Nonlinear Dyn (2013) 74:479–485DOI 10.1007/s11071-013-0970-0

O R I G I NA L PA P E R

Generalized finite-time synchronization between coupledchaotic systems of different orders with unknownparameters

Jiakun Zhao · Ying Wu · Yuying Wang

Received: 11 December 2012 / Accepted: 25 May 2013 / Published online: 18 September 2013© Springer Science+Business Media Dordrecht 2013

Abstract This letter investigates the adaptive finite-time synchronization of different coupled chaotic (orhyperchaotic) systems with unknown parameters. Thesufficient conditions for achieving the generalizedfinite-time synchronization of two chaotic systems arederived based on the theory of finite-time stability ofdynamical systems. By the adaptive control technique,the control laws and the corresponding parameters up-date laws are proposed such that the generalized finite-time synchronization of nonidentical chaotic (or hy-perchaotic) systems is to be obtained. These resultsobtained are in good agreement with the existing onein open literature and it is shown that the technique in-troduced here can be further applied to various finite-time synchronizations between dynamical systems. Fi-nally, numerical simulations are given to demonstratethe effectiveness of the proposed scheme.

Keywords Finite-time synchronization · Adaptivecontrol · Hyperchaotic system · Unknown parameter

J. Zhao (�) · Y. WangSchool of Mathematics and Statistics,Xi’an Jiaotong University, Xi’an, 710049, Chinae-mail: zhaojk@mail.xjtu.edu.cn

J. Zhao · Y. WuSchool of Aerospace, Xi’an Jiaotong University, Xi’an,710049, China

1 Introduction

Chaos synchronization has attracted an increasing in-terest due to its useful applications in secure communi-cation, power converters, biological systems, informa-tion processing, and chemical reactions [1]. The mainidea of synchronization between chaotic systems is todesign a suitable controller to control the response sys-tem so that the response system states follow the onesof a master system asymptotically. A number of differ-ent control schemes have been developed for achiev-ing chaos synchronization, such as the linear feedbackmethod [2], adaptive control [3, 4], impulsive method[5], stochastic control [6], H∞ approach [7], fuzzylogic control [8], etc.

The parameters of chaotic systems are inevitablyperturbed by various external factors such that thevalues of these parameters cannot be exactly knownin advance in many real applications. Therefore, theinvestigation into synchronization of chaotic systemswith known parameters becomes an important topic.A few results of the adaptive generalized synchro-nization with unknown parameters have been obtainedin [9, 10]. These methods have been used to guaran-tee the convergence of the synchronization procedurewith infinite settling time. Recent studies of chaos syn-chronization have more focused on the convergencerate of coupled chaotic systems. The main drive forstudying the convergence rate is that from a practicalpoint of view, however, it is more valuable that thesynchronization objective is realized in a finite time.

480 J. Zhao et al.

To obtain faster synchronization between chaotic sys-tems, an effective approach is using finite time con-trol techniques. Moreover, the finite time control tech-niques have demonstrated better robustness and distur-bance rejection properties [11]. Some works have in-vestigated finite-time chaos synchronization [12–15].However, most of above schemes are just presentedeither were more specific to synchronization of somesystems, or did not analyze the effects of the unknownparameters. In this letter, we propose a generalizedsynchronization scheme based on the finite-time sta-bility theory with unknown parameters. Inspired bythe former works and ideas, we use adaptive controlto implement a particular kind of synchronization withunknown parameters. The error systems of chaos syn-chronization are the functions of continuous systemswhere the finite time stability conditions are appliedfor. Based on the above reason and the so-called finitetime stability conditions for non autonomous contin-uous systems, we describe this kind of synchroniza-tion as generalized finite-time synchronization withunknown parameters, which is less restrictive but moreextensive.

The rest of this paper is organized as follows. Thepreliminary definition and lemma for the generalizedfinite-time synchronization is given in Sect. 2. Mainresults are provided in Sect. 3. An illustrative exampleis presented to show the effectiveness of the obtainedscheme in Sect. 4. Conclusion and discussion are fi-nally drawn in Sect. 5.

