Chaos generalized synchronization of new Mathieu-Van der Pol systems with new Duffing-Van der Pol systems as functional system by GYC partial region stability theory

Chaos generalized synchronization of new Mathieu–Van der Pol systems

with new Duffing–Van der Pol systems as functional system

by GYC partial region stability theory

Zheng-Ming Ge

, Shih-Yu Li

Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, People’s Republic of China

a r t i c l e

i n f o

Article history:

Received 30 August 2010

Received in revised form 25 February 2011 Accepted 8 March 2011

Available online 15 March 2011 Keywords:

Generalized synchronization Chaos

Synchronization

Partial region stability theory New Mathieu–van der Pol system New Duffing–van der Pol system

a b s t r a c t

In this paper, a new strategy by using GYC partial region stability theory is proposed to achieve generalized chaos synchronization. via using the GYC partial region stability the-ory, the new Lyapunov function used is a simple linear homogeneous function of states and the lower order controllers are much more simple and introduce less simulation error. Numerical simulations are given for new Mathieu–Van der Pol system and new Duffing– Van der Pol system to show the effectiveness of this strategy.

Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction

In the last few years, synchronization in chaotic dynamical systems has received a great deal of interest among scientists from various fields[1–8]. The phenomenon of synchronization of two chaotic systems is fundamental in science and has a wealth of applications in technology. In recent years, more and more applications of chaos synchronization were proposed. There are many control techniques to synchronize chaotic systems, such as linear error feedback control, adaptive control, active control[9–19].

In this paper, a new chaos generalized synchronization strategy by GYC partial region stability theory is proposed[20,21]. It means that there exists a given functional relationship between the states of the master and that of the slave. via using the GYC partial region stability theory, the new Lyapunov function is a simple linear homogeneous function of states and the lower order controllers are much more simple and introduce less simulation error.

The layout of the rest of the paper is as follows. In Section2, generalized chaos synchronization strategy by GYC partial region stability theory is proposed. In Section3, new Mathieu–Van der pol system and new Duffing–Van der pol system are introduced. In Section4, six simulation examples are given. In Section5, conclusions are given.

0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.03.022

⇑Corresponding author. Address: Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, People’s Republic of China. Tel.: +886 3 5712121 55119; fax: +886 3 5720634.

E-mail address:zmg@cc.nctu.edu.tw(Z.-M. Ge).

1

Tel.: +886 3 5712121 55179.

Contents lists available atScienceDirect

Applied Mathematical Modelling

2. Generalized Chaos Synchronization Strategy by GYC Partial Region Stability Theory 2.1. GYC Partial Region Stability Theory

Consider the differential equations of disturbed motion of a nonautonomous system in the normal form

dxs

dt ¼ Xsðt; x1; . . . ;xnÞ; ðs ¼ 1; . . . ; nÞ; ð1Þ

where the function Xsis defined on the intersection of the partial regionX(shown inFig. 1) and

X

s

x2

s 6H ð2Þ

and t > t0, where t0and H are certain positive constants. Xswhich vanishes when the variables xsare all zero, is a real valued

function of t, x1, . . . ,xn. It is assumed that Xsis smooth enough to ensure the existence, uniqueness of the solution of the initial

value problem. When Xsdoes not contain t explicitly, the system is autonomous.

Obviously, xs= 0 (s = 1, . . . n) is a solution of Eq.(1). We are interested to the asymptotical stability of this zero solution on

partial regionX(including the boundary) of the neighborhood of the origin which in general may consist of several subre-gions (Fig.1).

Definition 1. For any given number

e

X

s

x2

s06d; ðs ¼ 1; . . . ; nÞ ð3Þ

for all t P t0, the inequality

X

s

x2

s <

e

is satisfied for the solutions of Eq.(19)onX, then the disturbed motion xs= 0 (s = 1, . . . n) is stable on the partial regionX.

Definition 2. If the undisturbed motion is stable on the partial regionX, and there exists a d0_{> 0, so that on the given partial}

regionXwhen X s x2 s06d 0 ; ðs ¼ 1; . . . ; nÞ: ð5Þ The equality lim t!1 X s x2 s ! ¼ 0 ð6Þ

is satisfied for the solutions of Eq.(1)onX, then the undisturbed motion xs= 0 (s = 1, . . . n) is asymptotically stable on the

partial regionX. subregion 2 subregion 3 subregion 1 1 2 1 1 1

h

The intersection ofXand region defined by Eq.(2)is called the region of attraction.

