f
e
,
x
=
-
x
3
e
2
-
x
2
e
3
-
e
2
e
3
x
3
e
1
+
x
1
e
3
+
e
1
e
3
1
3
x
2
e
1
+
1
3
x
1
e
2
+
1
3
e
1
e
2
0.5
x
3
e
2
+
0.5
x
2
e
3
+
0.5
e
2
e
3
+
d
x
1
+
0.5
x
2
x
3
,
u
=
u
1
u
2
u
3
u
4
.Our aim is to design a suitable controller, in which we can achieve the finite-time synchronization of Lorenz system (3) and hyperchaotic Lorenz system (4). This problem can be converted to design a controller to attain finite-time stability of error system (10).For Lorenz system (3) and hyperchaotic Lorenz system (4) define the control functions u
1, u
2, u
3, and u
4 as follows:(12)
u
1
=
-
a
e
2
-
e
1
-
k
sgn
⁡
e
1
e
1
m
,
u
2
=
-
c
e
1
+
e
2
+
e
1
e
3
+
e
1
x
3
+
e
3
x
1
-
k
sgn
⁡
e
2
e
2
m
,
u
3
=
-
b
e
3
-
e
1
e
2
-
e
1
x
2
-
e
2
x
1
-
k
sgn
⁡
e
3
e
3
m
,
u
4
=
2
e
1
+
2
x
1
+
0.3
e
2
e
3
+
0.3
e
2
x
3
+
0.3
e
3
x
2
+
0.3
x
2
x
3
-
k
sgn
⁡
e
4
e
4
m
,where k
>
0
,
0
<
m
<
1. Substituting (12) into (10) yields(13)
e
˙
=
-
k
sgn
⁡
e
1
e
1
m
-
k
sgn
⁡
e
2
e
2
m
-
k
sgn
⁡
e
3
e
3
m
-
k
sgn
⁡
e
4
e
4
m
.From Lemma1, system (13) is stable in the finite time. That means(14)
t
>
t
s
i
=
e
0
i
1
-
m
k
1
-
m
,
e
i
t
i
s
=
0
,
i
=
1,2
,
3,4
.The synchronization of Lorenz system (3) and hyperchaotic Lorenz system (4) is achieved in a finite time.Theorem 2.
Suppose thatk
>
0
,
0
<
m
<
1. Let controller u
i
i
=
1,2
,
3,4 be defined as (12). Then error system (10) can be stabilized in finite time by the controller u
i
i
=
1,2
,
3,4. That is to say, slave system (4) will synchronize with master system (3) in finite time.


