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Communications in Nonlinear Science and Numerical Simulation

Volume 17, Issue 4, April 2012, Pages 1894-1907

Synchronization of non-autonomous chaotic systems with time-

varying delay via delayed feedback control

Botmart , T.a, Niamsup, P.ab , Liu, X.c

a Department of Mathematics, Chiang Mai University, Chiang Mai 50200, Thailand

b Center of Excellence in Mathematics, CHE, Si Ayutthaya Rd., Bangkok 10400, Thailand

c Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada

Abstract

In this paper, we investigate the synchronization of non-autonomous chaotic systems with time-

varying delay via delayed feedback control. Using a combination of Riccati differential equation

approach, Lyapunov-Krasovskii functional, inequality techniques, some sufficient conditions for

exponentially stability of the error system are formulated in form of a solution to the standard

Riccati differential equation. The designed controller ensures that the synchronization of non-

autonomous chaotic systems are proposed via delayed feedback control and intermittent linear

state delayed feedback control. Numerical simulations are presented to illustrate the

effectiveness of these synchronization criteria. © 2011 Elsevier B.V.

Author keywords

Chaotic systems; Delayed feedback control; Intermittent control; Non-autonomous;

Synchronization; Time-varying delay

Indexed Keywords

Delayed feedback control; Error systems; Inequality techniques; Intermittent control; Lyapunov-

Krasovskii functionals; Nonautonomous; Riccati differential equation; Sufficient conditions; Time-

varying delay

Engineering controlled terms: Differential equations; Feedback control; Lyapunov functions;

State feedback; Synchronization; Time varying control systems

Engineering main heading: Chaotic systems

ISSN: 10075704 Source Type: Journal Original language: English

DOI: 10.1016/j.cnsns.2011.07.038 Document Type: Article

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Li, N. , Xiang, W. , Liu, H. Function vector synchronization of uncertain chaotic systems with input nonlinearities and dead-zones (2012) Journal of Computational Information Systems

Li, N. , Liu, H. , Xiang, W. Function vector synchronization of uncertain chaotic systems with parameters variable (2012) Information Technology Journal

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Niamsup, P.; Department of Mathematics, Chiang Mai University, Chiang Mai 50200,

Thailand; email:[email protected]

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Commun Nonlinear Sci Numer Simulat 17 (2012) 1894–1907

Contents lists available at ScienceDirect

Commun Nonlinear Sci Numer Simulat

journal homepage: www.elsevier .com/locate /cnsns

Synchronization of non-autonomous chaotic systemswith time-varying delay via delayed feedback control

T. Botmart a, P. Niamsup a,b,⇑, X. Liu c

a Department of Mathematics, Chiang Mai University, Chiang Mai 50200, Thailandb Center of Excellence in Mathematics, CHE, Si Ayutthaya Rd., Bangkok 10400, Thailandc Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

a r t i c l e i n f o a b s t r a c t

Article history:Received 13 April 2010Received in revised form 28 April 2011Accepted 31 July 2011Available online 6 August 2011

Keywords:SynchronizationNon-autonomousChaotic systemsTime-varying delayDelayed feedback controlIntermittent control

1007-5704/$ - see front matter � 2011 Elsevier B.Vdoi:10.1016/j.cnsns.2011.07.038

⇑ Corresponding author at: Department of MatheE-mail address: [email protected] (P. N

In this paper, we investigate the synchronization of non-autonomous chaotic systems withtime-varying delay via delayed feedback control. Using a combination of Riccati differentialequation approach, Lyapunov–Krasovskii functional, inequality techniques, some sufficientconditions for exponentially stability of the error system are formulated in form of a solu-tion to the standard Riccati differential equation. The designed controller ensures that thesynchronization of non-autonomous chaotic systems are proposed via delayed feedbackcontrol and intermittent linear state delayed feedback control. Numerical simulationsare presented to illustrate the effectiveness of these synchronization criteria.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

The problem of chaos synchronization has attracted a wide range of research activity in recent years. A chaotic system hascomplex dynamical behaviors that possess some special features, such as being extremely sensitive to tiny variations of ini-tial conditions, having bounded trajectories in the phase space. The concept of chaos synchronization is making two or morechaotic systems oscillate in a synchronized manner. There are several schemes which can be used to achieve chaos synchro-nization of autonomous chaotic systems, for example linear feedback method [13,28], active control [1], adaptive control[1,22,30], impulsive control [3,23], back-stepping design [20] and time-delay feedback control [11], intermittent control[14–16,31,32], etc.

In fact, non-autonomous systems for modeling the behavior of many engineering systems, such as offshore platforms,earthquake dynamics, electronic circuits and so on have been widely explored [6,8,9,17,19]. In [5], Carroll and Pecora studiedsynchronization non-autonomous chaotic circuits by using a feedback device to correct the phase of the periodic forcing inthe response system. In [2], Cai et al. presented synchronizing two identical non-autonomous chaotic systems coupled bysinusoidal state error feedback control are derived by the Lyapunov direct method and the Gerschgorin disc theorem. Moreflexible criteria has been further obtained by the property that similar matrices have the same eigenvalues. In [28], Suykenset al. studied H1 synchronization criteria master–slave non-autonomous Lur’e synchronization schemes for the static linearstate error feedback control and dynamical output error feedback control.

. All rights reserved.

matics, Chiang Mai University, Chiang Mai 50200, Thailand.iamsup).

http://dx.doi.org/10.1016/j.cnsns.2011.07.038

mailto:[email protected]

http://dx.doi.org/10.1016/j.cnsns.2011.07.038

http://www.sciencedirect.com/science/journal/10075704

http://www.elsevier.com/locate/cnsns

T. Botmart et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 1894–1907 1895

Intermittent control has been used for a variety of purposes such as manufacturing, transportation and communication. In[32], the authors introduced intermittent control to nonlinear dynamical systems. However, results using intermittent con-trol to study synchronization are few. Recently, by using intermittent control, the authors of [14,15] investigated the syn-chronization of coupled chaotic systems with or without delay by using intermittent state feedback, the authors of [31]discussed the synchronization problem for a class of complex delayed dynamical networks by pinning periodically intermit-tent control, the authors of [16] studied the synchronization of coupled chaotic systems with time delay in the presence ofparameter mismatches by using intermittent linear state feedback control. However, to the best of our knowledge, few pub-lished papers deal with the problem of synchronization of non-autonomous chaotic systems with time-varying delay byusing intermittent linear state delayed feedback control. So, our paper presents a new non-autonomous system and we alsoapproach to establishing both delay and non-delay controller to the system.

