on the synchronization of identical and non-identical 4-d chaotic systems using arrow form matrix

Chaos, Solitons and Fractals 42 (2009) 101–112

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals

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

On the synchronization of identical and non-identical 4-Dchaotic systems using arrow form matrix

S. Hammami *, K. Ben Saad, M. BenrejebUR LARA Automatique, Ecole Nationale d’Ingénieurs de Tunis, BP 37, Tunis Le Belvédère 1002, Tunisia

a r t i c l e i n f o

Article history:Accepted 30 October 2008

Communicated by Prof. Ji-Huan He

0960-0779/$ - see front matter � 2008 Elsevier Ltddoi:10.1016/j.chaos.2008.10.038

* Corresponding author.E-mail addresses: [email protected] (S

a b s t r a c t

Using the Borne and Gentina practical criterion associated with the Benrejeb canonicalarrow form matrix, to derive the stability property of dynamic complex systems, a newstrategy of control is formulated for chaos synchronization of two identical Lorenz Stenflosystems and two new four-dimensional chaotic systems, namely the Qi chaotic systems.The designed controller ensures that the state variables of both controlled chaotic slaveLorenz Stenflo and Qi systems globally synchronizes with the state variables of the mastersystems, respectively. It is also shown that Qi system globally synchronizes with LorenzStenflo system under the afforded generalized strategy of control. Numerical simulationsare carried out to assess the performance of the proposed contributions in the importantfield of chaotic synchronization.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

In the past decades, chaos synchronization has received a tremendous increasing interest [1–6] and references therein.The phenomenon of synchronization is extremely wide spread in nature as well as in the realm technology. The fact that,various objects seek to achieve order and harmony in their behaviour, which is a characteristic of synchronization, seemsto be a manifestation of the natural tendency of self organization existing in nature. Considerable attention paid to such top-ics is due to the potential applications of synchronization in communication engineering, using chaos to mask the informa-tion bearing signal [1,7–9]. In biology, chemistry and medicine the phenomenon of synchronization is of interest to thosestudying rhythms, electric rhythms of the brain in particular, cardiac rhythms and wave reactions in chemistry. In industry,synchronization is also used in the power generation to ensure exact coincidence of frequencies of several alternating cur-rent generators operating in parallel mode for common loading. In medicine, the effects of synchronization have found wideapplication in the construction of various vibro-technical devices [10].

For a system of two coupled chaotic oscillators, the master ð _x ¼ f ðx; yÞÞ and the slave ð _y ¼ gðx; yÞÞ, where x and y are phasespace variables, f and g the corresponding nonlinear functions, synchronization in a direct sense implies j xðtÞ � yðtÞ j! 0 ast !1 [11]. When this occurs, the coupled system is said to be completely synchronized [12]. Other forms of synchronizationthat have been observed includes phase synchronization [13–16], lag synchronization, generalized synchronization [17] andsequential synchronization [18].

Synchronization using active control was proposed by Bai and Lonngren using the Lorenz system [19], and has been sub-sequently employed to synchronize other chaotic systems [18,20–24]. The technique was generalized to non-identical sys-tems, thus, extending its applicability beyond identical systems, and showing its pivotal role in other schemes [21–23].

Practical Borne and Gentina stability criterion [25–27], applied to continuous systems, generalizes the use of Kotelyanskilemma for nonlinear systems and defines large classes of processes for which linear Aizerman conjecture is satisfied [28–32].

. All rights reserved.

. Hammami), [email protected] (K. Ben Saad), [email protected] (M. Benrejeb).

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102 S. Hammami et al. / Chaos, Solitons and Fractals 42 (2009) 101–112

The stabilisability approach is usefully applied for nonlinear control systems, particularly, when the Benrejeb arrow formcharacteristic matrix is used [30–32].

Throughout the present paper, the Benrejeb arrow form matrix is applied to synchronize two identical Lorenz Stenflo sys-tems [32–35] and two identical new four-dimensional systems, hereafter called the Qi systems [36]. The paper deals alsowith synchronization behaviour between Lorenz Stenflo and Qi systems using the generalized active control. It is proved thatby applying the proposed control scheme, the variance of the synchronization error can converge to any arbitrarily smallbound around zero. The layout of the paper is as follows: synchronization behaviour of two identical Lorenz Stenflo systemsand two identical Qi systems are studied in Sections 2 and 3, respectively. Section 4 deals with non-identical synchronizationbetween Lorenz Stenflo and Qi systems. Simulation results show that the proposed method can be successfully used in syn-chronization of chaotic systems.