2 Preliminary definition and lemma

Consider the general m-dimensional chaotic (hyper-chaotic) system described by

x = f (x) + F(x)P, (1)

where x = (x1, x2, . . . , xm)T ∈ Rm is the state vectorof the system, f (x) ∈ C(Rm,Rm) including nonlinearterms, F ∈ C(Rm,Rm×k) and P ∈ Rk as the vector ofsystem parameters. It takes Eq. (1) as the drive system.The controlled response system is given by

y = g(y) + G(y)� + u, (2)

where y = (y1, y2, . . . , yn)T ∈ Rn is the state vector

of the response system, g(y) ∈ C(Rn,Rn) includingnonlinear terms, G ∈ C(Rn,Rn×l ) and � ∈ Rl as the

parameter vector of the response system. Then oneaims to design a controller u (u ∈ Rn), which is able tofinite-time synchronize the two chaotic (hyperchaotic)systems with identical or nonidentical orders. LetQ(x) = (Q1(x),Q2(x), . . . ,Qh(x))(∈ C(Rm,Rh))

and S(y) = (S1(y), S2(y), . . . , Sh(y))(∈ C(Rn,Rh))

be observable variables of the system (1) and the sys-tem (2), respectively. Then the generalized finite-timesynchronization error of the two chaotic (or hyper-chaotic) systems be

e(t) = Q(x) − S(y),

= (Q1(x) − S1(y),Q2(x) − S2(y), . . . ,

Qh(x) − Sh(y))T

. (3)

Then the error dynamical system between the drivesystem (1) and the response system (2) can be writtenas

e(t) = DQ(x) · (f (x) + F(x)P)

− DS(y) · (g(y) + G(y)� + u), (4)

where DQ(x) and DS(y) are the Jacobin matrices ofthe vector functions Q(x) and S(y), respectively, i.e.,

DQ(x) =

⎛

⎜⎜⎜⎜⎜⎝

∂Q1(x)∂x1

∂Q1(x)∂x2

· · · ∂Q1(x)∂xm

∂Q2(x)∂x1

∂Q2(x)∂x2

· · · ∂Q2(x)∂xm

......

. . ....

∂Qh(x)∂x1

∂Qh(x)∂x2

· · · ∂Qh(x)∂xm

⎞

⎟⎟⎟⎟⎟⎠

,

DS(y) =

⎛

⎜⎜⎜⎜⎜⎝

∂S1(x)∂y1

∂S1(x)∂y2

· · · ∂S1(x)∂yn

∂S2(x)∂y1

∂S2(x)∂y2

· · · ∂S2(x)∂yn

......

. . ....

∂Sh(x)∂y1

∂Sh(x)∂y2

· · · ∂Sh(x)∂yn

⎞

⎟⎟⎟⎟⎟⎠

.

Remark 1 The nonlinear dynamical systems (1) and(2) stand for a wide variety of chaotic or hyper-chaoticsystems including well-known Lorenz, hyperchaoticLorenz, Chen, hyperchaotic Chen, Rössler, and hyper-chaotic Rössler systems, etc.

Remark 2 h ≤ min{m,n} through this paper.

Finite-time synchronization of chaotic systemsmeans that the state of the drive system can converge

Generalized finite-time synchronization between coupled chaotic systems of different orders 481

to the state of the response system after a finite time.The definition of generalized finite-time synchroniza-tion between chaotic systems is given as follows.

Definition 1 For the above systems (1) and (2), it issaid that they are generalized finite-time synchroniza-tion. If there exists a constant T = T (e(0)) > 0, suchthat

limt→T

∥∥e(t)∥∥ = lim

t→T

∥∥Q(x) − S(y)∥∥ = 0, (5)

and ‖e(t)‖ ≡ 0, if t ≥ T , then the synchronizationabout the drive system (1) and the response system (2)is achieved within a finite time.

