Applied Mathematics and Computation 223 (2013) 180–190

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Applied Mathematics and Computation

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Synchronization of Lurie system based on contraction analysis

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.07.080

⇑ Corresponding author. Address: School of IoT Engineering, Jiangnan University, 1800 Lihu Road, Wuxi 214122, China.E-mail address: xiaojiao-abc@hotmail.com (X. Zhang).

Xiaojiao Zhang ⇑, Baotong CuiKey Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi 214122, ChinaSchool of IoT Engineering, Jiangnan University, Wuxi 214122, China

a r t i c l e i n f o

Keywords:ContractionVirtual systemPartial contractionUniformly negative definite

a b s t r a c t

In this paper, a new synchronization criterion for chaotic Lurie systems with sector andslope restricted nonlinearities is proposed. Without analyzing the stability of the error sys-tem, synchronization condition for general Lurie systems is firstly given based on the con-traction analysis. Furthermore, the result is extended to the Chua’s system whose stateequation is not differentiable. Finally, three representative examples are presented to sup-port the theoretical results.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Synchronization is a universal phenomenon in nature. Due to its wide application in many areas, more and more peoplehave paid attention on it and explored many synchronization methods for different systems, especially chaotic systems.These methods include back-stepping, OGY method, feedback controller, (parameters) adaptive control, observer-based con-troller and so on. Actually, the previous works about the synchronization of chaotic systems in the literature can be catego-rized into two schemes to follow. The first one is based on the form of master–slave (drive-response) dynamics [1,2]. Thesecond one translates synchronization into a state estimation problem [3,4]. However, all these schemes are establishedon Lyapunov stability analysis and linear matrix inequality (LMI) approach to guarantee the asymptotic synchronization.

In this paper, contraction theory is employed to ensure synchronization of system by designing a observed-based control-ler. The theory was firstly introduced in [5] and has been widely used in group cooperation and group agreement [6–8]. But itis different from the Lyapunov stability method due to that it does not require to know the specific movement of the systemstate. It only focuses on researching the convergence between different trajectories of the system, if all trajectories convergeto a unique trajectory. Although we know the trajectory exists, the properties of this unique trajectory are completely un-known to us. Based on this theory, Wang and Slotine [9,10] proposed a new method named partial contraction theory toextend the results of contraction theory which can be used more extensively to derive synchronization conditions fordynamical systems [11–13].

Recently, chaotic synchronization for Lurie system has been considered widely. Ji et al. [14] proposed a static error feed-back controller to solve asymptotic synchronization problem. Xiang et al. [15] used integral equality to consider the synchro-nization of Lurie system with time delay. Guo et al. [16] designed a PD controller for master–slave synchronization of chaoticLurie systems. Li et al. [17] studied the synchronization problem of a coupled delayed Lurie systems with hybrid coupling.Dun et al. [18] addressed the problem of dichotomy controller design for Lurie systems by using KYP lemma and two fre-quency equalities. Zecevic et al. [19] proposed an LMI-based approach for the design of static output feedback for thelarge-scale Lurie systems. However, all these results are based on analyzing the stability of error systems through LMI

X. Zhang, B. Cui / Applied Mathematics and Computation 223 (2013) 180–190 181

approach, so in this paper Lyapunov stability analysis is replaced by contraction theory to consider the synchronization ofLurie system. The approach proposed in this text is more simple than the ones described above.

The article is organized as follows. In Section 2, some concepts and results about contraction theory and partial contractionare briefly reviewed. In Section 3, synchronization criteria for general Lurie system and Chua’s system are derived. In Section 4,numerical examples are given to illustrate the effectiveness of theoretical analysis. Conclusions are finally drawn in Section 5.

2. Preliminaries

2.1. Contraction theory

Here, some definitions and main results about contraction theory are summarized in this section, and for more details canbe found in Ref. [5].