Definition of Functions V(t, x1, . . . , xn): Let us consider the functions V(t, x1, . . . , xn) given on the intersectionX1of the

partial regionXand the region

X

s

x2

s 6h; ðs ¼ 1; . . . ; nÞ ð7Þ

for t P t0> 0, where t0and h are positive constants. We suppose that the functions are single-valued and have continuous

partial derivatives and become zero when x1=  = xn= 0.

Definition 3. If there exists t0> 0 and a sufficiently small h > 0, so that on partial regionX1and t P t0, V P 0 (or 60), then V

is a positive (or negative) semidefinite, in general semidefinite, function on theX1and t P t0.

Definition 4. If there exists a positive (negative) definitive function W(x1   xn) onX1, so that on the partial regionX1and

t P t0

V  W P 0ðor  V  W P 0Þ; ð8Þ

then V(t, x1, . . . , xn) is a positive definite function on the partial regionX1and t P t0. Eq.(1)

Definition 5. If V(t, x1, . . . , xn) is neither definite nor semidefinite onX1and t P t0, then V(t, x1, . . . , xn) is an indefinite function

on partial regionX1and t P t0. That is, for any small h > 0 and any large t0> 0, V(t, x1, . . . , xn) can take either positive or

neg-ative value on the partial regionX1and t P t0.

Definition 6. Bounded function VIf there exist t0> 0, h > 0, so that on the partial regionX1, we have

Vðt; x1; . . . ;xnÞ

j j < L;

where L is a positive constant, then V is said to be bounded onX1.

Definition 7. Function with infinitesimal upper bound If V is bounded, and for any k > 0, there exists

l

whenP_sx2

s 6

l

jVðt; x1; . . . ;xnÞj 6 k

then V admits an infinitesimal upper bound onX1.

Theorem 1. If there can be found a definite function V(t, x1, . . . , xn) on the partial region for Eq.(1), and the derivative with respect

to time based on these equations:

dV dt ¼ @V @t þ Xn s¼1 @V @xs Xs ð9Þ

is a semidefinite function on the paritial region whose sense is opposite to that of V, or if it becomes zero identically, then the undisturbed motion is stable on the partial region.

Proof. Let us assume for the sake of definiteness that V is a positive definite function. Consequently, there exists a suffi-ciently large number t0and a sufficiently small number h< H, such that on the intersectionX1of partial regionXand

X

s

x2

s 6h; ðs ¼ 1; . . . ; nÞ

and t P t0, the following inequality is satisfied

Vðt; x1; . . . ;xnÞ P Wðx1; . . . ;xnÞ;

where W is a certain positive definite function which does not depend on t. Besides that, Eq.(9)may assume only negative or zero value in this region.

Let

e

e

x

X

s

x2

s ¼

e

and let us designate by l > 0 the precise lower limit of the function W under this condition. By virtue of Eq.(5), we shall have

We shall now consider the quantities xsas functions of time which satisfy the differential equations of disturbed motion. We

shall assume that the initial values xs0of these functions for t = t0lie on the intersectionX2ofX1and the region

X

s

x2

s 6d; ð12Þ

where d is so small that

Vðt0;x10; . . . ;xn0Þ < l: ð13Þ

By virtue of the fact that V(t0, 0, . . . , 0) = 0, such a selection of the number d is obviously possible. We shall suppose that in any

case the number d is smaller than

e

X

s

x2

s <

e

being satisfied at the initial instant will be satisfied, in the very least, for a sufficiently small t  t0, since the functions xs(t)

very continuously with time. We shall show that these inequalities will be satisfied for all values t > t0. Indeed, if these

inequalities were not satisfied at some time, there would have to exist such an instant t = T for which this inequality would become an equality. In other words, we would have

X

s

x2 sðTÞ ¼

e

and consequently, on the basis of Eq.(11)

VðT; x1ðTÞ; . . . ; xnðTÞÞ P l: ð15Þ

On the other hand, since

e

entire time intervaldV

dt60. This yields

VðT; x1ðTÞ; . . . ; xnðTÞÞ 6 Vðt0;x10; . . . ;xn0Þ;

which contradicts Eq.(14)on the basis of Eq.(13). Thus, the inequality (Eq.(1)) must be satisfied for all values of t > t0, hence

follows that the motion is stable.