2. Synchronization of two identical Lorenz Stenflo systems

In this subsection, let consider the problem of synchronization process of two identical chaotic Lorenz Stenflo systems.

2.1. The Lorenz Stenflo system

Here, we consider the following four coupled nonlinear autonomous first order differential equations, characterizing theLorenz Stenflo system [11,33]

_xðtÞ ¼ aðyðtÞ � xðtÞÞ þ cwðtÞ_yðtÞ ¼ xðtÞðr � zðtÞÞ � yðtÞ_zðtÞ ¼ xðtÞyðtÞ � bzðtÞ_wðtÞ ¼ �xðtÞ � awðtÞ

8>>><>>>:

ð1Þ

formulated by Stenflo from a low frequency short wavelength gravity wave equation. In (1), the strictly positive constantr;a; c and b are, respectively, the Rayleigh number, Prandtl number, rotation number and geometric parameter.

Some dynamical behaviours of the Lorenz Stenflo equations are reported in [33–35], including the familiar period-dou-bling route to chaos [34], to adaptive control and to synchronization. With the following parameters: a ¼ 1, b ¼ 0:7, c ¼ 1:5and r ¼ 26, the Lorenz Stenflo system exhibits the chaotic attractor shown in Fig. 1, in which the variable x is plotted againstthe variable y.

2.2. Problem statement

Let us consider a Lorenz Stenflo system given by

_x1ðtÞ ¼ aðy1ðtÞ � x1ðtÞÞ þ cw1ðtÞ_y1ðtÞ ¼ x1ðtÞðr � z1ðtÞÞ � y1ðtÞ_z1ðtÞ ¼ x1ðtÞy1ðtÞ � bz1ðtÞ_w1ðtÞ ¼ �x1ðtÞ � aw1ðtÞ

8>>><>>>:

ð2Þ

Fig. 1. Two-dimensional view of the Lorenz Stenflo attractors.

S. Hammami et al. / Chaos, Solitons and Fractals 42 (2009) 101–112 103

which drives a similar Lorenz Stenflo system given as

_x2ðtÞ ¼ aðy2ðtÞ � x2ðtÞÞ þ cw2ðtÞ þ u1ðtÞ_y2ðtÞ ¼ x2ðtÞðr � z2ðtÞÞ � y2ðtÞ þ u2ðtÞ_z2ðtÞ ¼ x2ðtÞy2ðtÞ � bz2ðtÞ þ u3ðtÞ_w2ðtÞ ¼ �x2ðtÞ � aw2ðtÞ þ u4ðtÞ

8>>><>>>:

ð3Þ

where UðtÞ ¼ ½u1ðtÞ u2ðtÞ u3ðtÞ u4ðtÞ �T is the active control function. Here, we aim at determining the controller Uwhich is required for system (3) to synchronize with system (2). For this purpose, let the error dynamics between (3) and(2) be

x3ðtÞ ¼ x2ðtÞ � x1ðtÞy3ðtÞ ¼ y2ðtÞ � y1ðtÞz3ðtÞ ¼ z2ðtÞ � z1ðtÞw3ðtÞ ¼ w2ðtÞ �w1ðtÞ

8>>><>>>:

ð4Þ

The previous expressions given by the system of Eqs. (4) lead to the following quantities:

_x3ðtÞ ¼ aðy3ðtÞ � x3ðtÞÞ þ cw3ðtÞ þ u1ðtÞ_y3ðtÞ ¼ rx3ðtÞ � y3ðtÞ � x2ðtÞz2ðtÞ þ x1ðtÞz1ðtÞ þ u2ðtÞ_z3ðtÞ ¼ x2ðtÞy2ðtÞ � x1ðtÞy1ðtÞ � bz3ðtÞ þ u3ðtÞ_w3ðtÞ ¼ �x3ðtÞ � aw3ðtÞ þ u4ðtÞ

8>>><>>>:

ð5Þ

which can be written under the following matrix description:

_XðtÞ ¼ Að:ÞXðtÞ þ BUðtÞ ð6Þ

with:

XðtÞ ¼ ½ x3ðtÞ y3ðtÞ z3ðtÞ w3ðtÞ �T ð7Þ

Að:Þ ¼

�a a 0 cðr � z1ðtÞÞ �1 �x2ðtÞ 0

y1ðtÞ x2ðtÞ �b 0�1 0 0 �a

26664

37775 ð8Þ

and

B ¼

1 0 0 00 1 0 00 0 1 00 0 0 1

26664

37775 ð9Þ

Let us try, in the next part of the paper, to design a state feedback control law assuring the stabilization of the error dy-namic system (6).