Consider a nonlinear system

_x ¼ f ðx; tÞ; ð1Þ

where x 2 Rn is a state vector and f 2 Rn is assumed a continuously differentiable function. Defining dx as a virtual displace-ment between two different neighboring trajectories, then we have

ddtðdxTdxÞ � 2dxTd _x ¼ 2dxT @f

@xdx 6 2kmaxðx; tÞdxTdx;

where kmaxðx; tÞ is the largest eigenvalue of the symmetric part of the Jacobian J ¼ @f@x. And then kdxk 6 kdx0ke

R t

0kmaxðx;tÞdt can be

easily concluded. So if kmaxðx; tÞ is uniformly strictly negative, any infinitesimal length kdxk will converge exponentially tozero. By path integration at fixed time, then all solutions of system (1) will converge exponentially to a single trajectory,independently of the initial conditions.

Definition 1 [5]. Uniformly negative definiteness of Jacobian @f ðx;tÞ@x means that there exists a scalar a > 0; 8x; 8t P 0 s.t.

@f@x 6 �aI < 0. As all matrix inequalities will refer to symmetric part of the square matrix involved, so we can further write

12 ð

@f@xþ

@f@x

TÞ 6 �aI < 0.

For more general case, introducing a coordinate transformation

dz ¼ Hdx;

where H :¼ Hðx; tÞ is a uniformly invertible square matrix. Then the derivative of the squared distance between trajectoriescan be written as:

ddtðdzTdzÞ ¼ 2dzTd _z ¼ 2dzT _HþH

@f@x

� �H�1dz:

So the exponential convergence of kdzk to zero can be guaranteed if the generalized Jacobian matrix

F ¼ _HþH@f@x

� �H�1

is UND (uniformly negative definite). This also implies that all solution trajectories of the system (1) converge exponentiallyto a single trajectory, independently of the initial conditions. Then the system (1) is contracting. f ðx; tÞ is called a contractingfunction and the absolute value of the largest eigenvalue of the symmetric part of F is called contraction rate of the systemwith respect to the uniformly positive definite metric M ¼ HTH.

Lemma 1 [5]. Given system (1) any trajectory which starts in a ball of constant radius centered about a given trajectory andcontained at all times in a contraction region, remains in that ball and converges exponentially to the given trajectory. Furthermore,global exponential convergence to this given trajectory is guaranteed if the whole state space region is contracting.

Lemma 2 [11]. For a nonlinear system _x ¼ f ðx; tÞ, if there exists a uniformly positive definite metric Mðx; tÞ ¼ HTðx; tÞHðx; tÞ such

that the associated generalized Jacobian matrix: F ¼ ð _HþH @f@xÞH

�1 is UND, then all system trajectories converge exponentially to a

single trajectory, with convergence rate jekmaxðx; tÞj, where ekmaxðx; tÞ is the largest eigenvalue of the symmetric part of F. Then thesystem is said to be contracting.

2.2. Partial contraction theory

Partial contraction theory is a mighty tool for studying synchronization behavior which makes the results of contractiontheory to be more simple and more direct. These results include the convergence of dynamical systems behaviors or specific

182 X. Zhang, B. Cui / Applied Mathematics and Computation 223 (2013) 180–190

properties such as equality of state components or convergence etc. [11]. We now introduce the concept of partial contrac-tion, which is the main theoretical basis of this paper.

Definition 2. Given a system _xðtÞ ¼ f ðx; x; tÞ, where x 2 Rn is state vector. Now introduce a new system _yðtÞ ¼ f ðy; x; tÞ, wherey 2 Rn is state vector which has the same constrains as the state vector x. Then the former is called an actual system, and thelatter is called a virtual system. Sometimes the virtual system is also referred as auxiliary system.

Lemma 3 [10]. Consider a nonlinear system of the form

_x ¼ f ðx; x; tÞ

and assume that the auxiliary system

_y ¼ f ðy; x; tÞ

is contracting with respect to y. If a particular solution of the auxiliary y�system verifies a smooth specific property, then all tra-jectories of the original x-system verify this property exponentially. The original system is said to be partially contracting.

