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Finite time synchronization of chaotic systems
Shihua Li *, Yu-Ping Tian
Department of Automatic Control, Southeast University, Nanjing 210096, PR China
Accepted 29 April 2002
Abstract
Using finite time control techniques, continuous state feedback control laws are developed to solve the synchro-
nization problem of two chaotic systems. We demonstrate that these two chaotic systems can be synchronized in finite
time. Examples of Duffing systems, Lorenz systems are presented to verify our method.
� 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction
Chaos synchronization has been of broad interest in recent years [1–7]. In [1–5], active control methods were used to
solve the synchronization problem of chaotic systems such as Rossler systems, Lorenz systems, Duffing systems, Chen
systems. The main idea of synchronization is to use the states of the master system to control the slave system so that
the states of the slave system follows the states of the master system asymptotically. The master system and the slave
system may have identical or completely different structures. The convergence of the synchronization procedure in [1–5]
is exponential with infinite settling time.
To achieve faster convergence in control systems, an effective method is using finite time control techniques. Finite
time stability means the optimality in settling time [8]. Moreover, the finite time control techniques have demonstrated
better robustness and disturbance rejection properties [9]. The problem we want to discuss here is that can these chaotic
systems mentioned above be synchronized in finite settling time? The answer is certain, which we will demonstrate in the
following sections.
So, our goal in this paper is to develop feedback control laws to make the synchronization procedure converging in
finite time, i.e., the states of the slave system follow the states of the master system in finite time. Discontinuous or open-
loop control techniques from finite time control theory may be available to solve this problem. However, considering
the convenience of continuous controllers in practical implementation and the robustness of closed-loop feedbacks to
system uncertainties, the finite time control technique we discussed here is based on continuous state feedbacks [8,9].
2. Finite time synchronization of second order chaotic systems
Some results of finite time control techniques using continuous feedbacks are given in [8,9], and are rephrased by the
following two lemmas.
Lemma 1 ([8,9]). The system
dxdt
¼ u ð1Þ
* Tel.: +86-25-3794168; fax: +86-25-7712719.
E-mail address: [email protected] (S. Li).
0960-0779/03/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.
PII: S0960-0779 (02 )00100-5
Chaos, Solitons and Fractals 15 (2003) 303–310www.elsevier.com/locate/chaos
can be globally stabilized in finite time under the feedback control law
u ¼ �k � signðxÞjxja ð2Þ
with k > 0, a 2 ð0; 1Þ.
The solution trajectory of (1) and (2) is
xðtÞ1�a ¼ xð0Þ1�a � ð1� aÞkt; xð0ÞP 0
½�xðtÞ1�a ¼ ½�xð0Þ1�a � ð1� aÞkt; xð0Þ < 0
�ð3Þ
For any initial value of state xðtÞ at t ¼ 0, i.e., xð0Þ, it is easily computed that the solution trajectory of (1) and (2) will
reach x ¼ 0 in finite time ts determined by ts ¼ jxð0Þj1�a=ð1� aÞk.
Lemma 2 ([9]). The system
dxðtÞdt
¼ vðtÞ;
dvðtÞdt
¼ uðtÞð4Þ
can be globally stabilized in finite time under the feedback control law
u ¼ �k1 signðxÞjxja1 � k2 signðvÞjvja2 ð5Þ
with k1, k2 > 0, a1 2 ð0; 1Þ, a2 ¼ 2a1=ð1þ a1Þ.
Now, let us first consider the synchronization problem of second order chaotic systems which can be written in the
following form [3]
dx1ðtÞdt
¼ v1ðtÞ;
dv1ðtÞdt
¼ f1 x1ðtÞ; v1ðtÞ; tð Þð6Þ
where f1ðx1ðtÞ; v1ðtÞ; tÞ is a nonlinear function. This system is called the master system. The equations that describe the
slave system are
dx2ðtÞdt
¼ v2ðtÞ þ u1ðtÞ;
dv2ðtÞdt
¼ f2 x2ðtÞ; v2ðtÞ; tð Þ þ u2ðtÞð7Þ
where f2ðx2ðtÞ; v2ðtÞ; tÞ is a nonlinear function and u1ðtÞ, u2ðtÞ are the control signals to be designed.
The difference between the two distances and the two velocities is described by
x3 ¼ x2 � x1;
v3 ¼ v2 � v1:ð8Þ
Thus, we get
dx3ðtÞdt
¼ v3ðtÞ þ u1ðtÞ;
dv3ðtÞdt
¼ f2 x2ðtÞ; v2ðtÞ; tð Þ � f1 x1ðtÞ; v1ðtÞ; tð Þ þ u2ðtÞð9Þ
Several possible choices for the control signals u1ðtÞ, u2ðtÞ can be taken to synchronize the slave system to the master
system.