2.3. Chaos synchronization: feedback control law

The concept of chaos synchronization emerged much later, not until the gradual realization of the usefulness of chaos byscientists and engineers. Synchrony is the simplest effect of coupled identical systems: two identical systems display thesame dynamical pattern in their common phase space. In this subsection, we apply feedback control technique to achievechaos synchronization of two identical Lorenz Stenflo systems.

At this stage, the work is focussed on the synthesis of a stabilizing state feedback controller of the following form:

UðtÞ ¼ �Kð:ÞXðtÞ ð10Þ

in such a way that the closed loop system

_XðtÞ ¼ Acð:ÞXðtÞ ð11Þ

Acð:Þ ¼ Að:Þ � BKð:Þ ¼

�a� k11ð:Þ a� k12ð:Þ �k13ð:Þ c� k14ð:Þðr � z1ðtÞÞ � k21ð:Þ �1� k22ð:Þ �x2ðtÞ � k23ð:Þ �k24ð:Þ

y1ðtÞ � k31ð:Þ x2ðtÞ � k32ð:Þ �b� k33ð:Þ �k34ð:Þ�1� k41ð:Þ �k42ð:Þ �k43ð:Þ �a� k44ð:Þ

26664

37775 ð12Þ

being described by a particular canonical matrix form, namely the Benrejeb arrow form matrix.


To satisfy this purpose, the parameters of correction kijð:Þ;8i; j ¼ 1;2;3; i–j, can be chosen as follows:

a� k12ð:Þ ¼ 0�k13ð:Þ ¼ 0ðr � z1ðtÞÞ � k21ð:Þ ¼ 0�x2ðtÞ � k23ð:Þ ¼ 0y1ðtÞ � k31ð:Þ ¼ 0x2ðtÞ � k32ð:Þ ¼ 0

8>>>>>>>><>>>>>>>>:

)

k12ð:Þ ¼ ak13ð:Þ ¼ 0k21ð:Þ ¼ r � z1ðtÞk23ð:Þ ¼ �x2ðtÞk31ð:Þ ¼ y1ðtÞk32ð:Þ ¼ x2ðtÞ

8>>>>>>>><>>>>>>>>:

ð13Þ

In fact, when the considered system (6) is stabilized by the feedback U, the error will converge to zero as t ! þ1, whichimplies that the systems (2) and (3) are globally synchronized. To achieve this goal, U is chosen such that the instantaneousgain matrix Kð:Þ defined by (10), is a 4� 4 matrix.

The application of the classical Borne and Gentina stability criterion [26], associated to the particular canonical Benrejebarrow form matrix leads to the following theorem.

Theorem 1. The process, described by (6) is stabilized by the control law defined by (10), if the matrix Acð:Þ, defined by (12), is inthe arrow form and such that:

(i) the nonlinear elements are isolated in either one row or one column of the matrix Acð:Þ,(ii) the diagonal elements, acii

ð:Þ, of the matrix Acð:Þ are such that:

aciið:Þ < 0 8 i ¼ 1;2; . . . ;n� 1; ð14Þ

(iii) there exist e > 0 such that:

acnn ð:Þ �Xn�1

i¼1

jacnið:Þacin

ð:Þj� �

a�1ciið:Þ 6 �e ð15Þ

Proof. The overvaluing system MðAcð:ÞÞ, associated to the vectorial norm pðzÞ ¼ ½ jz1j jz2j . . . jznj �T ; z ¼ ½ z1 z2 . . . zn �T ,is defined, in this case, by the following system of differential equations:

_z ¼ MðAcð:ÞÞz ð16Þ
such that the elements mijð:Þ of MðAcð:ÞÞ are deduced from the ones of the matrix Acð:Þ by substituting the off-diagonal ele-ments by their absolute values, which can be written as
miið:Þ ¼ aciið:Þ 8i ¼ 1; . . . ;n

mijð:Þ ¼ jacijð:Þj 8i; j ¼ 1; . . . ;n; i–j

(ð17Þ

The system (6) is then stabilized by (10) if the matrix MðAcð:ÞÞ is the opposite of an M – matrix [27], or if, by application ofthe practical criterion of Borne and Gentina [26], we have, for e > 0

aciið:Þ 6 �e 8 i ¼ 1;2; . . . ;n� 1

ð�1Þn detðMðAcð:ÞÞÞP e

�ð18Þ

The development of the first member of the last inequality announced by (18)

ð�1Þn detðMðAcð:ÞÞÞ ¼ ð�1Þ acnnð:Þ �Xn�1

i¼1

ðjacnið:Þacin

ð:ÞjÞa�1ciið:Þ

!ð�1Þn�1

Yn�1

j¼1

acjjð:Þ ð19Þ

achieves easily the proof of the theorem. h

Corollary 1. The process, described by (6) is stabilized by the control law defined by (10) if the characteristic matrix Acð:Þ, definedby (12), is under the arrow form and such that:

(i) all the nonlinearities are located in either one row or one column of Acð:Þ,(ii) the diagonal elements acii

ð:Þ;8i ¼ 1;2; . . . ;n� 1, of the matrix Acð:Þ are strictly negative,(iii) the products of the off-diagonal elements acni

ð:Þacinð:Þ;8i ¼ 1; . . . ;n� 1, of the matrix Acð:Þ are non-negative,

(iv) the characteristic instantaneous polynomial PAc ðk; :Þ, defined by

PAc ðk; :Þ ¼ detðkI� Acð:ÞÞ ð20Þ

is strictly positive for k ¼ 0.

Proof. The proof of this corollary is inferred of the previous theorem one by taking into account the new added hypothesis(iii) of this corollary, which guarantees, through a simple transformation, the identity of the matrix Acð:Þ and its overvaluingmatrix M; this specific case allows the satisfaction of the linear Aizerman conjecture.


For (6) to be asymptotically stable, the parameters of correction kijð:Þ; 8i; j ¼ 1; . . . ;4, can be chosen so that the followingconstraints are fulfilled:

(i) all the nonlinearities are located in either one row or one column of the matrix Acð:Þ,(ii) the diagonal elements of the matrix Acð:Þ are such that

�a� k11ð:Þ < 0�1� k22ð:Þ < 0�b� k33ð:Þ < 0

8><>: ð21Þ

(iii) there exist e > 0, such that

ð�a� k44ð:ÞÞ �ð�1� k41ð:ÞÞðc� k14ð:ÞÞð�a� k11ð:ÞÞ�1

þð�k42ð:ÞÞð�k24ð:ÞÞð�1� k22ð:ÞÞ�1

þð�k43ð:ÞÞð�k34ð:ÞÞð�b� k33ð:ÞÞ�1

0B@

1CA 6 �e ð22Þ

Various choices of the instantaneous gain matrix Kð:Þ, which can cancel the nonlinear part of the error system, are pos-sible. One good choice is

Kð:Þ ¼

�aþ 1 a 0 cðr � z1ðtÞÞ 1 �x2ðtÞ 0

y1ðtÞ x2ðtÞ �bþ 1 0�1 0 0 1

26664

37775 ð23Þ

Thus, yielding the feedback functions

u1ðtÞu2ðtÞu3ðtÞu4ðtÞ

26664

37775 ¼ �

�aþ 1 a 0 cðr � z1ðtÞÞ 1 �x2ðtÞ 0

y1ðtÞ x2ðtÞ �bþ 1 0�1 0 0 1

26664

37775

x3ðtÞy3ðtÞz3ðtÞw3ðtÞ

26664

37775 ð24Þ

We begin by illustrating the error states between systems (2) and (3) when the control is turned off (Fig. 2). It is obviousthat the error states grow with time chaotically.

With (23), the closed loop system (11) to be controlled is a linear system. These choices ensure that the error statesx3; y3; z3 and w3 asymptotically converge to zero as time t ! þ1 and therefore the synchronization between systems (2)and (3) is achieved.