### 3.2. Finite-Time Synchronization of Systems with Uncertain Parameters

We consider the more realistic and practical case, where some of the parameters of the master system and slave system are unknown. It is valuable because practical chaos systems are often disturbed by different factors. It is that the parameters of the systems have been disturbed.Consider the master system(15)
x
˙
1
t
=
a
x
2
t
-
x
1
t
,
x
˙
2
t
=
c
+
Δ
1
x
1
t
-
x
2
t
-
x
1
t
x
3
t
,
x
˙
3
t
=
x
1
t
x
2
t
-
b
+
Δ
2
x
3
tand the slave system(16)
y
˙
1
t
=
α
y
2
t
-
y
1
t
+
V
1
t
,
y
˙
2
t
=
γ
+
Δ
3
y
1
t
-
y
2
t
-
y
1
t
y
3
t
+
y
4
t
+
V
2
t
,
y
˙
3
t
=
-
β
+
Δ
4
y
3
t
+
y
1
t
y
2
t
+
V
3
t
,
y
˙
4
t
=
-
k
y
1
t
-
0.3
y
2
t
y
3
t
+
V
4
t
,where Δ
i
i
=
1,2
,
3,4 are the increments of the parameter, which is caused by the interruption, and V
1
t, V
2
t, V
3
t, and V
4
(
t
) are the control functions. Analogously, as in Section 3.1, a new artificial variable x
4 is given and x
˙
4
t
=
0, by defining the synchronization error in the following way:(17)
e
1
=
y
1
-
x
1
,
e
2
=
y
2
-
x
2
,
e
3
=
y
3
-
x
3
,
e
4
=
y
4
-
x
4
,and then the error system can be written as(18)
e
˙
1
=
α
e
2
-
e
1
+
α
-
a
x
2
-
x
1
+
V
1
,
e
˙
2
=
γ
+
Δ
3
e
1
+
γ
+
Δ
3
-
c
-
Δ
1
x
1
-
e
2
-
e
1
e
3
-
e
1
x
3
-
x
1
e
3
+
y
4
+
V
2
,
e
˙
3
=
-
β
+
Δ
4
e
3
-
β
+
Δ
4
-
b
-
Δ
2
x
3
+
e
1
e
2
+
e
1
x
2
+
x
1
e
2
+
V
3
,
e
˙
4
=
-
k
x
1
-
k
e
1
-
0.3
e
2
e
3
-
0.3
x
3
e
2
-
0.3
x
2
e
3
-
0.3
x
2
x
3
+
V
4
.In order to solve the finite-time master-slave synchronization problem between system equations (15) and (16), we select the Lyapunov function:(19)
V
=
1
2
e
1
2
+
1
2
e
2
2
+
1
2
e
3
2
+
1
2
e
4
2
.Differentiation of this function with respect to time yields(20)
V
˙
=
e
1
e
˙
1
+
e
2
e
˙
2
+
e
3
e
˙
3
+
e
4
e
˙
4
.Combining (18) and (20), we have(21)
V
˙
=
e
1
α
e
2
-
e
1
+
α
-
a
x
2
-
x
1
+
V
1
+
e
2
γ
+
Δ
3
e
1
+
γ
+
Δ
3
-
c
-
Δ
1
x
1
-
e
2
-
e
1
e
3
-
e
1
x
3
-
x
1
e
3
+
y
4
+
V
2
+
e
3
-
β
+
Δ
4
e
3
-
β
+
Δ
4
-
b
-
Δ
2
x
3
+
e
1
e
2
+
e
1
x
2
+
x
1
e
2
+
V
3
+
e
4
-
k
x
1
-
k
e
1
-
0.3
e
2
e
3
-
0.3
x
3
e
2
-
0.3
x
2
e
3
-
0.3
x
2
x
3
+
V
4
.We defined the active control functions V
1
t, V
2
t, V
3
t, and V
4
t as follows:(22)
V
1
=
-
α
-
a
x
2
-
x
1
-
α
e
2
-
e
1
m
sign
⁡
e
1
,
V
2
=
-
γ
e
1
-
γ
-
c
x
1
-
y
4
+
e
1
x
3
-
k
1
x
1
sign
⁡
e
2
-
k
2
e
1
sign
⁡
e
2
-
e
2
m
sign
⁡
e
2
,
V
3
=
-
e
1
x
2
+
β
-
b
x
3
-
k
3
x
3
sign
⁡
e
3
-
e
3
m
sign
⁡
e
3
,
V
4
=
k
x
1
+
k
e
1
+
0.3
e
2
e
3
+
0.3
e
2
x
3
+
0.3
e
3
x
2
+
0.3
x
2
x
3
-
e
4
m
sign
⁡
e
4
.One obtains(23)
V
˙
=
-
α
e
1
2
-
e
1
1
+
m
+
Δ
3
e
1
e
2
+
Δ
3
-
Δ
1
x
1
e
2
-
e
2
2
-
k
1
x
1
e
2
sign
⁡
e
2
-
k
2
e
1
e
2
sign
⁡
e
2
-
e
2
1
+
m
-
β
+
Δ
4
e
3
2
+
Δ
2
-
Δ
4
x
3
e
3
-
k
3
x
3
e
3
sign
⁡
e
3
-
e
3
1
+
m
-
e
4
1
+
m
=
-
α
e
1
2
-
k
2
-
Δ
3
sign
⁡
e
1
e
2
e
1
e
2
-
k
1
-
Δ
3
-
Δ
1
sign
⁡
x
1
e
2
x
1
e
2
-
β
+
Δ
4
e
3
2
-
k
3
-
Δ
2
-
Δ
4
s
i
g
n
x
3
e
3
e
3
x
3
-
e
1
1
+
m
-
e
2
1
+
m
-
e
3
1
+
m
-
e
4
1
+
m
≤
-
e
1
1
+
m
-
e
2
1
+
m
-
e
3
1
+
m
-
e
4
1
+
m
=
-
2
1
+
m
/
2
V
1
+
m
/
2
.From Lemma 1, it follows that (18) is finite-time stabilized. Thus, uncertain slave system (16) can synchronize uncertain master system (15) in finite time.Theorem 3.
Suppose thatk
>
0, 0
<
m
<
1. Let controller u
i
i
=
1,2
,
3,4 be defined as (22). Then uncertain slave system (16) synchronizes with uncertain master system (15) in finite time.