Finally, we must point out that from the view-point of mathematics, the stability on partial region in general does not be related logically to the stability on whole region. If an undisturbed solution is stable on a partial region, it may be either stable or unstable on the whole region and vice versa. In specific practical problems, we do not study the solution starting withinX2and running out ofX. h

Theorem 2. If in satisfying the conditions of theorem 1, the derivativedV

dtis a definite function on the partial region with opposite

sign to that of V and the function V itself permits an infinitesimal upper limit, then the undisturbed motion is asymptotically stable on the partial region.

Proof. Let us suppose that V is a positive definite function on the partial region and that consequently,dV

dtis negative definite.

Thus on the intersectionX1ofXand the region defined by Eq.(4)and t P t0there will be satisfied not only the inequality

(Eq.(5)), but the following inequality as will:

dV

dt 6W1ðx1; . . .xnÞ; ð16Þ

where W1is a positive definite function on the partial region independent of t. Let us consider the quantities xsas functions of

time which satisfy the differential equations of disturbed motion assuming that the initial values xs0= xs(t0) of these

quan-tities satisfy the inequalities (Eq.(12)). Since the undisturbed motion is stable in any case, the magnitude d may be selected so small that for all values of t P t0the quantities xsremain withinX1. Then, on the basis of Eq.(16)the derivative of

func-tion V(t, x1(t), . . . , xn(t)) will be negative at all times and, consequently, this function will approach a certain limit, as t

in-creases without limit, remaining larger than this limit at all times. We shall show that this limit is equal to some positive quantity different from zero. Then for all values of t P t0the following inequality will be satisfied:

Vðt; x1ðtÞ; . . . ; xnðtÞÞ >

a

where

a

Since V permits an infinitesimal upper limit, it follows from this inequality that

X

s

x2

sðtÞ P k; ðs ¼ 1; . . . ; nÞ; ð18Þ

where k is a certain sufficiently small positive number. Indeed, if such a number k did not exist, that is, if the quantityP_sxsðtÞ

were smaller than any preassigned number no matter how small, then the magnitude V(t, x1(t), . . . , xn(t)), as follows from the

If for all values of t P t0the inequality (Eq.(18)) is satisfied, then Eq.(16)shows that the following inequality will be

satisfied at all times:

dV dt 6l1;

where l1is positive number different from zero which constitutes the precise lower limit of the function W1(t, x1(t), . . . , xn(t))

under condition (Eq.(18)). Consequently, for all values of t P t0we shall have:

Vðt; x1ðtÞ; . . . ; xnðtÞÞ ¼ Vðt0;x10; . . . ;xn0Þ þ

Z t t0

dV

dtdt 6 Vðt0;x10; . . . ;xn0Þ  l1ðt  t0Þ;

which is, obviously, in contradiction with Eq.(17). The contradiction thus obtained shows that the function V(t, x1(t), . . . , xn(t))

approached zero as t increase without limit. Consequently, the same will be true for the function W(x1(t), . . . ,xn(t)) as well,

from which it follows directly that

lim

t!1xsðtÞ ¼ 0; ðs ¼ 1; . . . ; nÞ;

which proves the theorem. h

2.2. Generalized Chaos Synchronization Strategy

Consider the following unidirectional coupled chaotic systems

_x ¼ fðt; xÞ

_y ¼ hðt; yÞ þ u; ð19Þ

Where x = [x1, x2, . . . , xn]T2 Rn, y = [y1, y2, . . . , yn]T2 Rndenote the master state vector and slave state vector respectively, f and

h are nonlinear vector functions, and u = [u1, u2, . . . , un]T2 Rnis a control input vector.

The generalized synchronization can be accomplished when t ? 1, the limit of the error vector e = [e1, e2, . . . ,en]T

ap-proaches zero: lim t!1e ¼ 0; ð20Þ where e ¼ GðxÞ  y: ð21Þ G(x) is a given function of x.

By using the partial region stability theory, the Lyapunov function is easier to find, since the linear terms of the entries of e can be used to construct the definite Lyapunov function and the controllers can be designed in lower order.