0 10 20 30 40 50 60 70 80 90 100-20

0

20

x3

0 10 20 30 40 50 60 70 80 90 100-50

0

50

y3

0 10 20 30 40 50 60 70 80 90 100-50

0

50

z3

0 10 20 30 40 50 60 70 80 90 100-10

0

10

w3

0 10 20 30 40 50 60 70 80 90 1000

20

40

e

time (s)

Fig. 2. Error dynamics ðx3; y3; z3;w3 and eÞ of the coupled Lorenz Stenflo system when the active controller is deactivated.

0 5 10 15-0.02

-0.01

0

0.01

0.02

0.03

x3,y

3,z3

,w3

0 5 10 150

0.01

0.02

0.03

0.04

e

time (s)

x1

x2 x3x4

Fig. 3. Asymptotic convergence of the error states with the active controller functions activated at t ¼ 0.


In Fig. 3, we show the asymptotic convergence of the error states when the controller is switched on at t ¼ 0. One can seethat the master–slave system is globally synchronized. This is also confirmed by the exponential convergence of the synchro-nization quality defined by the error propagation on the error states

e ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

3 þ y23 þ z2

3 þw23

qð25Þ

To ascertain the stability of the synchronization, we activate the state feedback control law designed as shown in Fig. 3.Again, stable synchronization is guaranteed. h

3. Synchronization of two identical Qi systems

In this subsection, we focus on the problem of synchronization process of two identical chaotic Qi systems.

3.1. The Qi system

The second model of studied system, is the following 4-D autonomous system described by [11,36]

_xðtÞ ¼ aðyðtÞ � xðtÞÞ þ yðtÞzðtÞwðtÞ_yðtÞ ¼ bðxðtÞ þ yðtÞÞ � xðtÞzðtÞwðtÞ_zðtÞ ¼ �czðtÞ þ xðtÞyðtÞwðtÞ_wðtÞ ¼ �dwðtÞ þ xðtÞyðtÞzðtÞ

8>>><>>>:

ð26Þ

where x; y; z and w are the state variables of the system, a; b; c and d are all positive real constant parameters. System (26)was recently introduced by Qi and Al and has been shown to exhibit complex dynamical behaviour including the familiarperiod-doubling route to chaos as well as hopf bifurcations. For instance, in Fig. 4, we show the chaotic behaviour in theðy;wÞ plane for a ¼ 30; b ¼ 10; c ¼ 1 and d ¼ 10.


Following section 2, we choose a master Qi system given by

_x1ðtÞ ¼ aðy1ðtÞ � x1ðtÞÞ þ y1ðtÞz1ðtÞw1ðtÞ_y1ðtÞ ¼ bðx1ðtÞ þ y1ðtÞÞ � x1ðtÞz1ðtÞw1ðtÞ_z1ðtÞ ¼ �cz1ðtÞ þ x1ðtÞy1ðtÞw1ðtÞ_w1ðtÞ ¼ �dw1ðtÞ þ x1ðtÞy1ðtÞz1ðtÞ

8>>><>>>:

ð27Þ

which drives a slave Qi system given by

-6 -4 -2 0 2 4 60

1

2

3

4

5

6

7

8

9

y

w

Fig. 4. Chaotic behaviour of the chaotic Qi system.


_x2ðtÞ ¼ aðy2ðtÞ � x2ðtÞÞ þ y2ðtÞz2ðtÞw2ðtÞ þ u1ðtÞ_y2ðtÞ ¼ bðx2ðtÞ þ y2ðtÞÞ � x2ðtÞz2ðtÞw2ðtÞ þ u2ðtÞ_z2ðtÞ ¼ �cz2ðtÞ þ x2ðtÞy2ðtÞw2ðtÞ þ u3ðtÞ_w2ðtÞ ¼ �dw2ðtÞ þ x2ðtÞy2ðtÞz2ðtÞ þ u4ðtÞ

8>>><>>>:

ð28Þ

where uiðtÞ; i ¼ 1; . . . ;4, are the state feedback control functions to be determined.Subtracting (27) from (28) and using (4) gives the error dynamics equations below