2.3. Contraction theory and Krasovaskii’s classical theorem

The relation between contraction theory and Krasovaskii’s classical theorem is introduced in this section, and the moredetail discussions can be found in Ref. [20]. Now select a Lyapunov function as follows:

VðdxðtÞ; tÞ ¼ dxTðtÞMðxðtÞ; tÞdxðtÞ:

Just as in Section 2.1 discussed, we can easily get the following results:

VðdxðtÞ; tÞ ¼ dzTðtÞdzðtÞ; _VðdxðtÞ; tÞ ¼ 2dzTðtÞFdzðtÞ:

It yields global convergence condition F 6 �aI with a > 0.If the function f ðx; tÞ is autonomous and MðxðtÞ; tÞ keeps constant, then the Krasovaskii’s sufficient condition for asymp-

totic stability is equivalent to contraction of such systems [20].Consider the stability of the following system:

_x1 ¼ � 12 x1 þ x2

2;

_x2 ¼ �x1x2 � 12 x2:

(ð2Þ

From the viewpoint of Krasovaskii’s theorem, select Vðx1; x2Þ ¼ x21 þ x2

2. Obviously, _Vðx1; x2Þ ¼ �x21 � x2

2 � 0, the system (2)is stable. By using the contraction analysis, select M ¼ I, and constructing its virtual system shown as follows:

_y1

_y2

� �¼� 1

2 x2

�x2 � 12

!y1

y2

� �� �

Then Fs ¼� 1

2 00 � 1

2, system (2) is stable, where Fs is the symmetric part of the matrix F.

3. Main results

Consider the Lurie system as follows:

_xðtÞ ¼ AxðtÞ þ Bf ðCxðtÞÞ;pðtÞ ¼ HxðtÞ:

�ð3Þ

where xðtÞ 2 Rn is state vector and pðtÞ 2 Rl is output vector. The matrices A 2 Rn�n; B 2 Rn�m; C 2 Rm�n and H 2 Rl�n areknown matrices. The nonlinearity f ð�Þ : Rm�!Rm represents memoryless time-invariant nonlinearities.

We assume that rðtÞ ¼ CxðtÞ and f ðrðtÞÞ ¼ ½f1ðr1ðtÞÞ; f2ðr2ðtÞÞ; . . . ; fmðrmðtÞÞ�T and fið�Þ satisfy certain sector and slope re-stricted conditions:

k�i r2i 6 fiðriÞri 6 kþi r2

i ; 8ri 2 R; ð4Þ

~ki�6

dfiðriÞdri

6~kiþ; i ¼ 1;2; . . . ;m: ð5Þ

X. Zhang, B. Cui / Applied Mathematics and Computation 223 (2013) 180–190 183

3.1. Stability analysis

Theorem 1. Consider the asymptotic stability of the Lurie system, if the symmetric part of the Jacobian matrix eJ about the system(3) is UND in the whole state space region, then the Lurie system is stable and all states converge together, exponentially, whereeJ ¼ Aþ B @f

@r C.

Proof. Construct the following Lyapunov function:

VðdxðtÞ; tÞ ¼ dxTðtÞdxðtÞ:

The derivative of the Lyapunov function can be shown as:

_VðdxðtÞ; tÞ ¼ 2dxTðtÞeJdxðtÞ:

So if the symmetric part of eJ is UND, _VðdxðtÞ; tÞ < 0 can be guaranteed, system (3) is asymptotically stable.

3.2. Master–slave synchronization of Lurie system

Consider the master–slave scheme of two identical chaotic Lurie systems as follows:

Master � system :_xðtÞ ¼ AxðtÞ þ Bf ðCxðtÞÞ;pðtÞ ¼ HxðtÞ:

�ð6Þ

Slave� system :_yðtÞ ¼ AyðtÞ þ Bf ðCyðtÞÞ þ uðtÞ;qðtÞ ¼ HyðtÞ:

�ð7Þ

Controller : uðtÞ ¼ LðpðtÞ � qðtÞÞ ¼ LHðxðtÞ � yðtÞÞ; ð8Þ

where state vectors xðtÞ; yðtÞ 2 Rn, output vectors pðtÞ; qðtÞ 2 Rl. The matrices A 2 Rn�n; B 2 Rn�m; C 2 Rm�n and H 2 Rl�n areknown matrices, L 2 Rn�l is the gain matrix to be selected. The nonlinearity has the similar definition and limitations as (4)and (5).