## 3.1. Finite-Time Synchronization of Systems with the Certain Parameters

This subsection deals with finite-time synchronization of Lorenz system and hyperchaotic Lorenz system. Considering any initial conditions, system (3) is regarded as the master and system (4) as the slaver.The master system(5)
X
˙
=
-
a
1
0
c
-
1
0
0
0
-
b
X
+
0
-
x
1
x
3
x
1
x
2could be described by(6)
X
˙
=
A
X
+
F
x
,where X
=
x
1
x
2
x
3, A
=
-
a
1
0
c
-
1
0
0
0
-
b, F
(
x
)
=
0
-
x
1
x
3
x
1
x
2, and x
1, x
2, and x
3 are the state constants and a
,
b, and c are state parameters of system.Hyperchaotic Lorenz system (4) is regarded as the slaver system, and u
(
e
,
y
) is controllers, which is determined for achieving synchronization between master system (3) and slaver system (4). Described by (7)
Y
˙
=
B
Y
+
g
y
+
u
e
,
y
,where Y
=
y
1
,
y
2
,
y
3
,
y
4
T is the state vector of the responder system, g
y
=
0
,
-
y
1
y
3
,
y
1
y
2
,
-
0.3
y
2
y
3
T is a continuous smooth function and u
=
u
1
,
u
2
,
u
3
,
u
4
T is a control function. To make the same structures between (6) and (7), a new artificial variable x
4 is introduced into system (6) and set x
˙
4
t
=
0. Subtract (6) from (7) and define the state errors as(8)
e
1
=
y
1
-
x
1
,
e
2
=
y
2
-
x
2
,
e
3
=
y
3
-
x
3
,
e
4
=
y
4
-
x
4
.Then the error system between the Lorenz system and the controlled hyperchaotic Lorenz system could be written as(9)
e
˙
1
=
a
e
2
-
e
1
+
u
1
,
e
˙
2
=
c
e
1
-
e
2
-
e
1
e
3
-
e
1
x
3
-
e
3
x
1
+
u
2
,
e
˙
3
=
b
e
3
+
e
1
e
2
+
e
1
x
2
+
e
2
x
1
+
u
3
,
e
˙
4
=
-
2
e
1
-
2
x
1
-
e
2
e
3
-
e
2
x
3
-
e
3
x
2
-
x
2
x
3
+
u
4
.The above four equations merge into a unified form:(10)
e
˙
=
B
e
+
f
e
,
x
+
u
e
,
x
,where(11)
e
=
e
1
e
2
e
3
e
4
,
B
=
-
a
a
0
0
c
-
1
0
0
0
0
b
0
-
2
0
0
0
,
f
e
,
x
=
-
x
3
e
2
-
x
2
e
3
-
e
2
e
3
x
3
e
1
+
x
1
e
3
+
e
1
e
3
1
3
x
2
e
1
+
1
3
x
1
e
2
+
1
3
e
1
e
2
0.5
x
3
e
2
+
0.5
x
2
e
3
+
0.5
e
2
e
3
+
d
x
1
+
0.5
x
2
x
3
,
u
=
u
1
u
2
u
3
u
4
.Our aim is to design a suitable controller, in which we can achieve the finite-time synchronization of Lorenz system (3) and hyperchaotic Lorenz system (4). This problem can be converted to design a controller to attain finite-time stability of error system (10).For Lorenz system (3) and hyperchaotic Lorenz system (4) define the control functions u
1, u
2, u
3, and u
4 as follows:(12)
u
1
=
-
a
e
2
-
e
1
-
k
sgn
⁡
e
1
e
1
m
,
u
2
=
-
c
e
1
+
e
2
+
e
1
e
3
+
e
1
x
3
+
e
3
x
1
-
k
sgn
⁡
e
2
e
2
m
,
u
3
=
-
b
e
3
-
e
1
e
2
-
e
1
x
2
-
e
2
x
1
-
k
sgn
⁡
e
3
e
3
m
,
u
4
=
2
e
1
+
2
x
1
+
0.3
e
2
e
3
+
0.3
e
2
x
3
+
0.3
e
3
x
2
+
0.3
x
2
x
3
-
k
sgn
⁡
e
4
e
4
m
,where k
>
0
,
0
<
m
<
1. Substituting (12) into (10) yields(13)
e
˙
=
-
k
sgn
⁡
e
1
e
1
m
-
k
sgn
⁡
e
2
e
2
m
-
k
sgn
⁡
e
3
e
3
m
-
k
sgn
⁡
e
4
e
4
m
.From Lemma1, system (13) is stable in the finite time. That means(14)
t
>
t
s
i
=
e
0
i
1
-
m
k
1
-
m
,
e
i
t
i
s
=
0
,
i
=
1,2
,
3,4
.The synchronization of Lorenz system (3) and hyperchaotic Lorenz system (4) is achieved in a finite time.Theorem 2.
Suppose thatk
>
0
,
0
<
m
<
1. Let controller u
i
i
=
1,2
,
3,4 be defined as (12). Then error system (10) can be stabilized in finite time by the controller u
i
i
=
1,2
,
3,4. That is to say, slave system (4) will synchronize with master system (3) in finite time.