304 S. Li, Y.-P. Tian / Chaos, Solitons and Fractals 15 (2003) 303–310
2.1. Control strategy 1
One possible choice is given by
u1ðtÞ ¼ �v3ðtÞ � k1 signðx3Þjx3ja;
u2ðtÞ ¼ �f2 x2ðtÞ; v2ðtÞ; tð Þ þ f1 x1ðtÞ; v1ðtÞ; tð Þ
� k2 signðv3Þjv3jb; k1; k2 > 0; a; b;2 ð0; 1Þ:
ð10Þ
Both signals u1ðtÞ, u2ðtÞ are available in the synchronization procedure. With (10), (9) can be rewritten in the following
notation:
dx3ðtÞdt
¼ �k1 signðx3Þjx3ja;
dv3ðtÞdt
¼ �k2 signðv3Þjv3jb; k1; k2 > 0; a; b 2 ð0; 1Þ:ð11Þ
Due to Lemma 1, (11) implies that the two chaotic systems are synchronized with continuous state feedbacks in finite
time.
2.2. Control strategy 2
Another possible choice is given by
u1ðtÞ ¼ 0;
u2ðtÞ ¼ �f2 x2ðtÞ; v2ðtÞ; tð Þ þ f1ðx1ðtÞ; v1ðtÞ; tÞ � k1 signðx3Þjx3ja1
� k2 signðv3Þjv3ja2 ; k1; k2 > 0; a1 2 ð0; 1Þ; a2 ¼2a1
1þ a1
:
ð12Þ
Here, only control signal u2ðtÞ is made available while u1ðtÞ � 0.
With (12), the system (9) can be rewritten in the following notation:
dx3ðtÞdt
¼ v3;
dv3ðtÞdt
¼ �k1 signðx3Þjx3ja1 � k2 signðv3Þjv3ja2 ; k1; k2 > 0; a1 2 ð0; 1Þ; a2 ¼2a1
1þ a1
:
ð13Þ
According to Lemma 2, the differences will converge to zero in finite time.
Example 1 (Duffing systems). As an example, let us consider the synchronization of two Duffing systems [3]. The
master Duffing system is given by
dx1ðtÞdt
¼ v1;
dv1ðtÞdt
¼ 1:8x1ðtÞ � 0:1v1ðtÞ � ½x1ðtÞ3 þ 1:1 cosðtÞ:ð14Þ
The slave Duffing system is given as follows:
dx2ðtÞdt
¼ v2 þ u1ðtÞ;
dv2ðtÞdt
¼ 1:8x2ðtÞ � 0:1v2ðtÞ � ½x2ðtÞ3 þ 1:1 cosð0:4tÞ þ u2ðtÞ:ð15Þ
S. Li, Y.-P. Tian / Chaos, Solitons and Fractals 15 (2003) 303–310 305
Following the same procedure as outlined above, we write
x3 ¼ x2 � x1;
v3 ¼ v2 � v1:ð16Þ
The differences equations of x3, v3 satisfy
dx3ðtÞdt
¼ v3ðtÞ þ u1ðtÞ;
dv3ðtÞdt
¼ 1:8x3ðtÞ � 0:1v3ðtÞ þ ½x1ðtÞ3 � ½x2ðtÞ3 þ 1:1½cosð0:4tÞ � cosðtÞ þ u2ðtÞ:ð17Þ
According to Eq. (10), the control signals u1ðtÞ, u2ðtÞ are given by
u1ðtÞ ¼ �v3ðtÞ � k1 signðx3Þjx3ja;u2ðtÞ ¼ �1:8x3ðtÞ þ 0:1v3ðtÞ � ½x1ðtÞ3 þ ½x2ðtÞ3 � 1:1½cosð0:4tÞ � cosðtÞ
� k2 signðv3Þjv3jb; k1; k2 > 0; a; b 2 ð0; 1Þ:ð18Þ
We choose values for the constants k1 ¼ k2 ¼ 4, a ¼ b ¼ 13. After a time t ¼ 100, the slave Duffing system (15) is to be
controlled with the master Duffing system (14). The simulation results are shown in Fig. 1(a) and (b).
According to the second control strategy, the controllers can be described by
u1ðtÞ ¼ 0;
u2ðtÞ ¼ �1:8x3ðtÞ þ 0:1v3ðtÞ � ½x1ðtÞ3 þ ½x2ðtÞ3 � 1:1½cosð0:4tÞ � cosðtÞ
� k1 signðx3Þjx3ja1 � k2 signðv3Þjv3ja2 ; k1; k2 > 0; a1 2 ð0; 1Þ; a2 ¼2a1
1þ a1
:
ð19Þ
Fig. 1. Synchronization of a slave system to a master system after t ¼ 100.
306 S. Li, Y.-P. Tian / Chaos, Solitons and Fractals 15 (2003) 303–310
In numerical simulations, we select values for the constants k1 ¼ k2 ¼ 4, a1 ¼ 13. Then we get a2 ¼ 2a1=ð1þ a1Þ ¼ 1
2. The
controller was activated at t ¼ 100 (i.e., u2 ¼ 0 for all t < 100). Note that the slave Duffing system (15) is to be con-
trolled with the master Duffing system (14). The simulated response is shown in Fig. 2(a) and (b).