It is well known that the existence of time delay in a system may cause instability and oscillations system such as chem-ical engineering systems, biological modeling, electrical networks and many others, [4,10,12,21,27]. Stability of time-delaysystems has been studied for decades and many results on this subject have been reported, see, e.g. [24–26] and referencestherein. In [24], Niamsup et al. studied the exponential stability condition for a class of linear time-varying systems withnonlinear delayed perturbations was derived by using an improved LyapunovKrasovskii functional. The proposed exponen-tial stability conditions are formulated in terms of the solution of Lyapunov differential equations. In [25] Phat and Ha pre-sented H1 control and exponential stability of nonlinear non-autonomous systems with time-varying delay. By using theimproved Lyapunov–Krasovskii functional and the Razumikhin-type stability theorem, sufficient conditions for exponentialstabilization have been presented the delay-dependent and formulated by solving the standard Riccati differential equations.

In this paper, we shall investigate synchronization of non-autonomous chaotic systems with time-varying delay viadelayed feedback control. By utilizing Lyapunov–Krasovskii functional and combination of Riccati differential equation ap-proach. The sufficient conditions are obtained for the exponentially stable of the error system via solving Riccati differentialequation. The designed controller ensures that the synchronization of nonlinear non-autonomous chaotic systems are pro-posed via delayed feedback control and intermittent linear state delayed feedback control. Finally, we will provide numericalexamples to illustrate the effectiveness of these synchronization criteria.

2. Problem formulation

The following notation will be used in this paper: R+ denotes the set of all real non-negative numbers: Rn denotes then-dimensional Euclidean space; L2([0, t],Rn) denotes the Hilbert space of all L2-integrable and Rn-valued functions on [0, t];C([0, t],Rn) denotes the Banach space of all Rn-valued continuous functions mapping [ �h,0]; hx,yi or xTy denotes the scalarproduct of two vectors x, y; k � k denotes the Euclidean vector norm of x; AT denotes the transpose of the vector/matrix A; A issymmetric if A = AT; I denotes the identity matrix; xt :¼ {x(t + s) :s 2 [ �h,0]}, kxtk = sups2[�h,0]kx(t + s)k; BM+(0,1) denotes theset of all symmetric semi-positive definite matrix functions bounded on [0,1).

We consider the following nonlinear non-autonomous chaotic systems with time-varying delay systems:

_xðtÞ ¼ A1ðtÞxðtÞ þ A2ðtÞxðt � hðtÞÞ þ A3ðtÞf ðt; xðtÞ; xðt � hðtÞÞÞ; t P 0;xðt0 þ tÞ ¼ /ðtÞ; t 2 ½�h; 0�; h P 0;

ð2:1Þ

where x(t) 2 Rn is the state, A1(t), A2(t), A3(t) 2 Rn�n are given matrix function, which are continuous and bounded in t P 0, /(t) 2 C([ �h,0],Rn) is the initial function with the norm k/k = sups2[�h,0]k/(s)k and f(t,x(t),x(t � h(t))) 2 Rn are nonlinear func-tion vector satisfying f(t,0,0) = 0, and there exist c1, c2 > 0 such that

kf ðt; xðtÞ; xðt � hðtÞÞÞk 6 c1kxðtÞk þ c2kxðt � hðtÞÞk; 8t P 0: ð2:2Þ

The delays h(t) is a time-varying continuous function that satisfies

0 6 hðtÞ 6 h; _hðtÞ 6 d < 1: ð2:3Þ

This paper aims at proposing a procedure to design a control law to synchronize the master (drive) and slave (response)chaotic time-varying delays systems. According to the above drive-response concept, unidirectionally coupled chaotic time-varying delays systems can be described by the following equations:

_xmðtÞ ¼ A1ðtÞxmðtÞ þ A2ðtÞxmðt � hðtÞÞ þ A3ðtÞf ðt; xmðtÞ; xmðt � hðtÞÞÞ;xmðt0 þ tÞ ¼ /1ðtÞ; t 2 ½�h; 0�; h P 0; ð2:4Þ

and

_xsðtÞ ¼ A1ðtÞxsðtÞ þ A2ðtÞxsðt � hðtÞÞ þ A3ðtÞf ðt; xsðtÞ; xsðt � hðtÞÞÞ þ CðtÞUðtÞ;xsðt0 þ tÞ ¼ /2ðtÞ; t 2 ½�h;0�; h P 0;

ð2:5Þ

where xm(t) 2 Rn and xs(t) 2 Rn are the master system’s state and slave system’s state, respectively, C(t) 2 Rn�m are given ma-trix function, which are continuous and bounded in t P 0, /i(t), 2C([�h,0], Rn) are the initial function with the normk/ij = sups2[�h,0]k/i(s)k, i = 1, 2 and the admissible control U(.) 2 L2([0, t],Rm), for all t 2 R+. The synchronization error e(t), is

1896 T. Botmart et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 1894–1907

of the form e(t) = xm(t) � xs(t). Therefore, the chaotic time-varying delays systems of synchronization error between the mas-ter–slave systems given in (2.4) and (2.5) can be described by

_eðtÞ ¼ A1ðtÞeðtÞ þ A2ðtÞeðt � hðtÞÞ þ A3ðtÞ½f ðt; xmðtÞ; xmðt � hðtÞÞÞ � f ðt; xsðtÞ; xsðt � hðtÞÞÞ� � CðtÞUðtÞ; t P 0;eðt0 þ tÞ ¼ /1ðtÞ � /2ðtÞ ¼ /ðtÞ; 8t 2 ½�h;0�; h P 0;

ð2:6Þ

Definition 2.1. The system (2.6), where U(t) = K1(t)e(t) + K2(t)e(t � h(t)), is said to be exponentially stable, if there existpositive number a, N such that every solution e(t,/) of the system satisfies

keðt;/Þk 6 Ne�aðt�t0Þk/k 8t P t0 P 0:

The function U(t) = K1(t)e(t) + K2(t)e(t � h(t)) is called a stabilizing feedback control of the system.Denoting f1(t,e(t),e(t � h(t))) = f(t,xm(t),xm(t � h(t))) � f(t,xs(t),xs(t � h(t))). The state feedback controller U(t) satisfying

either (H1) or (H2):

ðH1Þ : UðtÞ ¼ K1ðtÞeðtÞ þ K2ðtÞeðt � hðtÞÞ; 8t P 0; ð2:7Þ

ðH2Þ : UðtÞ ¼K3ðtÞeðtÞ þ K4ðtÞeðt � hðtÞÞ; nx 6 t 6 nxþ d;

0; nxþ d < t 6 ðnþ 1Þx;