## 3.2. Finite-Time Synchronization of Systems with Uncertain Parameters

We consider the more realistic and practical case, where some of the parameters of the master system and slave system are unknown. It is valuable because practical chaos systems are often disturbed by different factors. It is that the parameters of the systems have been disturbed.Consider the master system(15)
x
˙
1
t
=
a
x
2
t
-
x
1
t
,
x
˙
2
t
=
c
+
Δ
1
x
1
t
-
x
2
t
-
x
1
t
x
3
t
,
x
˙
3
t
=
x
1
t
x
2
t
-
b
+
Δ
2
x
3
tand the slave system(16)
y
˙
1
t
=
α
y
2
t
-
y
1
t
+
V
1
t
,
y
˙
2
t
=
γ
+
Δ
3
y
1
t
-
y
2
t
-
y
1
t
y
3
t
+
y
4
t
+
V
2
t
,
y
˙
3
t
=
-
β
+
Δ
4
y
3
t
+
y
1
t
y
2
t
+
V
3
t
,
y
˙
4
t
=
-
k
y
1
t
-
0.3
y
2
t
y
3
t
+
V
4
t
,where Δ
i
i
=
1,2
,
3,4 are the increments of the parameter, which is caused by the interruption, and V
1
t, V
2
t, V
3
t, and V
4
(
t
) are the control functions. Analogously, as in Section 3.1, a new artificial variable x
4 is given and x
˙
4
t
=
0, by defining the synchronization error in the following way:(17)
e
1
=
y
1
-
x
1
,
e
2
=
y
2
-
x
2
,
e
3
=
y
3
-
x
3
,
e
4
=
y
4
-
x
4
,and then the error system can be written as(18)
e
˙
1
=
α
e
2
-
e
1
+
α
-
a
x
2
-
x
1
+
V
1
,
e
˙
2
=
γ
+
Δ
3
e
1
+
γ
+
Δ
3
-
c
-
Δ
1
x
1
-
e
2
-
e
1
e
3
-
e
1
x
3
-
x
1
e
3
+
y
4
+
V
2
,
e
˙
3
=
-
β
+
Δ
4
e
3
-
β
+
Δ
4
-
b
-
Δ
2
x
3
+
e
1
e
2
+
e
1
x
2
+
x
1
e
2
+
V
3
,
e
˙
4
=
-
k
x
1
-
k
e
1
-
0.3
e
2
e
3
-
0.3
x
3
e
2
-
0.3
x
2
e
3
-
0.3
x
2
x
3
+
V
4
.In order to solve the finite-time master-slave synchronization problem between system equations (15) and (16), we select the Lyapunov function:(19)
V
=
1
2
e
1
2
+
1
2
e
2
2
+
1
2
e
3
2
+
1
2
e
4
2
.Differentiation of this function with respect to time yields(20)
V
˙
=
e
1
e
˙
1
+
e
2
e
˙
2
+
e
3
e
˙
3
+
e
4
e
˙
4
.Combining (18) and (20), we have(21)
V
˙
=
e
1
α
e
2
-
e
1
+
α
-
a
x
2
-
x
1
+
V
1
+
e
2
γ
+
Δ
3
e
1
+
γ
+
Δ
3
-
c
-
Δ
1
x
1
-
e
2
-
e
1
e
3
-
e
1
x
3
-
x
1
e
3
+
y
4
+
V
2
+
e
3
-
β
+
Δ
4
e
3
-
β
+
Δ
4
-
b
-
Δ
2
x
3
+
e
1
e
2
+
e
1
x
2
+
x
1
e
2
+
V
3
+
e
4
-
k
x
1
-
k
e
1
-
0.3
e
2
e
3
-
0.3
x
3
e
2
-
0.3
x
2
e
3
-
0.3
x
2
x
3
+
V
4
.We defined the active control functions V
1
t, V
2
t, V
3
t, and V
4
t as follows:(22)
V
1
=
-
α
-
a
x
2
-
x
1
-
α
e
2
-
e
1
m
sign
⁡
e
1
,
V
2
=
-
γ
e
1
-
γ
-
c
x
1
-
y
4
+
e
1
x
3
-
k
1
x
1
sign
⁡
e
2
-
k
2
e
1
sign
⁡
e
2
-
e
2
m
sign
⁡
e
2
,
V
3
=
-
e
1
x
2
+
β
-
b
x
3
-
k
3
x
3
sign
⁡
e
3
-
e
3
m
sign
⁡
e
3
,
V
4
=
k
x
1
+
k
e
1
+
0.3
e
2
e
3
+
0.3
e
2
x
3
+
0.3
e
3
x
2
+
0.3
x
2
x
3
-
e
4
m
sign
⁡
e
4
.One obtains(23)
V
˙
=
-
α
e
1
2
-
e
1
1
+
m
+
Δ
3
e
1
e
2
+
Δ
3
-
Δ
1
x
1
e
2
-
e
2
2
-
k
1
x
1
e
2
sign
⁡
e
2
-
k
2
e
1
e
2
sign
⁡
e
2
-
e
2
1
+
m
-
β
+
Δ
4
e
3
2
+
Δ
2
-
Δ
4
x
3
e
3
-
k
3
x
3
e
3
sign
⁡
e
3
-
e
3
1
+
m
-
e
4
1
+
m
=
-
α
e
1
2
-
k
2
-
Δ
3
sign
⁡
e
1
e
2
e
1
e
2
-
k
1
-
Δ
3
-
Δ
1
sign
⁡
x
1
e
2
x
1
e
2
-
β
+
Δ
4
e
3
2
-
k
3
-
Δ
2
-
Δ
4
s
i
g
n
x
3
e
3
e
3
x
3
-
e
1
1
+
m
-
e
2
1
+
m
-
e
3
1
+
m
-
e
4
1
+
m
≤
-
e
1
1
+
m
-
e
2
1
+
m
-
e
3
1
+
m
-
e
4
1
+
m
=
-
2
1
+
m
/
2
V
1
+
m
/
2
.From Lemma 1, it follows that (18) is finite-time stabilized. Thus, uncertain slave system (16) can synchronize uncertain master system (15) in finite time.Theorem 3.
Suppose thatk
>
0, 0
<
m
<
1. Let controller u
i
i
=
1,2
,
3,4 be defined as (22). Then uncertain slave system (16) synchronizes with uncertain master system (15) in finite time.