Construct a virtual system as

_zðtÞ ¼ AzðtÞ þ Bf ðCzðtÞÞ þ LHðxðtÞ � zðtÞÞ;lðtÞ ¼ HzðtÞ;

�ð9Þ

systems (6) and (7) are particular solutions of (9).Denoting dz as the virtual displacement of the state vector z, the dynamics (9) can be written in differential form as

_dz ¼ Aþ B@f@r

@r@z� LH

� �dz ¼ Aþ B

@f@r

C � LH� �

dz ¼ Jdz; ð10Þ

where J is n� n dimensional Jacobian matrix. Select the available gain matrix L to ensure the uniform negative definiteness ofthe symmetric part

Js ¼12ðJ þ JTÞ ð11Þ

of Jacobian to be guaranteed. From the viewpoint of contraction theory, if the virtual system is contracting then its particularsolutions would converge to each other, exponentially. It means the master system and slave system will synchronize expo-nentially as special solutions for the system (9).

3.2.1. Synchronization of general Lurie system

Theorem 2. If (11) is uniformly negative definite in nature by suitable selection of L, then systems (6) and (7) are synchronous.In [11–13], synchronization of some special chaotic systems were considered by contraction analysis. All of these sys-

tems’ state equations are differentiable, non-differentiable case are not studied. So in the next section, non-differentiablecase will be discussed.

3.2.2. Synchronization of Chua’s systemHere the non-differentiable Chua’s circuit is considered which has the following form:

_x1ðtÞ ¼ ax2ðtÞ � af ðx1ðtÞÞ;_x2ðtÞ ¼ x1ðtÞ � x2ðtÞ þ x3ðtÞ;_x3ðtÞ ¼ �bx2ðtÞ;

8><>: ð12Þ

184 X. Zhang, B. Cui / Applied Mathematics and Computation 223 (2013) 180–190

where f ðx1ðtÞÞ ¼ m1x1ðtÞ þ 0:5ðm0 �m1Þðjx1ðtÞ þ cj � jx1ðtÞ � cjÞ. Obviously, f ðx1ðtÞÞ is not differentiable at points x1ðtÞ ¼ �c.The system can be rewritten in Lurie form as

A ¼0 a 01 �1 10 �b 0

0B@1CA; B ¼

�a

00

0B@1CA; C ¼

100

0B@1CA

T

and f ðrÞ ¼ m1rþ 0:5ðm0 �m1Þðjrþ cj � jr� cjÞ ¼ f ðx1ðtÞÞ.Now let hðxÞ ¼ 0:5ðjxþ cj � jx� cjÞ, its differentiability has been discussed in Refs. [1] and [14]. For convenience to under-

stand, we repeat it again. It is obviously that hðxÞ is not differentiable at x ¼ �c. Firstly, it is easy to get the slope bound ½0;1�for every differentiable points. Next, the sector bound for each non-differential point can also be found. Assuming that x ¼ c,then the following inequality can be obtained

0 6 hðxÞ � hðyÞ ¼ dðx� yÞ 6 2c; 8y:

The maximum value of d is 1 when y ¼ �c, and the minimum value of d is 0 for all y > �c. Similarly, we can find that

0 6 d 6 1 is satisfied for each non-differential point of hð�Þ. So the sector bound of the nonlinearity dhðxÞdx is ½0;1�.

Assume that H ¼ 1 0 0ð Þ; L ¼ l1 l2 l3ð ÞT , then the Jacobian matrix of system (12) in Lurie form is

J ¼ Aþ B@f@r

C � LH ¼�aM � l1 a 0

1� l2 �1 1�l3 �b 0

0B@1CA;

where M ¼ @f@r. Then we have

Js ¼�aM � l1

aþ1�l22

�l32

aþ1�l22 �1 1�b

2�l3

21�b

2 0

0BB@1CCA: ð13Þ

Hence, L is to be selected availably such that the matrix Js is UND in nature, which ensures the contracting behaviors of thevirtual system. As master system and slave system are particular solutions of the virtual system, so they will converge toeach other, exponentially. To ensure the UND nature of Js, the gain matrix should satisfy the following conditions:

(1) l1 > K;(2) l1 > K þ ðaþ1�l2Þ2

4 ;(3) ðaþ1�l2Þðb�1Þl3

4 þ l234 þ

ð1�bÞ2ðl1�KÞ4 < 0.