The finite-time synchronization problem can betransformed to the equivalent problem of the finite-time stabilization of the error system (4). The objectiveof this paper is to design a suitable feedback controllaw u(t) such that for any given drive system (1) andthe response system (2) with unknown parameters, thefinite-time stability of the resulting error system (4)can be achieved in the sense of Definition 1.

Lemma 1 [16] Suppose a1, a2, . . . , ak and 0 < r < 1are all real numbers, then the following inequalityholds:

(|a1|+ |a2|+ · · ·+ |ak|)r ≤ |a1|r +|a2|r +· · ·+ |ak|r .

(6)

Lemma 2 [17] Consider the system

x = f (t, x), x ∈ Rn, (7)

where f ∈ C([0,+∞) × D,Rn) and D is an openneighborhood D ⊂ Rn.

Suppose there exists a continuous differential posi-tive definite function V (t, x) : [0,+∞) × D →[0,+∞), V (t, x) is smooth and

V (t, x) = ∂V

∂t+ ∇V T · f (t, x). (8)

V (t, x) is radially unbounded, i.e.,

V (t, x) ≥ ϕ(‖x‖), ∀t ∈ [0,+∞), ∀x ∈ Rn, (9)

where the function ϕ(τ) ∈ C([0,+∞), [0,+∞)) andlimτ→+∞ ϕ(τ) = +∞,

V (t, x) ≤ −r(V (t, x)

)(10)

where the function r ∈ C([0,+∞), [0,+∞)) andr(0) = 0, and ∃ε > 0,∫ ε

0

dτ

r(τ )< +∞. (11)

Then the origin of system (6) is globally finite time sta-ble for the system (6), and the settling time, dependingon the initial state x(t0) = x0, satisfies

T (x0) ≤∫ V (t0,x0)

t0

dτ

r(τ ). (12)

3 Main results

Assumption 1 Real matrix DS is row full-rank,which means that the inverse matrix DS−1 existswhen the matrix is a square matrix. When the matrixis not a square matrix, we further assume the num-ber of rows less than the one of columns, then ac-cording to the generalized inverse matrix theory [18],the right inverse matrix DS−1

R exists, and DS−1R =

DST (DS · DST )−1. For simplicity, we use DS−1

to denote the inverse or right inverse matrix of DS

throughout this paper.

Assumption 2 P and � are the estimated vectors ofunknown parameters, and the updating laws of the es-timated parameters are given by

{ ˙P = (DQ · F(x))T · e − PH,

˙� = −(DS · G(y))T · e − �H,

(13)

where P = P−P, � = �−�, PH = (Pη1, Pη

2, . . . , Pη

k)T

and �H = (�η1, �η

2, . . . , �

ηl )

T , η = ba

where a, b areall odd numbers and a > b.

By using the adaptive control and the parametersidentification technique, the controller can be decidedas

u = −g(y) − G(y)�

+ DS−1(DQ(f (x) + F(x)P

) + eH), (14)

where eH = (eη1, eη

2, . . . , eη

h)T .

Remark 3 From the design of the controller u asEq. (10), the error dynamical system becomes

e(t) = −DQ · F(x)P + DS · G(y)� − eH. (15)

482 J. Zhao et al.

Theorem 1 Let e(t) be Eq. (15) for the drive system(1) and the response system (2). If the active controlu(x, y, t) under Assumption 1 is given by Eq. (14)and the updating laws of the estimated parameters aregiven by Assumption 2, then the error system (15) isglobally finite time stable at the origin, and the set-tling time, depending on the initial state x(t0) = x0,satisfies

T (e0) ≤ 21−η

2

1 − η

((V (t0, e0)

) 1−η2 − t

1−η2

0

). (16)

Proof Construct a continuous differential positive def-inite function

V (t, e) = 1

2

(eT e + PT P + �T �

)

= 1

2

(h∑

i=1

e2i +

k∑

i=1

P2i +

l∑

i=1

�2i

)