_x3ðtÞ ¼ aðy3ðtÞ � x3ðtÞÞ þ y2ðtÞz2ðtÞw2ðtÞ � y1ðtÞz1ðtÞw1ðtÞ þ u1ðtÞ_y3ðtÞ ¼ bðx3ðtÞ þ y3ðtÞÞ � x2ðtÞz2ðtÞw2ðtÞ þ x1ðtÞz1ðtÞw1ðtÞ þ u2ðtÞ_z3ðtÞ ¼ �cz3ðtÞ þ x2ðtÞy2ðtÞw2ðtÞ � x1ðtÞy1ðtÞw1ðtÞ þ u3ðtÞ_w3ðtÞ ¼ �dw3ðtÞ þ x2ðtÞy2ðtÞz2ðtÞ � x1ðtÞy1ðtÞz1ðtÞ þ u4ðtÞ

8>>><>>>:

ð29Þ

It should be pointed out that by considering, once again, the matrix description (6) and (7) to characterize the previous errordynamics system (29); it comes

Að:Þ ¼

�a ða� z1ðtÞw1ðtÞÞ y2ðtÞw1ðtÞ y2ðtÞz2ðtÞðbþ z2ðtÞw2ðtÞÞ b �x1ðtÞw2ðtÞ �x1ðtÞz1ðtÞ

y1ðtÞw1ðtÞ x2ðtÞw1ðtÞ �c x2ðtÞy2ðtÞy1ðtÞz1ðtÞ x2ðtÞz1ðtÞ x2ðtÞy2ðtÞ �d

26664

37775 ð30Þ

and B ¼ Id4�4.

3.3. Synchronization via state feedback active control law

To achieve the property of stable synchronization between the identical chaotic Qi systems (27) and (28) and by referringto the hypothesis mentioned in the theorem announced in section II2, the most suitable state feedback controller (10), allow-ing to put the matrix Acð:Þ (31)

Acð:Þ ¼

�a� k11ð:Þ ða� z1ðtÞw1ðtÞÞ � k12ð:Þ y2ðtÞw1ðtÞ � k13ð:Þ y2ðtÞz2ðtÞ � k14ð:Þðbþ z2ðtÞw2ðtÞÞ � k21ð:Þ b� k22ð:Þ �x1ðtÞw2ðtÞ � k23ð:Þ �x1ðtÞz1ðtÞ � k24ð:Þ

y1ðtÞw1ðtÞ � k31ð:Þ x2ðtÞw1ðtÞ � k32ð:Þ �c � k33ð:Þ x2ðtÞy2ðtÞ � k34ð:Þy1ðtÞz1ðtÞ � k41ð:Þ x2ðtÞz1ðtÞ � k42ð:Þ x2ðtÞy2ðtÞ � k43ð:Þ �d� k44ð:Þ

26664

37775 ð31Þ


under the Benrejeb arrow form, can be chosen as follows:

k12ð:Þ ¼ a� z1ðtÞw1ðtÞk13ð:Þ ¼ y2ðtÞw1ðtÞk21ð:Þ ¼ bþ z2ðtÞw2ðtÞk23ð:Þ ¼ �x1ðtÞw2ðtÞk31ð:Þ ¼ y1ðtÞw1ðtÞk32ð:Þ ¼ x2ðtÞw1ðtÞ

8>>>>>>>><>>>>>>>>:

ð32Þ

Furthermore, to satisfy the constraint (14) of the theorem, one possible choice is given by

k11 ¼ �29k22 ¼ 11k33 ¼ 0k44 ¼ �9

8>>><>>>:

ð33Þ

Now, in order to fulfil the inequality (15), we can easily check the stability of a linear controlled system, in the case wherethe parameters of correction k14ð:Þ; k24ð:Þ; k34ð:Þ; k41ð:Þ; k42ð:Þ and k43ð:Þ are selected as follows:

k14ð:Þ ¼ y2ðtÞz2ðtÞk24ð:Þ ¼ �x1ðtÞz1ðtÞk34ð:Þ ¼ x2ðtÞy2ðtÞk41ð:Þ ¼ y1ðtÞz1ðtÞk42ð:Þ ¼ x2ðtÞz1ðtÞk43ð:Þ ¼ x2ðtÞy2ðtÞ

8>>>>>>>><>>>>>>>>:

ð34Þ

Thus, when stabilized by the feedback uiðtÞ; i ¼ 1; . . . ;4, the error will converge to zero as t ! þ1 implying that system (28)will globally synchronize with system (27). This can be achieved by choosing a 4� 4 instantaneous gain matrix Kð:Þ such that

u1ðtÞu2ðtÞu3ðtÞu4ðtÞ

26664

37775 ¼ �

0 a� z1ðtÞw1ðtÞ y2ðtÞw1ðtÞ y2ðtÞz2ðtÞbþ z2ðtÞw2ðtÞ 15 �x1ðtÞw2ðtÞ �x1ðtÞz1ðtÞ

y1ðtÞw1ðtÞ x2ðtÞw1ðtÞ 9 x2ðtÞy2ðtÞy1ðtÞz1ðtÞ x2ðtÞz1ðtÞ x2ðtÞy2ðtÞ 5

26664

37775

x3ðtÞy3ðtÞz3ðtÞw3ðtÞ

26664

37775 ð35Þ

With Eq. (35), the identical chaotic Qi systems (27) and (28) will attain stable synchronization.Fig. 5 shows the error dynamics in the uncontrolled state, while Fig. 6 shows the error dynamics when control is switched

on. Obviously, the two chaotic Qi systems are globally asymptotically synchronized by means of the proposed state feedbackcontroller.

0 5 10 15 20 25

-505

10

x3

0 5 10 15 20 25-505

y3

0 5 10 15 20 25-2

0

2

z3

0 5 10 15 20 25-5

0

5

w3

0 5 10 15 20 2505

10

e

time (s)

Fig. 5. Error dynamics of the coupled Qi system when control is deactivated.

0 5 10 15 20 25-0.3

-0.2

-0.1

0

0.1

0.2

x3,y

3,z3

,w3

0 5 10 15 20 250

0.1

0.2

0.3

0.4

e

time (s)

x1

x2x4

x3

Fig. 6. Error dynamics of the coupled Qi system when control is switched on.


4. Synchronization between Lorenz Stenflo and Qi systems

In this subsection, we study the problem of synchronization process of two different chaotic systems, namely the LorenzStenflo system and the Qi system (Fig. 7).


We choose the Lorenz Stenflo system (2) as the drive system and the Qi system (28) as the response system. This impliesthat when the drive–response system is synchronized, the Qi system will follow the dynamics of the Lorenz Stenflo system.As in the previous sections, our aim is to design the most adequate structure of controllers uiðtÞ; i ¼ 1; . . . ;4, that will make

0 5 10 15-10

-5

0

5

x1,x

2 x1

x2

0 5 10 15-20

0

20

y1,y

2 y1

y2

0 5 10 150

20

40

z1,z

2 z1

z2

0 5 10 15

0

5

10

w1

,w2

time (s)

w1

w2

Fig. 7. Error dynamics between the Stenflo Lorenz system and the Qi system when controller is deactivated.


the response system achieve synchronism with the drive system. Subtracting Eqs. (2) from Eqs. (28), we find that the errorstates are not explicitly defined as before. Thus, we rearrange and obtain the following error dynamical system:

_x3ðtÞ ¼ �ðaþ aÞx3ðtÞ þ ðaþ aÞy3ðtÞ þ y2ðtÞw2ðtÞz3ðtÞ þ cw3ðtÞþaðy1ðtÞ � x1ðtÞÞ þ aðx2ðtÞ � y2ðtÞÞ þw2ðtÞðy2ðtÞz1ðtÞ � cÞ þ u1ðtÞ

_y3ðtÞ ¼ ðbþ rÞx3ðtÞ þ ðb� 1Þy3ðtÞ � x2ðtÞz2ðtÞw3ðtÞþbðx1ðtÞ þ y1ðtÞÞ � x2ðtÞðr þ z2ðtÞw1ðtÞÞ þ y2ðtÞ þ x1ðtÞz1ðtÞ þ u2ðtÞ

_z3ðtÞ ¼ �ðc þ bÞz3ðtÞ þ x2ðtÞy2ðtÞw3ðtÞ�cz1ðtÞ þ bz2ðtÞ � x1ðtÞy1ðtÞ þ x2ðtÞy2ðtÞw1ðtÞ þ u3ðtÞ

_w3ðtÞ ¼ �x3ðtÞ þ x2ðtÞy2ðtÞz3ðtÞ � ðdþ aÞw3ðtÞ�dw1ðtÞ þ x2ðtÞð1þ y2ðtÞz1ðtÞÞ þ aw2ðtÞ þ u4ðtÞ

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð36Þ

4.2. New control law synchronizing Lorenz Stenflo and Qi systems

Now, a control law must be designed to map asymptotically the trajectories of the master attractor into those of the slaveone.