Here K ¼ �aM. So any values of the gain L satisfying the above conditions would ensure the contracting behavior of virtualsystem. Then the master and slave (M–S) systems will converge to each other. From Theorem 1, we can easily get the fol-lowing result.Corollary 1. If (13) is uniformly negative definite in nature by suitable selection of L, then systems (6) and (7) in Chua’s circuit willsynchronize to each other.

4. Numerical examples

Example 1. Consider the general Lurie systems (6) and (7) with parameters

A ¼�1:5 0:10:1 �1:5

� �; B ¼

2 �0:1�5 3

� �; C ¼

1 00 1

� �; H ¼ 1 0ð Þ:� �

Then f ðrÞ ¼ tanhðrÞ which is differentiable at every point. With L ¼ l1

l2, we have !

J ¼�1:5þ 2M1 � l1 0:1� 0:1M1

0:1� l2 � 5M1 �1:5þ 3M1

� �; Js ¼

�1:5þ 2M1 � l10:2�5:1M1�l2

20:2�5:1M1�l2

2 �1:5þ 3M1

;

where M1 ¼ @fi@ri2 ½0;1�; ði ¼ 1;2Þ. According to Theorem 2, the synchronization of the M–S systems is ensured if Js is UND, so

the following conditions must be satisfied:

(i) �1:5þ 2M1 � l1 < 0;

(ii) ð�1:5þ 3M1Þð�1:5þ 2M1 � l1Þ � ð0:2�5:1M1�l2Þ24 > 0.

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

t/s

x i,i=1,

2

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

t/s

y i,i=1,

2

Fig. 1. The state responses of M–S systems.

0 5 10 15 20 25 30 35 40 45 50−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

t/s

e i,i=1,

2

e1e2

Fig. 2. Time responses of synchronization error in Example 1.

X. Zhang, B. Cui / Applied Mathematics and Computation 223 (2013) 180–190 185

So any values of the gain liði ¼ 1;2Þ satisfying the above conditions would ensure the contracting behavior of virtual sys-tem. That means M–S systems will converge to each other. Specially, we select l2 ¼ 0:2, then l1 > �1:5. These gains wouldensure the Jacobian matrix Js to be UND. With l1 ¼ 1; l2 ¼ 0:2 and the initial conditions are chosen randomly asx0 ¼ ð0:9355; 0:9169ÞT ; y0 ¼ ð0:4103;0:8936ÞT , simulations for the master–slave synchronization scheme are showed inFigs. 1 and 2. In Fig. 1, state responses about the master and slave systems are shown. Fig. 2 displays synchronization errorsbetween master and slave systems.

0 5 10 15 20 25 30 35 40−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t/s

x, y

x1x2y1y2

Fig. 3. State responses of M–S systems in Example 2.

0 5 10 15 20 25 30 35 40−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

t/s

e 1,2

e1e2

Fig. 4. The synchronization error in Example 2.

186 X. Zhang, B. Cui / Applied Mathematics and Computation 223 (2013) 180–190

Example 2. Consider the Lurie systems (6) and (7) with the following parameters:

A ¼�2 00 �0:9

� �; B ¼

�0:2�0:3

� �; C ¼ 0:6 0:8ð Þ; H ¼ 1 0ð Þ; f ðrÞ ¼ 1

2jrþ 1j � jr� 1jð Þ:� � � �

Taking L ¼ l1

l2, we have J ¼ Aþ B @f

@r C � LH ¼ �2� 0:12M2 � l1 �0:16M2

�0:18M2 � l2 �0:9� 0:24M2.

So

−3−2

−10

12

−0.4

−0.2

0

0.2

0.4−4

−3

−2

−1

0

1

2

3

4

x1

x2

x 3

Fig. 5. State trajectories of Master Chua’s circuit.

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−1

−0.5

0

0.5

−4

−3

−2

−1

0

1

2

3

4

y1

y2

y 3

Fig. 6. State trajectories of Slave Chua’s circuit.