## 4. Numerical Simulations

### 4.1. Case of the Certain Parameters

To verify the effectiveness of the proposed finite-time synchronization method, the 4th-order Runge-Kutta algorithm is used to solve the sets of differential equations in connection with the master and slave systems. The initial values of the two systems are taken asx
1
0
,
x
2
0
,
x
3
0
=
0.2,0.1,0.1, y
1
0
,
y
2
0
,
y
3
0
,
y
4
0
=
4,4
,
4,4, and the simulation parameters are a
=
10, b
=
-
8
/
3, c
=
28 and k
=
2, α
=
10, β
=
8
/
3, and γ
=
28. The error variables of system (4) and system (3) are shown in Figure 1. The error vector is achieved as zero, which implies that system (3) and system (4) have achieved finite-time synchronization. It is clear that synchronization time of the state variables is t
1
s
=
1.94935 S, t
2
s
=
1.94935 S, t
1
s
=
1.97484 S, t
3
s
=
1.9748 S, and t
1
s
=
2 S, respectively. These results are in line with t
s
=
e
0
1
-
m
/
k
(
1
-
m
). Figure 2 shows the synchronization of the Lorenz systems with the certain parameters.Figure 1
The error between system (3) and system (4).Figure 2
The state variables of system (3) and system (4).


### 4.2. Case of the Uncertain Parameters

In this subsection, we give numerical simulations to verify the results of Section3.2. We choose the initial conditions of master system (15) and controlled slave system (16) as follows: x
0
=
3,5
,
5, y
0
=
100,130,110,140, and the uncertain parameters of master system and slave system are adopted as Δ
1
=
0.5
sin
⁡
t, Δ
2
=
0.5
cos
⁡
t, Δ
3
=
0.1, Δ
2
=
cos
⁡
t, m
=
1
/
2, k
1
=
0.8, k
2
=
0.5, and k
3
=
2. The simulation results are given in Figures 3–6, and the master system and the slave system achieve synchronization at a finite time. Figure 6 shows clearly that the synchronization errors converge to zero very quickly.Figure 3
State trajectories of the uncertain masterx
1 and slave system y
1.Figure 4
State trajectories of the uncertain masterx
2 and slave system y
2.Figure 5
State trajectories of the uncertain masterx
3 and slave system y
3.Figure 6
Synchronization errors of uncertain Lorenz system.


## 4.1. Case of the Certain Parameters

To verify the effectiveness of the proposed finite-time synchronization method, the 4th-order Runge-Kutta algorithm is used to solve the sets of differential equations in connection with the master and slave systems. The initial values of the two systems are taken asx
1
0
,
x
2
0
,
x
3
0
=
0.2,0.1,0.1, y
1
0
,
y
2
0
,
y
3
0
,
y
4
0
=
4,4
,
4,4, and the simulation parameters are a
=
10, b
=
-
8
/
3, c
=
28 and k
=
2, α
=
10, β
=
8
/
3, and γ
=
28. The error variables of system (4) and system (3) are shown in Figure 1. The error vector is achieved as zero, which implies that system (3) and system (4) have achieved finite-time synchronization. It is clear that synchronization time of the state variables is t
1
s
=
1.94935 S, t
2
s
=
1.94935 S, t
1
s
=
1.97484 S, t
3
s
=
1.9748 S, and t
1
s
=
2 S, respectively. These results are in line with t
s
=
e
0
1
-
m
/
k
(
1
-
m
). Figure 2 shows the synchronization of the Lorenz systems with the certain parameters.Figure 1
The error between system (3) and system (4).Figure 2
The state variables of system (3) and system (4).