�ð2:8Þ

where K1(t), K2(t) are stabilizing gain matrix and K3(t), K4(t) are the intermittent linear state stabilizing gain matrix, x > 0 isthe control period and d > 0 is called the control width (control duration) and n is a non-negative integer. In this paper, ourgoal is to design suitable x, d, Ki(t), i = 1, 2, 3, 4 such that system (2.5) synchronizes with system (2.4). Then, substituting itinto (2.6), it is easy to get the following:

_eðtÞ ¼ ðA1ðtÞ � CðtÞK1ðtÞÞeðtÞ þ ðA2ðtÞ � CðtÞK2ðtÞÞeðt � hðtÞÞ þ A3ðtÞf1ðeðtÞ; eðt � hðtÞÞÞ; t P 0;eðt0 þ tÞ ¼ /ðtÞ; t 2 ½�h; 0�; h P 0:

ð2:9Þ

Namely, the error system is governed by the following system:

_eðtÞ ¼ ðA1ðtÞ � CðtÞK3ðtÞÞeðtÞ þ ðA2ðtÞ � CðtÞK4ðtÞÞeðt � hðtÞÞ þ A3ðtÞf1ðt; eðtÞ; eðt � hðtÞÞÞ; nx 6 t 6 nxþ d;

_eðtÞ ¼ A1ðtÞeðtÞ þ A2ðtÞeðt � hðtÞÞ þ A3ðtÞf1ðt; eðtÞ; eðt � hðtÞÞÞ; nxþ d < t 6 ðnþ 1Þx;eðt0 þ tÞ ¼ /ðtÞ; t 2 ½�h; 0�; h P 0:

ð2:10Þ

where /(t) 2 C([�h,0],Rn) is the initial function with the norm k/k = sups2[�h,0]k/(s)k. The following proposition will be usedin the proof of the main results.

Proposition 2.2 (Completing the Square, [24]). Let Q, S be matrices of appropriate dimensions, and S > 0 be symmetric. Then

2hQy; xi � hSy; yi 6 hQS�1Q T x; xi; 8ðx; yÞ:
3. Main results
3.1. Linear delayed feedback control

In this section, we will given some sufficient conditions for the synchronization of system (2.4) and (2.5), firstly, somepropositions are introduced. Then, we present exponential stability criteria for the system (2.9) and thus the system (2.4)synchronize with the system (2.5). Given positive number b, h, a, c, c1, c2, �i, i = 1, 2, 3, 4, we set

p ¼ supt2RþkPðtÞk; g ¼ �1 þ �2he2ah þ �3; �4 ¼

�1ð1� dÞe�2ah

3;

SðtÞ ¼ CðtÞCTðtÞ � 6�1ð1� dÞe�2ah

ðA2ðtÞAT2ðtÞ � CðtÞCTðtÞPbðtÞPbðtÞCðtÞCTðtÞÞ � ��1

3 c21 þ

3c22


� �A3ðtÞAT

3ðtÞ;

M ¼ pþ bþ h�1 þ 2h2�2; N ¼

ffiffiffiffiffiMb

s; PbðtÞ ¼ PðtÞ þ bI;

QðtÞ ¼ 2aPbðtÞ þ gI þ wðtÞ:

Theorem 3.1. Assume that the condition (2.3) hold and there exist positive numbers a, b, c, ei, i = 1, 2, 3, 4, and a matrix functionP(t), w(t) 2 BM+(0,1) satisfying the following Riccati differential equation:

ðRDEÞ _PðtÞ þ PbðtÞA1ðtÞ þ AT1ðtÞPbðtÞ � PbðtÞSðtÞPbðtÞ þ QðtÞ ¼ 0: ð3:1Þ

Then, the system (2.9) is exponentially stable and has the synchronization. Moreover, the stabilizing feedback control is given by


UðtÞ ¼ 12

CTðtÞPbðtÞeðtÞ þ CTðtÞPbðtÞeðt � hðtÞÞ; 8t 2 Rþ: ð3:2Þ

Proof. Let U(t) = K1(t)e(t) + K2e(t � h(t))), where

K1ðtÞ ¼12

CTðtÞPbðtÞ; K2ðtÞ ¼ CTðtÞPbðtÞ 8t 2 Rþ:

We define the following Lyapunov–Krasovskii functional for system (2.9)

VðeðtÞÞ ¼ V1ðeðtÞÞ þ V2ðeðtÞÞ þ V3ðeðtÞÞ; ð3:3Þ

where

V1ðeðtÞÞ ¼ hPðtÞeðtÞ; eðtÞi þ bkeðtÞk2;

V2ðeðtÞÞ ¼ �1

Z t

t�hðtÞe2aðs�tÞkeðsÞk2ds;


Z 0

�h

Z t

tþs�hðtþsÞe2aðsþh�tÞkeðhÞk2dhds:

It is easy to see that

bkeðtÞk26 VðeðtÞÞ; 8t P 0: ð3:4Þ

By taking the derivative of V(t,et) along the trajectories of system (2.9) is given by

_V1ðeðtÞÞ ¼ h _PðtÞeðtÞ; eðtÞi þ 2hPðtÞ _eðtÞ; eðtÞi þ 2bh _eðtÞ; eðtÞi ¼ h _PðtÞeðtÞ; eðtÞi þ 2hðPðtÞ þ bIÞ _eðtÞ; eðtÞi

¼ h _PðtÞeðtÞ; eðtÞi þ 2hPbðtÞ _eðtÞ; eðtÞi

¼ _PðtÞ þ PbðtÞA1ðtÞ þ AT1ðtÞPbðtÞ � PbðtÞCðtÞCTðtÞPbðtÞ

h ieðtÞ; eðtÞ

D Eþ 2h½PbðtÞA2ðtÞeðt � hðtÞÞ; eðtÞi

� 2hPbðtÞCðtÞCTðtÞPbðtÞeðt � hðtÞÞ; eðtÞi þ 2hPbðtÞA3ðtÞf1ðeðtÞ; eðt � hðtÞÞÞ; eðtÞi

_V2ðeðtÞÞ ¼ �2a�1

Z t

t�hðtÞe2aðs�tÞkxðsÞk2dsþ �1keðtÞk2 � �1ð1� _hðtÞÞe�2ahðtÞkeðt � hðtÞÞk2

6 �2aV2ðt; etÞ þ �1keðtÞk2 � �1ð1� dÞe�2ahkeðt � hðtÞÞk2;