. (17)

Then, according to Eq. (8), it can obtain

V (t, e) = Vt + Ve

(−DQ · F(x)P

+ DS · G(y)� − eH)

= PT ˙P + �T ˙� + eT

(−DQ · F(x)P

+ DS · G(y)� − eH)

= −PT PH − �T �H − eT eH

= −(

k∑

i=1

P1+ηi +

l∑

i=1

�1+ηi +

h∑

i=1

e1+ηi

)

= −21+η

2

(k∑

i=1

(P2

i

2

) 1+η2 +

l∑

i=1

(�2

i

2

) 1+η2

+h∑

i=1

(e2i

2

) 1+η2

)

≤ −21+η

2

(k∑

i=1

P2i

2+

l∑

i=1

�2i

2+

h∑

i=1

e2i

2

) 1+η2

.

(18)

Let r(τ ) = 21+η

2 τ1+η

2 . The function r ∈ C([0,+∞),

[0,+∞)) and r(0) = 0, and when ε > 0,

∫ ε

0

dτ

r(τ )= (2ε)

1−η2

1 − η< +∞.

Moreover, ∃ϕ(τ) ∈ C([0,+∞), [0,+∞)), ϕ(τ) =12τ 2, then

V (t, e) = 1

2

(eT e + PT P + �T �

) ≥ 1

2eT e = 1

2‖e‖2

= ϕ(‖e‖).

From Lemma 2, the origin of system (6) is globallyfinite time stable for the system (6), and the settlingtime, depending on the initial state x(0) = x0, satisfies

T (e0) ≤∫ V (t0,e0)

t0

dτ

r(τ )

= 21−η

2

1 − η

((V (t0, e0)

) 1−η2 − t

1−η2

0

). �

Remark 4 If t0 = 0, then

T = T(e(0)

) ≤ 21−η

2

1 − η

(V

(0, e(0)

)) 1−η2 . (19)

If S(y) = y, then it obtains the following corollary.

Corollary 1 If m = n, Q(x) = x, and S(y) = y, thenthe error system is

e(t) = x − y, (20)

which is so-called complete synchronization (CS) in fi-nite time. Moreover, the updating laws of the estimatedparameters are given by{ ˙P = (F (x))T · e − PH,

˙� = −(G(y))T · e − �H.

(21)

The controller becomes

u = −g(y) − G(y)� + f (x) + F(x)P + eH.

Remark 5 Cai et al. [19] studied the finite-time syn-chronization of the error system e(t) = y − φ(x), butthe author did not consider unknown parameters, andthe error system should be the non-autonomous sys-tem. Nevertheless, we appreciate this idea.

4 An illustrative example

The following example will demonstrate the efficiencyand applicability of the proposed method and validatethe theoretical results.

It is selected that Lorenz system [20] is the drivesystem and the response system is the Chen hyper

Generalized finite-time synchronization between coupled chaotic systems of different orders 483

chaotic system [21] with the controller. The drive dy-namic equations and the response ones are given by

⎧⎪⎨

⎪⎩

x1 = a(x2 − x1),

x2 = −x2 − x1x3 + bx1,

x3 = x1x2 − cx3

(22)

and

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

y1 = a1(y2 − y1) + y4 + u1,

y2 = d1y1 − y1y3 + c1y2 + u2,

y3 = y1y2 − b1y3 + u3,

y4 = y2y3 + r1y4 + u4

(23)

respectively.Assume the observable variables of systems (22)

and (23) are

Q(x) =(

x1x2

x1 − x3

)and S(y) = 1

2

(y1 − y2

y3 − y4

),

respectively. Then

e(t) = Q(x) − S(y) =(

x1x2 − 12 (y1 − y2)

(x1 − x3) − 12 (y3 − y4)

)

.

It holds

DQ =(

x2 x1 01 0 −1

),

DS = 1

2

(1 −1 0 00 0 1 −1

)and

DS−1 =

⎛

⎜⎜⎝

1 0−1 00 10 −1

⎞

⎟⎟⎠ .