By putting in prominent position the practical criterion of Borne and Gentina, associated to the Benrejeb arrow form ma-trix, we redefine the active control functions uiðtÞ; i ¼ 1; . . . ;4, as follows:

uiðtÞ ¼ �fið:Þ � kijð:ÞXi 8i; j ¼ 1; . . . ;4 ð37Þ

with:

f1ð:Þ ¼ aðy1ðtÞ � x1ðtÞÞ þ aðx2ðtÞ � y2ðtÞÞ þw2ðtÞðy2ðtÞz1ðtÞ � cÞf2ð:Þ ¼ bðx1ðtÞ þ y1ðtÞÞ � x2ðtÞðr þ z2ðtÞw1ðtÞÞ þ y2ðtÞ þ x1ðtÞz1ðtÞf3ð:Þ ¼ �cz1ðtÞ þ bz2ðtÞ � x1ðtÞy1ðtÞ þ x2ðtÞy2ðtÞw1ðtÞf4ð:Þ ¼ �dw1 þ x2ðtÞð1þ y2ðtÞz1ðtÞÞ þ aw2ðtÞ

8>>><>>>:

ð38Þ

With both Eqs. (37) and (38), the error system (36) is reduced, in this particular case, to a nonlinear system with controlinputs: �kijð:ÞXi; 8 i; j ¼ 1; . . . ;4, as functions of the error states x3; y3; z3 and w3.

Proceeding as before, we make the most appropriate choice for the instantaneous gain matrix Kð:Þ ¼ fkijð:Þg; 8 i; j ¼ 1; . . . ;4,so that the synchronization, between the two non-identical systems (2) and (28), is guaranteed. Among various choices of thegain matrix Kð:Þ, one possible solution is the following:

0 50 100 150-10

0

10

x1

0 50 100 150-10

0

10

x2

0 50 100 150-50

0

50

y1

0 50 100 150-50

0

50

y2

0 50 100 1500

50

z1

0 50 100 1500

50

z2

0 50 100 150-10

0

10

w1

time (s)0 50 100 150

-10

0

10

w2

time (s)

Fig. 8. Synchronization dynamics between the Stenflo Lorenz system and the Qi system when controller is activated.

0 5 10 15 20 25-4

-2

0

2

4

6

x3,y

3,z3

,w3

0 5 10 15 20 250

2

4

6

8

e

time (s)

x1 x2

x3x4

Fig. 9. Error dynamics of the coupled Lorenz Stenflo and Qi chaotic systems when controller is switched on.


Kð:Þ ¼

�ðaþ aÞ þ 1 ðaþ aÞ y2w2 cðbþ rÞ b 0 �x2z2

0 0 �ðc þ bÞ þ 1 x2y2

�1 x2y2 0 �ðdþ aÞ þ 1

26664

37775 ð39Þ

With imposed conditions (38) and (39), it is clear that the Qi system (28) should trace the dynamics of the Stenflo Lorenzsystem (2) as they achieve synchronous states shown in Fig. 8, in which both systems oscillate in a synchronized manner andthe error dynamics presented by Fig. 9 below. Thus, the required synchronization has been provided thanks to our designedcontroller.

5. Conclusion

This paper has examined the synchronization of identical Lorenz Stenflo systems, identical Qi systems, and non-identicalsystems consisting of the Lorenz Stenflo system, as drive, and the Qi system, as response, using the practical stability crite-rion of Borne and Gentina, associated to the particular matrix description, namely the Benrejeb arrow form matrix. The dif-ferent active controllers are designed for the chosen slave system states to be synchronized with the target master systemstates. Numerical simulations were also employed to illustrate the effective performance of the proposed control schemethat is outlined in this paper, in order to synchronize both chaotic identical and non-identical systems.

In a forthcoming work, we will be concerned with the effectiveness and feasibility of the new afforded approach of sta-bilization, which can also be applied to chaotic secure communication, extensively.

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on the synchronization of identical and non-identical 4-d chaotic systems using arrow form matrix

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