_V3ðeðtÞÞ ¼ �2a�2

Z 0

�h

Z t

tþs�hðtþsÞe2aðsþh�tÞkxðhÞk2dhdsþ �2

Z 0

�he2ahkeðtÞk2ds� �2

Z 0

�he�2ahð1� _hðtÞÞkeðt þ s� hðt þ sÞÞk2ds

6 �2a�2V3ðt; etÞ þ �2he2ahkeðtÞk2

because of the last integral term is non-negative. Therefore, it follows that

_VðeðtÞÞ þ 2aVðeðtÞÞ ¼ _V1ðeðtÞÞ þ 2aV1ðeðtÞÞ þ _V2ðeðtÞÞ þ 2aV2ðeðtÞÞ þ _V3ðeðtÞÞ þ 2aV3ðeðtÞÞ

6 _PðtÞ þ PbðtÞA1ðtÞ þ AT1ðtÞPbðtÞ � PbðtÞCðtÞCTðtÞPbðtÞ

hDþ2aPbðtÞ

�eðtÞ; eðtÞi

þ 2h½PbðtÞA2ðtÞeðt � hðtÞÞ; eðtÞi � 2hPbðtÞCðtÞCTðtÞPbðtÞeðt � hðtÞÞ; eðtÞiþ 2hPbðtÞA3ðtÞf1ðeðtÞ; eðt � hðtÞÞÞ; eðtÞi þ �1keðtÞk2 � �1ð1� dÞe�2ahkeðt � hðtÞÞk2

þ �2he2ahkeðtÞk2 ¼ _PðtÞ þ PbðtÞA1ðtÞ þ AT1ðtÞPbðtÞ � PbðtÞCðtÞCTðtÞPbðtÞ

hDþ 2aPbðtÞ�eðtÞ; eðtÞi

þ 2h½PbðtÞA2ðtÞeðt � hðtÞÞ; eðtÞi � 2hPbðtÞCðtÞCTðtÞPbðtÞeðt � hðtÞÞ; eðtÞi

þ 2hPbðtÞA3ðtÞf1ðeðtÞ; eðt � hðtÞÞÞ; eðtÞi � 2�1ð1� dÞe�2ah

6heðt � hðtÞÞ; eðt � hðtÞÞi

þ ð�1 þ �2he2ahÞkeðtÞk2 � 2�1ð1� dÞe�2ah

3keðt � hðtÞÞk2

: ð3:5Þ

Applying Proposition (2.2) gives


2hPbðtÞA2ðtÞeðt � hðtÞÞ; eðtÞi � �1ð1� dÞe�2ah


66

�1ð1� dÞe�2ahPbðtÞA2ðtÞAT

2ðtÞPbeðtÞ; eðtÞD E

; ð3:6Þ

2hPbðtÞCðtÞCTðtÞPbðtÞeðt � hðtÞÞ; eðtÞi � �1ð1� dÞe�2ah


66

�1ð1� dÞe�2ahhPbðtÞCðtÞCTðtÞPbðtÞPbðtÞCðtÞCTðtÞPb:ðtÞeðtÞ; eðtÞi: ð3:7Þ

On the other hand, by taking the condition (2.2) into account, we have

2hPbðtÞA3ðtÞf1ðeðtÞ; eðt � hðtÞÞÞ; eðtÞi 6 2kPbðtÞA3ðtÞeðtÞkkf1ðeðtÞ; eðt � hðtÞÞÞk6 2c1kPbðtÞA3ðtÞeðtÞkkeðtÞk þ 2c2kPbðtÞA3ðtÞeðtÞkkeðt � hðtÞÞk: ð3:8Þ

Applying Proposition (2.2), we have

2hPbðtÞA3ðtÞf1ðeðtÞ; eðt � hðtÞÞÞ; eðtÞi 6 ��13 c2

1kPbðtÞA3ðtÞeðtÞk2 þ �3keðtÞk2 þ ��14 c2

2kPbðtÞA3ðtÞeðtÞk2 þ �4keðt � hðtÞÞk2:

For the chosen �4 ¼ �1ð1�dÞe�2ah

3 > 0, we have

2hPbðtÞA3ðtÞf1ðeðtÞ; eðt � hðtÞÞÞ; eðtÞi 6 ��13 c2

1kPbðtÞA3ðtÞeðtÞk2 þ �3keðtÞk2 þ 3c22

�1ð1� dÞe�2ahkPbðtÞA3ðtÞeðtÞk2

þ �1ð1� dÞe�2ah


¼ ��13 c2

1 þ3c2

2


� �hPbðtÞA3ðtÞAT

3ðtÞPbðtÞeðtÞ; eðtÞi þ �3keðtÞk2

þ �1ð1� dÞe�2ah


: ð3:9Þ

Therefore, from (3.5)–(3.8) it follows that

_VðeðtÞÞ þ 2aVðeðtÞÞ 6 _PðtÞ þ PbðtÞA1ðtÞ þ AT1ðtÞPbðtÞ � PbðtÞSðtÞPbðtÞ þ 2aPbðtÞ þ gI

h ieðtÞ; eðtÞ

D E

� �1ð1� dÞe�2ah


: ð3:10Þ

Since P(t) is a solution of RDE, we have

_VðeðtÞÞ þ 2aVðeðtÞÞ 6 �kwðtÞkkeðtÞk2 � �1ð1� dÞe�2ah

3keðt � hðtÞÞk2: ð3:11Þ

Thus, we obtain

_VðeðtÞÞ þ 2aVðeðtÞÞ 6 0; 8t P 0: ð3:12Þ

Integrating both sides of (3.11) from 0 to t, we have

VðeðtÞÞ 6 Vðeð0ÞÞe�2at ; 8t P 0:

On the other hand, using the condition (3.4), we have

bkeðtÞk26 VðeðtÞÞ; 8t P 0

and then

keðt;/Þk 6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVðeð0ÞÞ

b

se�at; 8t P 0:

Estimating V(0,e0) gives

Vðeð0ÞÞ ¼ ðpþ bþ h�1Þk/k2 þ �2

Z 0

�h

Z 0

s�hðsÞe2aðsþhÞkeðhÞk2dhds 6 ðpþ bþ h�1Þk/k2 þ 2h2�2k/k2

6 Mk/k2;

we have

keðt;/Þk 6 Nk/ke�at ; 8t P 0;


which implies the exponential stability of the error system (2.9) under the controller H1. Consequentially, the controlledslave system (2.5) is synchronized with the master system (2.4). This completes the proof. h

From the proof of Theorem 3.1, we have the following step-by-step procedure for finding the output feedback controllerof system (2.6).