X. Zhang, B. Cui / Applied Mathematics and Computation 223 (2013) 180–190 187

Js ¼�2� 0:12M2 � l1

�0:34M2�l22

�0:34M2�l22 �0:9� 0:24M2

!;

where M2 ¼ @f@r. To ensure Js is UND, the following conditions should be satisfied:

0 10 20 30 40 50 60 70 80−4

−2

0

2

4

t/s

x i,i=1,

2,3

0 10 20 30 40 50 60 70 80−4

−2

0

2

4

t/s

y i,i=1,

2,3

Fig. 7. State responses of M–S systems.

0 1 2 3 4 5 6 7 8 9 10

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

t/s

e i,i=1,

2,3

e1e2e3

Fig. 8. The synchronization error of Chua’s circuit.

188 X. Zhang, B. Cui / Applied Mathematics and Computation 223 (2013) 180–190

(i) �2� 0:12M2 � l1 < 0;(ii) ð0:9þ 0:24M2Þð2þ 0:12M2 þ l1Þ � ð�0:34M2�l2Þ2

4 > 0.

So any values of the gain li ði ¼ 1;2Þ satisfying the above conditions would ensure the contracting behavior of virtual system.Then M–S will converge to each other. Because 0 6 M2 6 1, if we let l2 ¼ �0:34, then l1 > �2. These gains ensure the Jacobianmatrix Js to be UND. Take the initial states of M–S systems as ð0:4154;0:3050;0:8744;0:0150ÞT and supposel1 ¼ �1; l2 ¼ �0:34, Fig. 3 depicts the state responses of the M–S systems. Synchronization errors are revealed in Fig. 4.

X. Zhang, B. Cui / Applied Mathematics and Computation 223 (2013) 180–190 189

Example 3. Here the Chua’s circuit is considered with parameters

a ¼ 9; b ¼ 14:28; c ¼ 1; m0 ¼ �17; m1 ¼

27:

The system can be rewritten in Lurie form by

A ¼0 a 01 �1 10 �b 0

0B@1CA ¼ 0 9 0

1 �1 10 �14:28 0

0B@1CA; B ¼

�a

00

0B@1CA ¼ �9

00

0B@1CA; C ¼ 1 0 0ð Þ; f ðCxðtÞÞ ¼ f ðx1ðtÞÞ:

From the discussion of differentiability of the function hðxÞ, we can conclude that

�17

x21ðtÞ 6 f ðx1ðtÞÞx1ðtÞ 6

27

x21ðtÞ; �

176

df ðx1ðtÞÞdt

627:

From the assumption H ¼ 1 0 0ð Þ and calculating (13), if we choose l2 ¼ 12, then other gains can be chosen as� 11

7 6 l1 6167 ; l3 ¼ 13:28. These setting gains ensure the Jacobian (13) to be UND. Fig. 5 shows the chaotic behavior of the

master Chua’s system with initial condition x0 ¼ ð�0:2;�0:33;0:2ÞT . Fig. 6 shows the state trajectories of controlled slave

system with initial condition y0 ¼ ð0:5;�0:6;0:8ÞT . Fig. 7 shows the state responses of M–S systems respectively. The syn-

chronization error between master and slave systems using the controller (8) with gain matrix L ¼ 2 12 13:28ð ÞT isshown in Fig. 8. It shows that the synchronization error converges to zero asymptotically.

Remark 1. Comparing with these examples, we can know that the system in Example 2 is stable, but those in Examples 1and 3 are not stable. The reason is that the symmetric part of the matrixeJ ¼ Aþ B @f

@r C is UND in Example 2. This phenomenonalso illustrates the efficiency of Theorem 1.

5. Conclusion

In this brief, the synchronization problem of Lurie system was investigated based on the contraction analysis. Differentfrom previous studies, non-differentiable case was considered and some simple criteria for synchronization were given. Fur-thermore, simulation results were provided to show the effectiveness of theoretical results. Meanwhile, there are still a num-ber of related interesting problems deserving further investigation. For example, it is necessary to study Lurie systems withtime delay or hybrid coupling by contraction analysis.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 61174021, No. 61104155), the Fun-damental Research Funds for the Central Universities (No. JUDCF12033), the Jiangsu Innovation Program for Graduates (No.CXZZ12_0742), the Fundamental Research Funds for the Central Universities (No. JUSRP51322B) and the 111 Project(B12018).

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