## 4.2. Case of the Uncertain Parameters

In this subsection, we give numerical simulations to verify the results of Section3.2. We choose the initial conditions of master system (15) and controlled slave system (16) as follows: x
0
=
3,5
,
5, y
0
=
100,130,110,140, and the uncertain parameters of master system and slave system are adopted as Δ
1
=
0.5
sin
⁡
t, Δ
2
=
0.5
cos
⁡
t, Δ
3
=
0.1, Δ
2
=
cos
⁡
t, m
=
1
/
2, k
1
=
0.8, k
2
=
0.5, and k
3
=
2. The simulation results are given in Figures 3–6, and the master system and the slave system achieve synchronization at a finite time. Figure 6 shows clearly that the synchronization errors converge to zero very quickly.Figure 3
State trajectories of the uncertain masterx
1 and slave system y
1.Figure 4
State trajectories of the uncertain masterx
2 and slave system y
2.Figure 5
State trajectories of the uncertain masterx
3 and slave system y
3.Figure 6
Synchronization errors of uncertain Lorenz system.


## 5. Design of Circuit

### 5.1. Case of the Certain Parameters

Because the values of the state variables are out of the scope of the amplifier, the linear compressor of the system variable is necessary; to be specific, variables of system are multiplied by a factorζ
=
0.1 [17]. The transform has no influence on the dynamics of the system, so the circuit equation of the master system can be written as(24)
d
x
1
d
t
=
1
ζ
c
1
R
2
x
2
-
1
ζ
c
1
R
4
x
1
,
d
x
2
d
t
=
1
ζ
c
2
R
6
x
1
-
1
ζ
c
2
R
9
x
2
-
1
ζ
c
2
R
7
x
1
x
3
,
d
x
3
d
t
=
1
ζ
c
3
R
11
x
2
x
3
-
1
ζ
c
3
R
12
x
3
.Compared with system equation (3), we can obtain, 1
/
ζ
c
1
R
2
=
1
/
ζ
c
1
R
4
=
10, 1
/
ζ
c
2
R
6
=
28, 1
/
ζ
c
2
R
9
=
1, 1
/
ζ
c
3
R
11
=
1
/
ζ
c
2
R
7
=
1, and 1
/
ζ
c
3
R
12
=
8
/
3. From here we have different values of the components R
2
=
R
4
=
R
7
=
R
11
=
1
K
Ω, R
1
=
R
3
=
R
5
=
R
8
=
R
10
=
10
K
Ω, R
6
=
357.143
K
Ω, R
12
=
3.57
K
Ω, and C
1
=
C
2
=
C
3
=
C
4
=
10
n
F.As shown in Figure7, we have reduced the number of circuits and further reduced the design complexity of system circuit. The operational amplifiers LM741 and associated circuitry perform the basic implementation of addition, subtraction, and integration. The nonlinear terms of system are implemented with the analog multipliers AD633.Figure 7
The circuit of the master system.For the slave system, we take the same design as the master system. The circuit equation of system (4) can be written as follows:(25)
d
y
1
d
t
=
1
ξ
c
4
R
14
y
2
-
1
ξ
c
4
R
1
5
y
1
,
d
y
2
d
t
=
1
ξ
c
5
R
22
y
1
-
1
ξ
c
5
R
21
y
2
-
1
ξ
c
5
R
23
y
1
y
3
,
d
y
3
d
t
=
1
ξ
c
6
R
26
y
1
y
2
-
1
ξ
c
6
R
28
y
3
,
d
y
4
d
t
=
0
.Slave circuit is implemented via operational amplifiers. u
16
A constitutes a subtractor circuit, with y
i minus x
i and voltage comparator LM339DG implement sign operational. Average absolute value detector circuit consists of u
16
B, u
16
C, and others. Root circuit consists of u
17
A and A
5. When the parameters k
=
0.5 and m
=
0.5, we can get the value of the component:(26)
1
ξ
c
4
R
14
=
1
ξ
c
4
R
1
5
=
10
,
1
ξ
c
5
R
22
=
28
,
n
1
ξ
c
5
R
21
=
1
,
1
ξ
c
5
R
23
=
1
,
1
ξ
c
6
R
26
=
1
,
1
ξ
c
6
R
28
=
8
3
,
R
14
=
R
15
=
R
23
=
R
26
=
1
K
Ω
,
R
16
=
R
17
=
R
24
=
R
25
=
R
29
=
R
30
=
R
32
=
R
33
=
10
K
Ω
,
R
20
=
R
21
=
100
K
Ω
,
R
22
=
357.14
K
Ω
,
R
28
=
3.75
K
Ω
,
R
18
=
R
19
=
R
27
=
R
31
=
5
K
Ω
,
C
4
=
C
5
=
C
6
=
C
7
=
10
n
F
,and Figure 8 shows the circuit diagram of the slave system.Figure 8
The circuit of slave system.The circuit of the control functionu consists of subtractor circuit, sign function circuit, average absolute value detector circuit, and root circuit. By adjusting the value of the R
i
i
=
18,19,27,31, the master system and the slaver system can realize the synchronization quickly and monotonically through controller circuits. The circuit of the control function is shown in Figure 9. The result of the simulation is shown in Figure 10. It can see that the ratio of y
i : x
i
i
=
1,2
,
3 is in close proximity to 1 : 1, which shows that the master system and the slaver system are synchronized.Figure 9
Function circuit.Figure 10
The ratio of the state variables of different systems.