Let η = 0.2. One obtains the controller u = (u1, u2,

u3, u4)T , i.e.,

u =

⎛

⎜⎜⎜⎜⎝

−y4 − (y2 − y1)a1 + (x22

− x1x2)a − x1x2 − x21x3 + x2

1 b + e0.21

y1y3 − y1d1 − y2c1 + (x1x2 − x22)a + x1x2 + x2

1x3 − x2

1 b − e0.21

−y1y2 + y3b1 − x1x2 + (x2 − x1)a + x3c + e0.22

−y2y3 − y4r1 + x1x2 + (x1 − x2)a − x3c − e0.22

⎞

⎟⎟⎟⎟⎠

. (24)

The updating laws are

⎛

⎜⎝

˙a˙b

˙c

⎞

⎟⎠ =

⎛

⎜⎝

(x2 − x1)(x2e1 + e2) − a0.2

x21e1 − b0.2

x3e2 − c0.2

⎞

⎟⎠ (25)

and

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

˙a1

˙b1

˙c1

˙d1

˙r1

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

= 1

2

⎛

⎜⎜⎜⎜⎜⎜⎝

(y1 − y2)e1 − 2a0.21

y3e2 − 2b0.21

y2e1 − 2c0.21

y1e1 − 2d0.21

y4e2 − 2r0.21

⎞

⎟⎟⎟⎟⎟⎟⎠

. (26)

From Theorem 1, one can obtain the generalizedfinite-time synchronization between the system (22)and the system (23). It selects the parameters of thedrive system (22) as a = 10.0, b = 28.0, c = 8/3 to en-

sure the chaotic behavior and the parameters of the re-sponse system (23) as a1 = 35.0, b1 = 3.0, c1 = 12.0,d1 = 7.0, r1 = 0.5. The initial values of two systems

Fig. 1 Dynamics of generalized finite-time synchronization er-rors states for systems (22) and (23) with time t

484 J. Zhao et al.

Fig. 2 Estimate values ofparameters with the updatelaws of these parameters:(a) a, b, c, (b) a1, b1, c1, d1and r1

are x1(0) = 4.8, x2(0) = 2.2, x3(0) = 1.6 and y1(0) =3.2, y2(0) = 4.3, y3(0) = 1.2, y4(0) = 5.6. The esti-mated parameters start from a(0) = b(0) = c(0) = 0.1and a1(0) = b1(0) = c1(0) = d1(0) = r1(0) = 0.2, re-spectively. The corresponding synchronization errorstates for the system (22) and (23) are shown in Fig. 1.The dynamics of the parameter estimations are plottedin Fig. 2.

From (19), T ∗ = T (e(0)) ≤ 21−η

2

1−η(V (0, e(0)))

1−η2 .

5 Conclusion and discussion

The present paper investigated the fast synchroniza-tion of directionally coupled chaotic systems under achained interaction topology. It was shown that suchdirectionally coupled systems can achieve synchro-nization in finite time as long as the coupling controllersatisfied the stability conditions. The optimization ofthe synchronization time is essential in a number ofreal-time data applications, which is currently a chal-lenging task in control theory. To handle this problemwith the simplest control input, we have studied thegeneralized finite-time synchronization with unknownparameters based on the adaptive laws and the finite-time stability theory of dynamics. Moreover, suitableadaptive laws were obtained to tackle the estimatedparameters. The proposed technique has been provedto be effective by an illustrate example. In general,the proposed control method is very simple and couldbe easily realized experimentally compared with pre-vious techniques that use complex nonlinear controlfunctions. An illustrative example and its numericalsimulation were used to interpret the theoretical re-sults.

Acknowledgements This work was supported by theChina Postdoctoral Science Foundation funded project(No. 2013M532030) and the Natural Science Foundation ofChina (NSFC) under Grant 11272242.

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