3. Finite time synchronization of Lorenz systems
Now, let us consider the synchronization of two Lorenz systems [2]. The master Lorenz system is given by
dx1ðtÞdt
¼ rðy1 � x1Þ;
dy1ðtÞdt
¼ rx1 � y1 � x1z1;
dz1ðtÞdt
¼ x1y1 � bz1:
ð20Þ
The slave Lorenz system is given as follows:
dx2ðtÞdt
¼ rðy2 � x2Þ þ u1;
dy2ðtÞdt
¼ rx2 � y2 � x2z2 þ u2;
dz2ðtÞdt
¼ x2y2 � bz2 þ u3:
ð21Þ
Letting the error state be
x3 ¼ x2 � x1;
y3 ¼ y2 � y1;
z3 ¼ z2 � z1;
ð22Þ
Fig. 2. Synchronization of a slave system to a master system after t ¼ 100.
S. Li, Y.-P. Tian / Chaos, Solitons and Fractals 15 (2003) 303–310 307
then the error state dynamic equations satisfy
dx3ðtÞdt
¼ rðy3 � x3Þ þ u1ðtÞ;
dy3ðtÞdt
¼ rx3 � y3 � x2z2 þ x1z1 þ u2ðtÞ;
dz3ðtÞdt
¼ x2y2 � x1y1 � bz3 þ u3ðtÞ:
ð23Þ
Similarly, one possible choice of control signals u1ðtÞ, u2ðtÞ, u3ðtÞ is
u1ðtÞ ¼ �rðy3 � x3Þ � k1 signðx3Þjx3ja;
u2ðtÞ ¼ �rx3 þ y3 þ x2z2 � x1z1 � k2 signðy3Þjy3jb;
u3ðtÞ ¼ �x2y2 þ x1y1 þ bz3 � k3 signðz3Þjz3jc; k1; k2; k3 > 0; a; b; c 2 ð0; 1Þ:
ð24Þ
which implies that the error state described by the following equations
dx3ðtÞdt
¼ �k1 signðx3Þjx3ja;
dy3ðtÞdt
¼ �k2 signðy3Þjy3jb;
dz3ðtÞdt
¼ �k3 signðz3Þjz3jc:
ð25Þ
Thus, the three error states will converge to zero in finite time.
For the two Lorenz systems, the parameters are r ¼ 10, b ¼ 83, r ¼ 28. We choose values for the constants
k1 ¼ k2 ¼ 20, k3 ¼ 4, a ¼ b ¼ c ¼ 12. After a time t ¼ 100, the slave Lorenz system (21) is to be controlled with the
master Duffing system (20). The simulation results are shown in Fig. 3(a)–(c).
Another choice is only taking u2ðtÞ, u3ðtÞ as available control signals while u1ðtÞ � 0. The control signals are designed
as follows:
u1ðtÞ ¼ 0;
u2ðtÞ ¼ �ðr þ 1Þx3 þ 2y3 þ x2z2 � x1z1 �k1rsignðx3Þjx3ja1
� k2rsignðy3 � x3Þ½rjy3 � x3ja2 ;
u3ðtÞ ¼ �x2y2 þ x1y1 þ bz3 � k3 signðz3Þjz3ja3 ; k1; k2; k3 > 0; a1; a3 2 ð0; 1Þ; a2 ¼2a1
1þ a1
:
ð26Þ
which implies that
d2x3ðtÞdt2
¼ �k1 signðx3Þjx3ja1 � k2 signdx3ðtÞdt
� �dx3ðtÞdt
��������a2
;
dz3ðtÞdt
¼ �k3 signðz3Þjz3ja3 k1; k2; k3 > 0; a1; a3 2 ð0; 1Þ; a2 ¼2a1
1þ a1
:
ð27Þ
Hence, the closed-loop system (25) is stabilized in finite time.
For the two Lorenz systems, we choose values for the constants k1 ¼ 40, k2 ¼ 20, k3 ¼ 4, a1 ¼ a3 ¼ 12. Thus we get
a2 ¼ 2a1=ð1þ a1Þ ¼ 23. After a time t ¼ 100, the slave Lorenz system (21) is to be controlled with the master Duffing
system (20). The simulation result is shown in Fig. 4(a)–(c).
308 S. Li, Y.-P. Tian / Chaos, Solitons and Fractals 15 (2003) 303–310
Fig. 4. Synchronization of a slave system to a master system after t ¼ 100.
Fig. 3. Synchronization of a slave system to a master system after t ¼ 100.
S. Li, Y.-P. Tian / Chaos, Solitons and Fractals 15 (2003) 303–310 309
4. Conclusion
In this paper, the synchronization problem of chaotic systems is studied. Based on finite time control techniques,
continuous state feedback control laws are designed for the synchronization of chaotic systems such as Duffing systems,
Lorenz systems. The convergence of the synchronization procedure is in finite time.
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