### 5.2. Case of the Uncertain Parameters

In this subsection, combining with the numerical simulations of the Section4.2, master system (15) is implemented in Figure 11. Figure 11 shows the circuit diagram of the master system and the values of the circuit components; the frequency of the sinusoidal signal is 2000 Hz.Figure 11
Circuit of the master system with uncertain parameters.Figure12 shows the circuit diagram of the state variables for slave system (16). The value of the components in Figures 11 and 12 are scaled by a factor 0.1.Figure 12
Circuit of the slave system with uncertain parameters.The circuit schematic is composed of four submodule control circuit. Figure13 shows the circuit diagram of control function V
2, and the design methods of the other circuit is identical to it. Figure 14 illustrates the synchronization is achieved between master system (15) and slave system (16).Figure 13
Circuit diagram of the control functionV
2.Figure 14
The ratio of the state variables of different systems.


## 5.1. Case of the Certain Parameters

Because the values of the state variables are out of the scope of the amplifier, the linear compressor of the system variable is necessary; to be specific, variables of system are multiplied by a factorζ
=
0.1 [17]. The transform has no influence on the dynamics of the system, so the circuit equation of the master system can be written as(24)
d
x
1
d
t
=
1
ζ
c
1
R
2
x
2
-
1
ζ
c
1
R
4
x
1
,
d
x
2
d
t
=
1
ζ
c
2
R
6
x
1
-
1
ζ
c
2
R
9
x
2
-
1
ζ
c
2
R
7
x
1
x
3
,
d
x
3
d
t
=
1
ζ
c
3
R
11
x
2
x
3
-
1
ζ
c
3
R
12
x
3
.Compared with system equation (3), we can obtain, 1
/
ζ
c
1
R
2
=
1
/
ζ
c
1
R
4
=
10, 1
/
ζ
c
2
R
6
=
28, 1
/
ζ
c
2
R
9
=
1, 1
/
ζ
c
3
R
11
=
1
/
ζ
c
2
R
7
=
1, and 1
/
ζ
c
3
R
12
=
8
/
3. From here we have different values of the components R
2
=
R
4
=
R
7
=
R
11
=
1
K
Ω, R
1
=
R
3
=
R
5
=
R
8
=
R
10
=
10
K
Ω, R
6
=
357.143
K
Ω, R
12
=
3.57
K
Ω, and C
1
=
C
2
=
C
3
=
C
4
=
10
n
F.As shown in Figure7, we have reduced the number of circuits and further reduced the design complexity of system circuit. The operational amplifiers LM741 and associated circuitry perform the basic implementation of addition, subtraction, and integration. The nonlinear terms of system are implemented with the analog multipliers AD633.Figure 7
The circuit of the master system.For the slave system, we take the same design as the master system. The circuit equation of system (4) can be written as follows:(25)
d
y
1
d
t
=
1
ξ
c
4
R
14
y
2
-
1
ξ
c
4
R
1
5
y
1
,
d
y
2
d
t
=
1
ξ
c
5
R
22
y
1
-
1
ξ
c
5
R
21
y
2
-
1
ξ
c
5
R
23
y
1
y
3
,
d
y
3
d
t
=
1
ξ
c
6
R
26
y
1
y
2
-
1
ξ
c
6
R
28
y
3
,
d
y
4
d
t
=
0
.Slave circuit is implemented via operational amplifiers. u
16
A constitutes a subtractor circuit, with y
i minus x
i and voltage comparator LM339DG implement sign operational. Average absolute value detector circuit consists of u
16
B, u
16
C, and others. Root circuit consists of u
17
A and A
5. When the parameters k
=
0.5 and m
=
0.5, we can get the value of the component:(26)
1
ξ
c
4
R
14
=
1
ξ
c
4
R
1
5
=
10
,
1
ξ
c
5
R
22
=
28
,
n
1
ξ
c
5
R
21
=
1
,
1
ξ
c
5
R
23
=
1
,
1
ξ
c
6
R
26
=
1
,
1
ξ
c
6
R
28
=
8
3
,
R
14
=
R
15
=
R
23
=
R
26
=
1
K
Ω
,
R
16
=
R
17
=
R
24
=
R
25
=
R
29
=
R
30
=
R
32
=
R
33
=
10
K
Ω
,
R
20
=
R
21
=
100
K
Ω
,
R
22
=
357.14
K
Ω
,
R
28
=
3.75
K
Ω
,
R
18
=
R
19
=
R
27
=
R
31
=
5
K
Ω
,
C
4
=
C
5
=
C
6
=
C
7
=
10
n
F
,and Figure 8 shows the circuit diagram of the slave system.Figure 8
The circuit of slave system.The circuit of the control functionu consists of subtractor circuit, sign function circuit, average absolute value detector circuit, and root circuit. By adjusting the value of the R
i
i
=
18,19,27,31, the master system and the slaver system can realize the synchronization quickly and monotonically through controller circuits. The circuit of the control function is shown in Figure 9. The result of the simulation is shown in Figure 10. It can see that the ratio of y
i : x
i
i
=
1,2
,
3 is in close proximity to 1 : 1, which shows that the master system and the slaver system are synchronized.Figure 9
Function circuit.Figure 10
The ratio of the state variables of different systems.