Step 1 Define the matrices A1(t), A2(t), A3(t) and C(t).Step 2 Verify that all condition of Theorem 3.1 and find a solution P(t), w(t) 2 BM+(0,1) to the (RDE) (3.1).Step 3 The output feedback control is defined by (4.5), where UðtÞ ¼ 1

2 CTðtÞPbðtÞeðtÞ þ CTðtÞPbðtÞeðt � hðtÞÞ.

3.2. Intermittent linear state delayed feedback control

In this section, we first study the state error system (2.10) and given some sufficient conditions for the synchronization ofsystem (2.4) and (2.5) through intermittent linear state delayed feedback control using Lyapunov–Krasovskii function ap-proach. Then, we present exponential stability criteria for the system (2.10) and thus system (2.4) synchronize with system(2.5). For this section, let us now set:

R1ðtÞ ¼ �6

�1ð1� dÞe�2ahA2ðtÞAT

2ðtÞ � CðtÞK4ðtÞKT4ðtÞC

TðtÞ� �

� ��13 c2

1 þ3c2

2


� �A3ðtÞAT

3ðtÞ;

R2ðtÞ ¼ �6

�1ð1� dÞe�2ahA2ðtÞAT

2ðtÞ � ��13 c2

1 þ3c2

2


� �A3ðtÞAT

3ðtÞ;

M ¼ pþ bþ h�1 þ 2h2�2; N ¼

ffiffiffiffiffiMb

s; PbðtÞ ¼ PðtÞ þ bI; q ¼ k� a;

Q 1ðtÞ ¼ 2aPbðtÞ þ gI þ lðtÞ; Q 2ðtÞ ¼ 2aPbðtÞ � 2kPbðtÞ þ gI þ mðtÞ:

Theorem 3.2. Assume that the condition (2.3) hold and there exist positive numbers a, b, k, ei, i = 1, 2, 3, 4, 0 < a 6 k and a matrixfunction P(t), l(t), m(t) 2 BM+(0,1) satisfying the following Riccati differential equation and inequality:

(1) _PðtÞ þ PbðtÞA1ðtÞ þ AT1ðtÞPbðtÞ � PbðtÞCðtÞK3ðtÞ � KT

3ðtÞCTðtÞPbðtÞ � PbðtÞR1ðtÞPbðtÞ þ Q 1ðtÞ ¼ 0;

(2) _PðtÞ þ PbðtÞA1ðtÞ þ AT1ðtÞPbðtÞ � PbðtÞR2ðtÞPbðtÞ þ Q2ðtÞ ¼ 0;

(3) ad� ðk� aÞðx� dÞ > 0:Then, system (2.10) is exponentially stable and has the synchronization.

Proof. We define the following Lyapunov–Krasovskii functional for system (2.10)

VðeðtÞÞ ¼ V1ðeðtÞÞ þ V2ðeðtÞÞ þ V3ðeðtÞÞ; ð3:13Þ

where

V1ðeðtÞÞ ¼ hPðtÞeðtÞ; eðtÞi þ bkeðtÞk2;


Z t

t�hðtÞe2aðs�tÞkeðsÞk2ds;


Z 0

�h

Z t

tþs�hðtþsÞe2aðsþh�tÞkeðhÞk2dhds:

By taking the derivative of V(t,et) along the trajectories of system (2.10).For nx 6 t 6 nx + d, we are able to do similar estimation as we did for the theorem 3.1. We have the following:

_VðeðtÞÞ þ 2aVðeðtÞÞ 6 _PðtÞ þ PbðtÞA1ðtÞ þ AT1ðtÞPbðtÞ � PbðtÞCðtÞK3ðtÞ

hD� KT

3ðtÞCTðtÞPbðtÞ � PbðtÞR1ðtÞPbðtÞ þ 2aPbðtÞ þ gI

ieðtÞ; eðtÞ

E� �1ð1� dÞe�2ah


: ð3:14Þ

Since P(t) is a solution of condition (1), we have

_VðeðtÞÞ þ 2aVðeðtÞÞ 6 �klðtÞkkeðtÞk2 � �1ð1� dÞe�2ah


: ð3:15Þ


Thus, we obtain
_VðeðtÞÞ þ 2aVðeðtÞÞ 6 0; for nx 6 t 6 nxþ d: ð3:16Þ
Thus, by the above differential inequality, we have

VðeðtÞÞ 6 VðeðnxÞÞe�2aðt�nxÞ; for nx 6 t 6 nxþ d: ð3:17Þ

For nx + d 6 t 6 (n + 1)x, we are able to do similar estimation as we did for the theorem 3.1. We have the following:

_VðeðtÞÞ þ 2aVðeðtÞÞ 6 _PðtÞ þ PbðtÞA1ðtÞ þ AT1ðtÞPbðtÞ � PbðtÞR2ðtÞPbðtÞ þ 2aPbðtÞ þ gI

h ieðtÞ; eðtÞ

D E

� �1ð1� dÞe�2ah


6 _PðtÞ þ PbðtÞA1ðtÞ þ AT1ðtÞPbðtÞ � PbðtÞR2ðtÞPbðtÞ þ 2aPbðtÞ � 2kPbðtÞ þ gI

h ieðtÞ; eðtÞ

D E

þ 2kVðeðtÞÞ � �1ð1� dÞe�2ah


: ð3:18Þ

Since P(t) is a solution of condition (2), we have

_VðeðtÞÞ þ 2aVðeðtÞÞ 6 2kVðeðtÞÞ � kmðtÞkkeðtÞk2 � �1ð1� dÞe�2ah


: ð3:19Þ

Thus, we have
_VðeðtÞÞ 6 2ðk� aÞVðeðtÞÞ; for nxþ d < t 6 ðnþ 1Þx: ð3:20Þ
From the above differential inequality, we have

VðeðtÞÞ 6 Vðeðnxþ dÞÞe2ðk�aÞðt�nx�dÞ; for nxþ d < t 6 ðnþ 1Þx: ð3:21Þ

By (3.17) and (3.21), we have

Vðeððnþ 1ÞxÞÞ 6 Vðeðnxþ dÞÞe2qðx�dÞ

6 VðeðnxÞÞe�2ade2qðx�dÞ

¼ VðeðnxÞÞe�2adþ2qðx�dÞ

6 Vðeððn� 1Þxþ dÞÞe2qðx�dÞe�2adþ2qðx�dÞ

6 Vðeððn� 1ÞxÞÞe�2adþ2qðx�dÞe�2adþ2qðx�dÞ

¼ Vðeððn� 1ÞxÞÞe2ð�2adþ2qðx�dÞÞ

..

.

6 Vðeð0ÞÞeð�2adþ2qðx�dÞÞðnþ1Þ: ð3:22Þ

For any t > 0, there is a n0 P 0, such that n0x 6 t 6 (n0 + 1)x.