3. New Chaotic Mathieu–Van der Pol System and New Chaotic Duffing–Van der Pol System

This section introduces new Mathieu–van der Pol system and new Duffing–van der Pol system, respectively.

3.1. New Mathieug–van der Pol system

Mathieu equation and van der Pol equation are two typical nonlinear non-autonomous systems:

_x1¼ x2

_x2¼ ða þ b sin

x

x

x

 ð22Þ _x3¼ x4 _x4¼ ex3þ f ð1  x23Þx4þ g sin

x

Exchanging sin

x

x

system: _x1¼ x2 _x2¼ ða þ bx3Þx1 ða þ bx3Þx31 cx2þ dx3 _x3¼ x4 _x4¼ ex3þ f ð1  x23Þx4þ gx1; 8 > > > < > > > : ð24Þ

where a, b, c, d, e, f, g are uncertain parameter. This system exhibits chaos when the parameters of system are a = 10, b = 3, c = 0.4, d = 70, e = 1, f = 5, g = 0.1 and the initial states of system are (x10, x20, x30, x40)=(0.1, -0.5, 0.1, -0.5), its phase portraits

3.2. New Duffing–van der Pol system

Duffing equation and van der Pol equation are two typical nonlinear non-autonomous systems:

_z1¼ z2

_z2¼ z1 z31 hz2þ i sin

x



ð25Þ Fig. 2. Phase portraits of new chaotic Mathieu–Van der Pol System.

_z3¼ z4 _z4¼ jz3þ k 1  z23   z4þ l sin

x

Exchanging sin

x

x

Pol system: _z1¼ z2 _z2¼ z1 z31 hz2þ iz3 _z3¼ z4 _z4¼ jz3þ k 1  z23   z4þ lz1; 8 > > > < > > > : ð27Þ

where h, i, j, k, l are uncertain parameter. This system exhibits chaos when the parameters of system are h = 0.0006, j = 1, k = 5, i = 0.67 and l = 0.05 and initial states is (2, 2.4, 5, 6), its phase portraits and Lyapunov exponents as shown inFigs. 4 and 5.

Fig. 4. Phase portraits of new chaotic Duffing–Van der Pol System.

4. Numerical simulations

The two unidirectional coupled new chaotic Mathieu–Van der pol systems are shown as follows:

_x1¼ x2 _x2¼ ða þ bx3Þx1 ða þ bx3Þx31 cx2þ dx3 _x3¼ x4 _x4¼ ex3þ f 1  x23   x4þ gx1; ð28Þ _y1¼ y2þ u1

_y2¼ ða þ by3Þy1 ða þ by3Þy31 cy2þ dy3þ u2

_y3¼ y4þ u3 _y4¼ ey3þ f 1  y 2 3   y4þ gy1þ u4:

CASE I. The generalized synchronization error function is ei= (xi yi+ 100), (i = 1, 2, 3, 4.).

The addition of 100 makes the error dynamics always happens in first quadrant. Our goal isyi= xi+ 100, i.e.

lim

t!1ei¼ limt!1ðxi yiþ 100Þ ¼ 0 ði ¼ 1; 2; 3; 4Þ: ð29Þ

The error dynamics becomes:

_e1¼ _x1 _y1¼ x2 y2 u1;

_e2¼ _x2 _y2¼ ðða þ bx3Þx1 ða þ by3Þy1Þ  ða þ bx3Þx31 ða þ by3Þy31

   cðx2 y2Þ þ dðx3 y3Þ  u2; _e3¼ _x3 _y3¼ x4 y4 u3; _e4¼ _x4 _y4¼ eðx3 y3Þ þ f 1  x23   x4 1  y23   y4   þ gðx1 y1Þ  u4: ð30Þ

System parameters are chosen as a = 10, b = 3, c = 0.4, d = 70,e = 1, f = 5, g = 0.1 and initial states are (x10, x20

, x30, x40)=(0.1,  0.5, 0.1,  0.5), (y10, y20, y30, y40)=(0.3,  0.1, 0.3,  0.1). Before control action, the error dynamics always

hap-pens in first quadrant as shown inFig. 6. By GYC partial region stability, one can choose a Lyapunov function in the form of a positive definite function in first quadrant:

V ¼ e1þ e2þ e3þ e4: ð31Þ

Its time derivative through Eq.(29)is

_

V ¼ _e1þ _e2þ _e3þ _e4¼ ðx2 y2 u1Þ þ ððða þ bx3Þx1 ða þ by3Þy1Þ  ðða þ bx3Þx31 ða þ by3Þy31Þ  cðx2 y2Þ

þdðx3 y3Þ  u2Þ þ ðx4 y4 u3Þ þ ðeðx3 y3Þ þ f ðð1  x23Þx4 ð1  y23Þy4Þ þ gðx1 y1Þ  u4Þ:

ð32Þ

Choose

u1¼ ðx2 y2Þ þ e1;

u2¼ ððða þ bx3Þx1 ða þ by3Þy1Þ  ðða þ bx3Þx31 ða þ by3Þy31Þ  cðx2 y2Þ þ dðx3 y3ÞÞ þ e2;

u3¼ ðx4 y4Þ þ e3; u4¼ ðeðx3 y3Þ þ f ðð1  x23Þx4 ð1  y23Þy4Þ þ gðx1 y1ÞÞ þ e4: ð33Þ We obtain _ V ¼ e1 e2 e3 e4<0; ð34Þ

which is negative definite function in the first quadrant. Four state errors versus time and time histories of states are shown inFigs. 7 and 8.

CASE II. The generalized synchronization error function is ei= (xi yi+ Fisin

x

The addition of 100 makes the error dynamics always happens in first quadrant. Our goal is yi= xi+ Fisin

x

lim

t!1ei¼ limt!1ðxi yiþ Fisin

x

Where F1= F2= F3= F4= F = 10,

x

The error dynamics becomes

_e1¼ x2 y2 u1þ F

x

x

_e2¼ ðða þ bx3Þx1 ða þ by3Þy1Þ  ðða þ bx3Þx31 ða þ by3Þy31Þ  cðx2 y2Þ þ dðx3 y3Þ  u2þ F

x

x

_e3¼ x4 y4 u3þ F

x

x

_e4¼ eðx3 y3Þ þ f ðð1  x23Þx4 ð1  y32Þy4Þ þ gðx1 y1Þ  u4þ F

x

x

ð36Þ

System parameters are chosen as a = 10, b = 3, c = 0.4, d = 70,e = 1, f = 5, g = 0.1 and initial states are (x10, x20, x30, x40)=

(0.1,  0.5, 0.1,  0.5), (y10, y20, y30, y40)=(0.3,  0.1, 0.3,  0.1). Before control action, the error dynamics always happens in

first quadrant as shown inFig. 9. By GYC partial region stability, one can choose a Lyapunov function in the form of a positive definite function in first quadrant:

Fig. 8. Time histories of x1, x2, x3, y1, y2, y3for Case I.

V ¼ e1þ e2þ e3þ e4: ð37Þ

Its time derivative through Eq.(35)is

_

V ¼ _e1þ _e2þ _e3þ _e4

¼ ðx2 y2 u1þ F

x

x

1Þ  cðx2 y2Þ

þ dðx3 y3Þ  u2þ F

x

x

x

x

þ gðx1 y1Þ  u4þ F

x

x

Choose

u1¼ ðx2 y2Þ þ F

x

x

u2¼ ððða þ bx3Þx1 ða þ by3Þy1Þ  ðða þ bx3Þx31 ða þ by3Þy31Þ  cðx2 y2Þ þ dðx3 y3ÞÞ þ F

x

x

u3¼ ðx4 y4Þ þ F

x

x

u4¼ ðeðx3 y3Þ þ f ðð1  x23Þx4 ð1  y32Þy4Þ þ gðx1 y1ÞÞ þ F

x

x

ð39Þ

We obtain

_

V ¼ e1 e2 e3 e4<0 ð40Þ

which is a negative definite function in the first quadrant. Three state errors versus time and time histories of xi yi+ 100

and  Fisin wt are shown inFigs. 10 and 11.

CASE III. The generalized synchronization error function is ei= xi yi+ Fiesinxt+ 100, (i=1 2, 3, 4).