## 5.2. Case of the Uncertain Parameters

In this subsection, combining with the numerical simulations of the Section4.2, master system (15) is implemented in Figure 11. Figure 11 shows the circuit diagram of the master system and the values of the circuit components; the frequency of the sinusoidal signal is 2000 Hz.Figure 11
Circuit of the master system with uncertain parameters.Figure12 shows the circuit diagram of the state variables for slave system (16). The value of the components in Figures 11 and 12 are scaled by a factor 0.1.Figure 12
Circuit of the slave system with uncertain parameters.The circuit schematic is composed of four submodule control circuit. Figure13 shows the circuit diagram of control function V
2, and the design methods of the other circuit is identical to it. Figure 14 illustrates the synchronization is achieved between master system (15) and slave system (16).Figure 13
Circuit diagram of the control functionV
2.Figure 14
The ratio of the state variables of different systems.


## 6. Experiment on Secure Communication

Most secure communication schemes using chaotic dynamics are based on identical synchronization. These systems employing chaos are partly due to their high unpredictability and simplicity of implementation. Chaotic Masking (CM), Chaos Shift Keying (CSK), Chaos On-Off Keying (COOK), and Differential Chaos Shift Keying (DCSK) are the most common chaos modulation techniques. However, the technology of the finite-time synchronization and unpredictability of uncertain parameters can enhance the security of communication. In the paper, based on the method of finite-time synchronization, simulation test of the digital secure communication adopting Chaotic Masking (CM) technology is performed. The information signals
T
(
t
) is chosen, which is a square signal with frequency of 4 khz and 500 mVp. The signal is modulated with the variable of the master system. When x
1
t is chosen as a marking signal, the transmitting system can be written as(27)
x
˙
1
t
=
a
x
2
t
-
x
1
t
+
s
T
t
,
x
˙
2
t
=
c
x
1
t
-
x
2
t
-
x
1
t
x
3
t
,
x
˙
3
t
=
x
1
t
x
2
t
-
b
x
3
t
.The receiving system is written as(28)
y
˙
1
t
=
α
y
2
t
-
y
1
t
+
u
1
,
y
˙
2
t
=
γ
y
1
t
-
y
2
t
-
y
2
t
y
3
t
+
y
4
t
+
u
2
,
y
˙
3
t
=
-
β
y
3
t
+
y
1
t
y
2
t
+
u
3
,
y
˙
4
t
=
-
k
y
1
t
-
0.3
y
2
t
y
3
t
+
u
4
.At the transmitter, the information signal s
T
t is added into the master system and chaos signal mask the information signal. One obtained the encryption signal:(29)
x
′
t
=
x
1
t
+
s
T
t
.At the receive end, for maintaining synchronization between y
1
(
t
) and x
1
t
, the information signal s
R
t is decoded by a subtractor:(30)
s
R
t
=
x
′
t
-
y
1
t
=
x
1
t
+
s
T
t
-
y
1
t
=
s
T
t
.Modulation-demodulation of the chaotic communication system is shown in Figure 15.Figure 15
Chaotic communication system.The result of circuit simulation proves that the encryption signals
′
t has chaotic characteristics and contains information. The synchronous demodulation signal s
R
t is consistent with the modulating signal s
T
t after the finite time t
s; Figure 16 shows the waves of the signals and state variables of the system. Chaos communication system showed good reducibility.Figure 16
The simulation result of secure communication.


## 7. Conclusion

In this paper, the finite-time chaos synchronization between the 3D Lorenz system and the 4D hyperchaotic system is investigated. We proposed an accurate finite-time synchronization control strategy and implemented finite-time synchronization between the 3D Lorenz system and the 4D hyperchaotic system based on the finite-time stability theory. An electronic circuit is designed to realize the controller using Multisim 12. Numerical simulations have verified the effectiveness of these methods. Furthermore, the experimental results are in agreement with numerical simulation results and demonstrate the effectiveness and feasibility of the proposed method.



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*Source: 7693547-2016-09-04.xml*