Case 1. For n0x + d 6 t 6 (n0 + 1)x, using condition (3), we have

VðeðtÞÞ 6 Vðeðnxþ dÞÞe2qðt�ðnxþdÞÞ

6 VðeðnxÞÞe�2ade2qðt�ðnxþdÞÞ

6 Vðeð0ÞÞeð�2adþ2qðx�dÞÞne�2ade2qðt�ðnxþdÞÞ

6 Vðeð0ÞÞeð�2adþ2qðx�dÞÞne�2ade2qððnþ1Þx�ðnxþdÞÞ

¼ Vðeð0ÞÞeð�2adþ2qðx�dÞÞðnþ1Þ

¼ Vðeð0ÞÞeð�2adþ2qðx�dÞÞðnþ1Þx

x

6 Vðeð0ÞÞeð�2adþ2qðx�dÞÞt

x : ð3:23Þ

Case 2. For n0x 6 t 6 n0x + d, using condition (3), we have

VðeðtÞÞ 6 VðeðnxÞÞe�2aðt�nxÞ

6 Vðeð0ÞÞeð�2adþ2qðx�dÞÞne�2aðt�nxÞ

6 Vðeð0ÞÞeð�2adþ2qðx�dÞÞn

¼ Vðeð0ÞÞe�ð�2adþ2qðx�dÞÞeð�2adþ2qðx�dÞÞðnþ1Þ

¼ Vðeð0ÞÞe�ð�2adþ2qðx�dÞÞeð�2adþ2qðx�dÞÞðnþ1Þx

x

6 Vðeð0ÞÞe�ð�2adþ2qðx�dÞÞeð�2adþ2qðx�dÞÞt

x : ð3:24Þ


Let L = Mk/k2e�(�2ad+2q(x�d)). By (3.22) and (3.23), we have

VðeðtÞÞ 6 Leð�2adþ2qðx�dÞÞt

x ; 8t P 0: ð3:25Þ

On the other hand, using the condition (3.4), we have

bkeðtÞk26 VðeðtÞÞ; 8t P 0:

Thus, we have obtained the following:

keðt;/Þk 6

ffiffiffiLb

seð�adþqðx�dÞÞt

x ; 8t P 0:

which implies the exponential stability of the error system (2.10) under the controller (H2). Consequentially, the controlledslave system (2.5) is synchronized with the master system (2.4). This completes the proof. h

Remark 3.3. It is clear that as d ? x the intermittent feedback control will reduce to a continuous feedback. In this case,presented in Theorem 3.1.

Remark 3.4. The stability conditions are given in terms of the solution of some RDEs. Although the problem of solving RDEsis in general still not easy, various effective approaches for finding the solutions of RDEs can be found in [7,18,29].

Remark 3.5. In most results on synchronization problem for chaotic systems, authors have considered only autonomoussystem. For example, in [15], the authors have considered synchronization problem for autonomous chaotic systems withtime delay by using intermittent linear state feedback control. On the other hands, we have considered more complicatedproblem, namely, synchronization of non-autonomous chaotic systems with time-varying delay using both state-delayedfeedback control as well as intermittent linear state-delayed feedback control. To the best of our knowledge, our resultsare among the first results on synchronization of non-autonomous system with time-varying delay using intermittent linearstate-delayed feedback control.

4. Numerical examples

In this section, we now provide an example to show the effectiveness of the result in Theorems 3.1 and 3.2

Example 4.1. We assume that there are two non-autonomous chaotic systems with time-varying delay systems and thestate feedback controller satisfying (H1) such that the master system (with the subscript m) and the slave system (withsubscript s). The master and slave systems are given, respectively, by

_xmðtÞ ¼ A1ðtÞxmðtÞ þ A2ðtÞxmðt � hðtÞÞ þ A3ðtÞf ðt; xmðtÞ; xmðt � hðtÞÞÞ;xmðtÞ ¼ /1ðtÞ; t 2 ½�h;0�; h P 0

ð4:1Þ

and

_xsðtÞ ¼ A1ðtÞxsðtÞ þ A2ðtÞxsðt � hðtÞÞ þ A3ðtÞf ðt; xsðtÞ; xsðt � hðtÞÞÞ þ CðtÞUðtÞ;xsðtÞ ¼ /2ðtÞ; t 2 ½�h; 0�; h P 0;

ð4:2Þ

where xm(t) = [x1(t) y1(t)]T, xs(t) = [x2(t) y2(t)]T, h(t) = 0.5sin 2t, /1(t) = [x1(t) y1(t)]T = [0.1cos t � 0.2cos t]T, /2(t) = [x2(t)y2(t)]T = [0.2cos t � 0.1cos t]T, "t 2 [�1,0],

A1ðtÞ ¼�1þ 0:5 sin t � e� cos t 0

0 �1þ 0:5 sin t � e� cos t

;

A2ðtÞ ¼ð0:1þ e

1:28Þe� cos t 00 ð0:1þ e

1:28Þe� cos t

" #;

A3ðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðe� cos tþe�2 cos tÞ10:562þ156:366e

q0

0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðe� cos tþe�2 cos tÞ10:562þ156:366e

q264

375;

Fig


CðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:1þ e

1:28

pe� cos t 0

0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:1þ e

1:28

pe� cos t

" #;

f ðt; xmðtÞ; xmðt � hðtÞÞÞ ¼ 2y1 þ jy1j 0:5 sin tx1 � x31 � 0:2y2

1

� �T þ g1ðxmðt � hðtÞÞÞg2ðxmðt � hðtÞÞÞ½ �T� �

;

g1ðxmðt � hðtÞÞÞ ¼ x1ðt � hðtÞÞy1ðt � hðtÞÞ � 0:1þ e1:28

� �ðe� cos t � 1Þx1ðt � hðtÞÞ;

g2ðxmðt � hðtÞÞÞ ¼ �x1ðt � hðtÞÞjy1ðt � hðtÞÞj � 0:1þ e1:28

� �ðe� cos t � 1Þy1ðt � hðtÞÞ:

The model (4.1) has a chaotic attractor with initial condition /1(t), which can be seen in Fig. 1. We define the state errorsbetween the master system (4.1) and the slave system (4.2) as:

exðtÞ ¼ x1ðtÞ � x2ðtÞ;eyðtÞ ¼ y1ðtÞ � y2ðtÞ: ð4:3Þ

By using the notation in (4.3) and e(t) = [ex(t),ey(t)]T, we have

_eðtÞ ¼ A1ðtÞeðtÞ þ A2ðtÞeðt � hðtÞÞ þ A3ðtÞ½f ðt; xmðtÞ; xmðt � hðtÞÞÞ � f ðt; xsðtÞ; xsðt � hðtÞÞÞ� � CðtÞUðtÞ; t P 0;eðtÞ ¼ /1ðtÞ � /2ðtÞ; 8t 2 ½�h;0�; h P 0:

ð4:4Þ

We see that h = 1, and _hðtÞ ¼ sin t cos t and then d = 0.5. From Fig. 1. we know that the bounds of the chaotic attractor�1 < x1 < 1.5, �0.9 < y1 < 0.3. For the positive numbers �1 ¼ 1; �2 ¼ 1

1:28 ; �3 ¼ 1100 ; c1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10:562p

; c2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi26:062p

; a ¼ 0:5;b ¼ e�1, we can verify that all conditions of Theorem 3.1 hold and that the RDE has a solution

PðtÞ ¼ ecos t � e�1 00 ecos t � e�1

" #2 BPþð0;1Þ

and

wðtÞ ¼0:09 0

0 0:09

2 BPþð0;1Þ:

−1 −0.5 0 0.5 1 1.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

x1

y1

−1 −0.5 0 0.5 1 1.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

x2

y2

0 200 400 600 800 1000−1

−0.5

0

0.5

1

1.5

Time t

x1,y

1

0 200 400 600 800 1000−1

−0.5

0

0.5

1

1.5

Time t

x2,y

2

Master Slave

Master Slave x1 y1

x2 y2

0.4

. 1. The chaotic attractor of nonlinear non-autonomous chaotic systems with time-varying delay systems (4.1) in the xy-plane and xy-Time t.


Therefore, by Theorem 3.1 the error system (4.4) is exponentially stable with an exponential a = 0.5 and intermittent feed-back linear state delayed feedback stabilizing control is given by

Fig. 2.feedbac

u1ðtÞ ¼12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:1þ e

1:28

� �rðx1ðtÞ � x2ðtÞÞ þ


1:28

� �rðx1ðt � hðtÞÞ � x2ðt � hðtÞÞÞ;

u2ðtÞ ¼12


1:28

� �rðy1ðtÞ � y2ðtÞÞ þ


1:28

� �rðy1ðt � hðtÞÞ � y2ðt � hðtÞÞÞ:

ð4:5Þ

The numerical simulations are carried out using the explicit Runge–Kutta-like method (dde45), interpolation and extrap-olation by spline of the third order. The initial function /1(t), /2(t) for the master system and slave system, respectively.Fig. 1. show that the chaotic attractor of nonlinear non-autonomous chaotic systems with time-varying delay systems(4.1) in the xy-plane and xy-Time t. The results of the two identical nonlinear non-autonomous chaotic systems withtime-varying delay systems with delay feedback control (4.5) are shown in Fig. 2: synchronization error curves of the mastersystem (4.1) and the slave system (4.2), (a), (c) without delay feedback control and (b), (d) with delay feedback control (4.5)input and Fig. 3: the time response of states for master system (x1,y1) and slave system (x2,y2): (a) signals x1and x2, (c) signalsy1 and y2 without delay feedback control; (b) signals x1 and x2, (d) signals y1 and y2 with delay feedback control (4.5) input.

Example 4.2. We assume that the non-autonomous chaotic systems with time-varying delay and the state feedbackcontroller satisfying (H2) such that the master system (with the subscript m) and the slave system (with subscript s). Themaster and slave systems are given, respectively, by

_xmðtÞ ¼ A1ðtÞxmðtÞ þ A2ðtÞxmðt � hðtÞÞ þ A3ðtÞf ðt; xmðtÞ; xmðt � hðtÞÞÞ;xmðtÞ ¼ /1ðtÞ; t 2 ½�h;0�; h P 0 ð4:6Þ

and

0 50 100 150 200−2

−1

0

1

2

3

Time t(a)

The

sta

te e

rror

(ex)

0 10 20 30 40 50−0.1

−0.05

0

0.05

Time t(b)

The

sta

te e

rror

(ex)

0 50 100 150 200−1

−0.5

0

0.5

1

Time t(c)

The

sta

te e

rror

(ey)

0 10 20 30 40 50−0.1

−0.05

0

0.05

Time t(d)

The

sta

te e

rror

(ey)

Synchronization error curves of the master system (4.1) and the slave system (4.2), (a), (c) without delay feedback control and (b), (d) with delayk control (4.5) input.

0 50 100 150 200−1

−0.5

0

0.5

1

1.5

Time t(a)

The

sta

te [

x1, x

2]

0 50 100 150 200−1

−0.5

0

0.5

1

1.5

Time t(b)

The

sta

te [

x1, x

2]

0 50 100 150 200−1

−0.5

0

0.5

Time t(c)

The

sta

te [

y1, y

2]

0 50 100 150 200−1

−0.5

0

0.5

Time t(d)

The

sta

te [

y1, y

2]

x1 x2

x1 x2

y1 y2

y1 y2

Fig. 3. The time response of states for master system (x1,y1) and slave system (x2,y2): (a) signals x1 and x2, (c) signals y1 and y2 without delay feedbackcontrol; (b) signals x1 and x2, (d) signals y1 and y2 with delay feedback control (4.5) input.


_xsðtÞ ¼ ðA1ðtÞ � CðtÞK3ðtÞÞxsðtÞ þ ðA2ðtÞ � CðtÞK4ðtÞÞxsðt � hðtÞÞ þ A3ðtÞf ðt; xsðtÞ; xsðt � hðtÞÞÞ; nx 6 t 6 nxþ d;

_xsðtÞ ¼ A1ðtÞxsðtÞ þ A2ðtÞxsðt � hðtÞÞ þ A3ðtÞf ðt; xsðtÞ; xsðt � hðtÞÞÞ þ A3ðtÞf ðt; xsðtÞ; xsðt � hðtÞÞÞ; nxþ d < t 6 ðnþ 1Þx;

ð4:7Þ

where xm(t) = [x1(t) y1(t)]T, xs(t) = [x2(t) y2(t)]T, h(t) = 0.4 + 0.5sin2 t, /1(t) = [x1(t) y1(t)]T = [�0.3cos t 0.4cos t]T, /2(t) = [x2(t)y2(t)]T = [�0.8cos t 0.7cos t]T, "t 2 [�1,0],

A1ðtÞ ¼0:5ðsin t � 1Þ � ð2:01þe�1Þ

2 e� cos t 0

0 �0:5ð1þ cos tÞ � ð2:01þe�1Þ2 e� sin t

" #;

A2ðtÞ ¼

ffiffiffiffiffiffiffiffiffie� cos tp

2ffiffi3p 0

0ffiffiffiffiffiffiffiffiffie� sin tp

2ffiffi3p

24

35; A3ðtÞ ¼

ffiffiffiffiffiffiffiffiffie� cos tp

6 0

0ffiffiffiffiffiffiffiffiffie� sin tp

6

24

35;

CðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffie�3 cos t4p ffiffiffiffiffiffiffi

2ffiffi3pp 0

0ffiffiffiffiffiffiffiffiffiffiffie�3 sin t4p ffiffiffiffiffiffiffi

2ffiffi3pp

2664

3775;

f ðt; xmðtÞ; xmðt � hðtÞÞÞ ¼y1 þ 1

1þx21ðt�hðtÞÞ þ

ffiffiffiffiffiffiffiffiffiffiffiffi1:58ep

�ffiffiffiffiffiffiffiffiffie� cos t

12

q� �x1ðt � hðtÞÞ

0:5 sin tx1 � x31 � 0:2y2

1 þ 11þy2

1ðt�hðtÞÞ þ

ffiffiffiffiffiffiffiffiffiffiffiffi1:58ep

�ffiffiffiffiffiffiffiffiffie� sin t

12

q� �y1ðt � hðtÞÞ

2664

3775:

The model (4.6) has a chaotic attractor with initial condition /1(t), which can be seen in Fig. 4. By using the notation in(4.3), it is easy to have the following:

−3 −2 −1 0 1 2−6

−4

−2

0

2

4

x1 (a)

y1

−3 −2 −1 0 1 2−6

−4

−2

0

2

4

x2 (b)

y2

0 200 400 600 800 1000

−4

−2

0

2

4

Time t (c)

The

sta

te (

x1, y

1)

0 200 400 600 800 1000

−4

−2

0

2

4

Time t (d)

The

sta

te (

x2, y

2)

Master Slave

x1 y1

x2 y2

evalS retsaM

Fig. 4. The chaotic attractor of nonlinear non-autonomous chaotic systems with time-varying delay systems (4.6) in the xy-plane and xy-Time t.


_eðtÞ ¼ ðA1ðtÞ � CðtÞK3ðtÞÞeðtÞ þ ðA2ðtÞ � CðtÞK4ðtÞÞeðt � hðtÞÞþ A3ðtÞf1ðt; eðtÞ; eðt � hðtÞÞÞ; nx 6 t 6 nxþ d;

_eðtÞ ¼ A1ðtÞeðtÞ þ A2ðtÞeðt � hðtÞÞ ð4:8Þþ A3ðtÞf1ðt; eðtÞ; eðt � hðtÞÞÞ; nxþ d < t 6 ðnþ 1Þx;

eðt0 þ tÞ ¼ /ðtÞ; t 2 ½�h;0�; h P 0:

h = 1, and _hðtÞ ¼ sin t cos t and then d = 0.5. From Fig. 4. we know that the bounds of the chaotic attractor �2.2 < x1 < 1.5,�5 < y1 < 3. For the positive numbers �1 ¼ e; �2 ¼ e�1; �3 ¼ 1; c1 ¼

ffiffiffiffiffiffi19p

; c2 ¼ 1; a ¼ 0:5; b ¼ e�1; x ¼ 5; d ¼ 4; k ¼ 1;n ¼ 0;1;2; . . ., we can verify that all conditions of Theorem 3.2 hold and that the condition (1) � (2) has a solution

PðtÞ ¼ ecos t � e�1 00 esin t � e�1

" #2 BPþð0;1Þ;

lðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffie3 cos tp

00

ffiffiffiffiffiffiffiffiffiffiffie3 sin tp

" #2 BPþð0;1Þ;

mðtÞ ¼0:01 0

0 0:01

2 BPþð0;1Þ:

Therefore, by Theorem 3.2 the error system (4.8) is exponentially stable with an exponential a = 0.5 and we chosen the inter-mittent linear state stabilizing feedback control K3 and K4 are same as that in Theorem 3.1 such that K3 ¼ 1

2 CTðtÞPðtÞ andK4 = CT(t)P(t), we obtain

0 50 100 150 200−2

−1

0

1

2

Time t(a)

The

sta

te e

rror

(ex)

0 10 20 30−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Time t(b)

The

sta

te e

rror

(ex)

0 50 100 150 200−4

−2

0

2

4

Time t(c)

The

sta

te e

rror

(ey)

0 10 20 30−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Time t(d)

The

sta

te e

rror

(ey)

Fig. 5. Synchronization error curves of the master system (4.6) and the slave system (4.7), (a), (c) without delay feedback control and (b), (d) with delayfeedback control (4.9) input.


u1ðtÞ ¼ffiffiffiffiffiffiffiffiffiecos t4p

2ffiffiffiffiffiffiffiffiffiffi2ffiffiffi3pp ðx1ðtÞ � x2ðtÞÞ þ

ffiffiffiffiffiffiffiffiffiecos t4p

2ffiffiffiffiffiffiffiffiffiffi2ffiffiffi3pp ðx1ðt � hðtÞÞ � x2ðt � hðtÞÞÞ;

u2ðtÞ ¼ffiffiffiffiffiffiffiffiffiesin t4p

2ffiffiffiffiffiffiffiffiffiffi2ffiffiffi3pp ðy1ðtÞ � y2ðtÞÞ þ

ffiffiffiffiffiffiffiffiffiesin t4p

2ffiffiffiffiffiffiffiffiffiffi2ffiffiffi3pp ðy1ðt � hðtÞÞ � y2ðt � hðtÞÞÞ: ð4:9Þ

The numerical simulations are carried out using the explicit Runge–Kutta-like method (dde45), interpolation and extrapo-lation by spline of the third order. The initial function /1(t),/2(t) for the master system and slave system, respectively. Fig. 4.show that the chaotic attractor of nonlinear non-autonomous chaotic systems with time-varying delay systems (4.6) in thexy-plane and xy-Time t. The results of the two identical nonlinear non-autonomous chaotic systems with time-varying delaysystems with intermittent linear state delayed feedback control (4.9) are shown in Fig. 5: synchronization error curves of themaster system (4.6) and the slave system (4.7), (a), (c) without delay feedback control and (b), (d) with delay feedback con-trol (4.9) input.

5. Conclusions

This paper has investigated synchronization of non-autonomous chaotic systems with time-varying delay via delayedfeedback control. We have obtained some sufficient conditions for the exponential stability of the error system via solvingRiccati differential equation. The delay feedback controller H1 and H2 designed can guarantee exponential stability of theerror system. The validity of the approach has been demonstrated by numerical examples.

Acknowledgment

The authors thank anonymous reviewers for their valuable comments and suggestions. The first author is supported bythe Graduate School, Chiang Mai University and Thai Government Scholarships in the Area of Science and Technology (Min-istry of Science and Technology). The second author is supported by the Center of Excellence in Mathematics, CHE, Thailand.The third author was supported by NSERC Canada.


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