The addition of 100 makes the error dynamics always happens in first quadrant. Our goal is yi= xi+ Fiesinxt+ 100, i.e.

lim

t!1ei¼ limt!1ðxi yiþ Fie

sinxt_{þ 100Þ ¼ 0 ði ¼ 1; 2; 3; 4Þ: ð41Þ}

The error dynamics becomes

_e1¼ x2 y2 u1þ F

x

x

_e2¼ ðða þ bx3Þx1 ða þ by3Þy1Þ  ðða þ bx3Þx31 ða þ by3Þy13Þ  cðx2 y2Þ þ dðx3 y3Þ  u2þ F

x

x

_e3¼ x4 y4 u3þ F

x

x

_e4¼ eðx3 y3Þ þ f ðð1  x23Þx4 ð1  y23Þy4Þ þ gðx1 y1Þ  u4þ F

x

x

ð42Þ

System parameters are chosen as a = 10, b = 3, c = 0.4, d = 70, e = 1, f = 5, g = 0.1, F1= F2= F3= F4= F = 10,

x

states are (x10, x20, x30, x40)=(0.1,  0.5, 0.1,  0.5), (y10, y20, y30, y40)=(0.3,  0.1, 0.3,  0.1). Before control action, the error

dynamics always happens in first quadrant as shown inFig. 12. By GYC partial region stability, one can choose a Lyapunov function in the form of a positive definite function in first quadrant:

V ¼ e1þ e2þ e3þ e4: ð43Þ

Its time derivative through Eq.(41)is

_

V ¼ _e1þ _e2þ _e3þ _e4¼ ðx2 y2 u1þ F

x

x

1 ða þ by3Þy31Þ

 cðx2 y2Þ þ dðx3 y3Þ  u2þ F

x

x

x

x

þ ðeðx3 y3Þ þ f ðð1  x23Þx4 ð1  y23Þy4Þ þ gðx1 y1Þ  u4þ F

x

x

Choose

u1¼ ðx2 y2Þ þ F

x

x

u2¼ ððða þ bx3Þx1 ða þ by3Þy1Þ  ðða þ bx3Þx31 ða þ by3Þy13Þ  cðx2 y2Þ þ dðx3 y3ÞÞ þ F

x

x

u3¼ ðx4 y4Þ þ F

x

x

u4¼ ðeðx3 y3Þ þ f ðð1  x23Þx4 ð1  y23Þy4Þ þ gðx1 y1ÞÞ þ F

x

x

ð45Þ

We obtain

_

V ¼ e1 e2 e3 e4<0 ð46Þ

which is a negative definite function in the first quadrant. Three state errors versus time and time histories of xi yi+ 100

and  Fiesinwtare shown inFigs. 13 and 14.

CASE IV. The generalized synchronization error function is ei¼1₂x2i  yiþ 100, (i=1, 2, 3, 4). The addition of 100 makes the

error dynamics always happens in first quadrant.

Fig. 12. Phase portraits of error dynamics for Case III.

Our goal is y_i¼1 2x 2 i þ 100, i.e. lim t!1ei¼ limt!1 1 2x 2 i  yiþ 100   ði ¼ 1; 2; 3; 4Þ ð47Þ

The error dynamics becomes

_e1¼ x1_x1 _y1¼ x1x2 y2 u1;

_e2¼ x2_x2 _y2¼ ðða þ bx3Þx2x1 ða þ by3Þy1Þ  ðða þ bx3Þx2x31 ða þ by3Þy31Þ  cðx22 y2Þ þ dðx2x3 y3Þ  u2;

_e3¼ x3_x3 _y3¼ x3x4 y4 u3;

_e4¼ x4_x4 _y4¼ eðx4x3 y3Þ þ f ðð1  x23Þx24 ð1  y23Þy4Þ þ gðx4x1 y1Þ  u4:

ð48Þ

System parameters are chosen as a = 10, b = 3, c = 0.4, d = 70,e = 1, f = 5, g = 0.1 and initial states are (x10, x20, x30, x40)=

(0.1,  0.5, 0.1,  0.5), (y10, y20, y30, y40)=(0.3,  0.1, 0.3,  0.1). Before control action, the error dynamics always happens in

first quadrant as shown inFig. 15. By GYC partial region stability, one can choose a Lyapunov function in the form of a po-sitive definite function in first quadrant:

V ¼ e1þ e2þ e3þ e4: ð49Þ

Its time derivative through Eq.(47)is

_

V ¼ _e1þ _e2þ _e3þ _e4¼ ðx1x2 y2 u1Þ þ ððða þ bx3Þx2x1 ða þ by3Þy1Þ  ðða þ bx3Þx2x31 ða þ by3Þy31Þ  cðx22 y2Þ

þ dðx2x3 y3Þ  u2Þ þ ðx3x4 y4 u3Þ þ ðeðx4x3 y3Þ þ f ðð1  x23Þx 2 4 ð1  y 2 3Þy4Þ þ gðx4x1 y1Þ  u4Þ: ð50Þ Choose u1¼ x1x2 y2þ e1;

u2¼ ðða þ bx3Þx2x1 ða þ by3Þy1Þ  ðða þ bx3Þx2x31 ða þ by3Þy31Þ  cðx22 y2Þ þ dðx2x3 y3Þ þ e2;

u3¼ x3x4 y4þ e3; u4¼ eðx4x3 y3Þ þ f ðð1  x23Þx24 ð1  y23Þy4Þ þ gðx4x1 y1Þ þ e4: ð51Þ We obtain _ V ¼ e1 e2 e3 e4<0; ð52Þ

which is a negative definite function in the first quadrant. Three state errors versus time is shown inFig. 16.

CASE V. The generalized synchronization error function is ei¼1₃x3i  yiþ 10000 (i=1, 2, 3, 4).

The addition of 10000 makes the error dynamics always happens in first quadrant. Our goal is yi¼13x3i þ 10000, i.e.

Fig. 15. Phase portraits of error dymanics for Case IV.

lim t!1ei¼ limt!1 1 3x 3 i  yiþ 10000   ði ¼ 1; 2; 3; 4Þ: ð53Þ

The error dynamics becomes

_e1¼ x21_x1 _y1¼ x21x2 y2 u1;

_e2¼ x22_x2 _y2¼ ðða þ bx3Þx22x1 ða þ by3Þy1Þ  ðða þ bx3Þx22x 3 1 ða þ by3Þy31Þ  cðx 3 2 y2Þ þ dðx22x3 y3Þ  u2; _e3¼ x23_x3 _y3¼ x23x4 y4 u3; _e4¼ x24_x4 _y4¼ eðx24x3 y3Þ þ f ðð1  x32Þx34 ð1  y23Þy4Þ þ gðx24x1 y1Þ  u4: ð54Þ

System parameters are chosen as a = 10, b = 3, c = 0.4, d = 70,e = 1, f = 5, g = 0.1 and initial states are (x10, x20, x30 , x40)=

(0.1,  0.5, 0.1,  0.5), (y10, y20, y30, y40)=(0.3,  0.1, 0.3,  0.1). Before control action, the error dynamics always happens in

first quadrant as shown inFig. 17. By GYC partial region stability, one can choose a Lyapunov function in the form of a po-sitive definite function in first quadrant:

V ¼ e1þ e2þ e3þ e4: ð55Þ

Its time derivative through Eq.(53)is

_

V ¼ _e1þ _e2þ _e3þ _e4¼ ðx21x2 y2 u1Þ þ ððða þ bx3Þx22x1 ða þ by3Þy1Þ  ðða þ bx3Þx22x13 ða þ by3Þy31Þ  cðx32 y2Þ

þ dðx2 2x3 y3Þ  u2Þ þ ðx23x4 y4 u3Þ þ ðeðx24x3 y3Þ þ f ðð1  x23Þx 3 4 ð1  y 2 3Þy4Þ þ gðx24x1 y1Þ  u4ÞÞ: ð56Þ Choose u1¼ x21x2 y2þ e1;

u2¼ ðða þ bx3Þx22x1 ða þ by3Þy1Þ  ðða þ bx3Þx22x13 ða þ by3Þy31Þ  cðx32 y2Þ þ dðx22x3 y3Þ þ e2;

u3¼ x23x4 y4þ e3; u4¼ eðx24x3 y3Þ þ f ðð1  x23Þx34 ð1  y23Þy4Þ þ gðx24x1 y1Þ þ e4: ð57Þ We obtain _ V ¼ e1 e2 e3 e4<0: ð58Þ

which is a negative definite function in the first quadrant. Three state errors versus time is shown inFig. 18.

CASE VI. The generalized synchronization error function is ei= xi yi+ zi+ 100,zi(i=1, 2, 3, 4) is the states of new chaotic

Duffing–Van der pol system.

The functional system for synchronization is new Duffing-Van der pol system and initial states is (2, 2.4, 5, 6), system parameters h = 0.0006, j = 1, k = 5, i = 0.67 and l = 0.05. _z1¼ z2; _z2¼ z1 z31 hz2þ iz3; _z3¼ z4; _z4¼ jz3þ kð1  z23Þz4þ lz1: ð59Þ We have lim t!1ei¼ limt!1ðxi yiþ ziþ 100Þ ¼ 0ði ¼ 1; 2; 3; 4Þ ð60Þ

The error dynamics becomes

_e1¼ _x1þ _z1 _y1¼ x2þ z2 y2 u1;

_e2¼ _x2þ _z2 _y2¼ ðða þ bx3Þx1 ða þ by3Þy1Þ  ðða þ bx3Þx31 ða þ by3Þy31Þ  cðx2 y2Þ

þdðx3 y3Þ þ ðz1 z31 hz2þ iz3Þ  u2;

_e3¼ _x3þ _z3 _y3¼ x4þ z4 y4 u3;

_e4¼ _x4þ _z4 _y4¼ eðx3 y3Þ þ f ðð1  x23Þx4 ð1  y32Þy4Þ þ gðx1 y1Þ  u4þ ðjz3þ kð1  z23Þz4þ lz1Þ:

ð61Þ

System parameters are chosen as a = 10, b = 3, c = 0.4, d = 70,e = 1, f = 5, g = 0.1 and initial states are (x10, x20, x30,

x40)=(0.1,  0.5, 0.1,  0.5), (y10, y20, y30, y40)=(0.3,  0.1, 0.3,  0.1). Before control action, the error dynamics always happens

in first quadrant as shown inFig. 19. By GYC partial region stability, one can choose a Lyapunov function in the form of a positive definite function in first quadrant:

V ¼ e1þ e2þ e3þ e4: ð62Þ

Its time derivative through Eq.(60)is

_

V ¼ _e1þ _e2þ _e3þ _e4¼ ðx2þ z2 y2 u1Þ þ ððða þ bx3Þx1 ða þ by3Þy1Þ  ðða þ bx3Þx31

 ða þ by3Þy31Þ  cðx2 y2Þ þ dðx3 y3Þ þ ðz1 z31 hz2þ iz3Þ  u2Þ þ ðx4þ z4 y4 u3Þ

þ ðeðx3 y3Þ þ f ðð1  x23Þx4 ð1  y23Þy4Þ þ gðx1 y1Þ  u4þ ðjz3þ kð1  z23Þz4þ lz1ÞÞ: ð63Þ

Fig. 19. Phase portraits of error dymanics for Case VI.

Choose

u1¼ x2þ z2 y2þ e1;

u2¼ ðða þ bx3Þx1 ða þ by3Þy1Þ  ðða þ bx3Þx31 ða þ by3Þy31Þ  cðx2 y2Þ þ dðx3 y3Þ þ ðz1 z31 hz2þ iz3Þ þ e2;

u3¼ x4þ z4 y4þ e3; u4¼ eðx3 y3Þ þ f ðð1  x23Þx4 ð1  y23Þy4Þ þ gðx1 y1Þ þ e4þ ðjz3þ kð1  z23Þz4þ lz1Þ: ð64Þ We obtain _ V ¼ e1 e2 e3 e4<0; ð65Þ

which is a negative definite function in the first quadrant. Four state errors versus time and time histories of xi yi+ 100 and

 ziare shown inFigs. 20 and 21.

5. Conclusions

In this paper, a new strategy by using GYC partial region stability theory is proposed to achieve generalized chaos syn-chronization. via using the GYC partial region stability theory, the new Lyapunov function used is a simple linear homoge-neous function of states and the lower order controllers are much simpler and introduce less simulation error. The new chaotic Mathieu–Van der pol system and new chaotic Duffing–Van der pol system are used as simulation examples which confirm the scheme effectively.

Acknowledgment

This research was supported by the National Science Council, Republic of China, under Grant No. NSC 96-2221-E-009-145-MY3.

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