Chaos synchronization of double Duffing systems with parameters excited by a chaotic signal

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JOURNAL OF

SOUND AND

VIBRATION

Journal of Sound and Vibration 317 (2008) 449–455

Rapid Communication

Chaos synchronization of double Dufﬁng systems with

parameters excited by a chaotic signal

Zheng-Ming Ge

, Chien-Hao Li, Shih-Yu Li, Ching Ming Chang

Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, ROC Received 8 May 2008; accepted 8 May 2008

Handling Editor: L.G. Tham Available online 20 June 2008

Abstract

Chaos synchronization by driving parameter for two uncoupled identical chaotic double Dufﬁng systems is presented.

Replacing two corresponding parameters of the identical systems by the same function of chaotic state variables of a third

chaotic system, the synchronization or anti-synchronization of two uncoupled systems can be obtained. Numerical

simulations are illustrated for either synchronization or anti-synchronization of which the occurrence depends signiﬁcantly

on initial conditions and on driving strength. Alternative complete synchronization and anti-synchronization is also

discovered.

r

2008 Published by Elsevier Ltd.

1. Introduction

Various synchronization phenomena are being reported for chaotic systems, such as complete

synchronization (CS), anti-synchronization (AS), phase synchronization (PS), lag synchronization, and

generalized synchronization

[1–20,29–38]

. However, most of synchronizations can only realize under the

condition that there exists coupling between two chaotic systems. In practice, such as in physical and electrical

systems, sometimes, it is difﬁcult even impossible to couple two chaotic systems. In comparison with coupled

chaotic systems, synchronization between the uncoupled chaotic systems has many advantages

[20–29]

.

In this paper, synchronization of two double Dufﬁng systems whose corresponding parameter is driven by a

chaotic signal of a third system is analyzed. The chaos synchronizations of two uncoupled double Dufﬁng

systems are obtained by replacing their corresponding parameters by the same function of chaotic state

variables of a third chaotic system. It is noted that whether CS or AS appear depends on the initial conditions.

Besides, CS and AS are also characterized by great sensitivity to initial conditions and on the strengths of the

substituted variable. It is found that CS or AS alternatively occurs under certain conditions

[38–42]

.

This paper is organized as follows. In Section 2, a brief description of synchronization scheme based on the

substitution of the strengths of the mutual coupling term of two identical chaotic double Dufﬁng systems by

www.elsevier.com/locate/jsvi

0022-460X/$ - see front matter r 2008 Published by Elsevier Ltd. doi:10.1016/j.jsv.2008.05.019

Corresponding author. Fax: +886 35720634. E-mail address:zmg@cc.nctu.edu.tw (Z.-M. Ge).

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the variable of a third double Dufﬁng system is presented. In Section 3, numerical simulations are given for

illustration. It is found that one can obtain CS or AS by adjusting the driving strength and initial conditions.

Finally, in Section 4 conclusions are drawn.

2. Synchronization of two double Dufﬁng systems

The famous Dufﬁng system is

€

x þ a _

x þ bx þ cx

3

¼

d cos ot

(1)

where a, b are constant parameters, d cos ot is an external excitation. It can be written as two ﬁrst-order

differential equations:

dx

dt

¼

y

dy

dt

¼ ay bx cx

3

_þ

_{d cos ot}

8 >

>

<

>

:

(2)

Consider the following double Dufﬁng system:

dx

dt

¼

y

dy

dt

¼ ay bx cx

3

_þ

_du

du

dt

¼

v

dv

dt

¼ ev gu hu

3

_þ

_kx

8 >

>

<

>

:

(3)

It consists of two Dufﬁng systems in which two external excitations are replaced by two coupling terms. It is

an autonomous system with four states where a, b, c, d, e, g, h, and k are constant parameters of the systems.

Two identical double Dufﬁng systems to be synchronized are

dx

1

dt

¼

y

1

dy

₁

dt

¼ ay

1

bx

1

cx

3 1

þ

d

1

u

1

du

1

dt

¼

v

1

dv

1

dt

¼ ev

1

gu

1

hu

3 1

þ

kx

1

8 >

>

<

>

:

(4)

dx

2

dt

¼

y

2

dy

2

dt

¼ ay

2

bx

2

cx

3 2

þ

d

2

u

2

du

2

dt

¼

v

2

dv

2

dt

¼ ev

2

gu

2

hu

3 2

þ

kx

2

8 >

>

<

>

:

(5)

(3)

where a, b, c, d, e, g, h, and k are positive scalars, and d

1

¼

d

2

are the control inputs to be designed. The third

system is also a double Dufﬁng system:

dx

3

dt

¼

y

3

dy

₃

dt

¼ ay

3

bx

3

cx

3 3

þ

du

3

du

3

dt

¼

v

3

dv

3

dt

¼ ev

3

gu

3

hu

3 3

þ

kx

3

8 >

>

<

>

:

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In order to obtain the chaos synchronization of systems (4) and (5), the corresponding parameters d

1

¼

d

2

of

two systems are replaced by a chaotic signal px

3

+qy

3

of the third system (6), where p, q are constant driving

strengths. The error state variables are deﬁned:

e

1

¼

x

1

x

2

e

2

¼

y

1

y

2

e

3

¼

u

1

u

2

e

4

¼

v

1

v

2

8 >

>

<

>

:

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Giving suitable values for p, q and initial conditions, the synchronization or anti-synchronization of systems

(4) and (5) can be obtained.

3. Numerical simulations

Matlab method is used to all of our simulations with time step 0.01. The parameters of two systems (4) and

(5) are given as a ¼ 0.5, b ¼ 1, c ¼ 3, d ¼ 2, e ¼ 5, g ¼ 1, h ¼ 2, k ¼ 2 to ensure the chaotic behavior. To

verify CS and AS, it is convenient to introduce the following coordinate transformation: E

1

¼

(x

1

+x

2

) and

e

1

¼

(x

1

x

2

) and the same transformation for y, u, and v variables. Therefore, the new coordinate systems (E

1

,

E

2

, E

3

, E

4

) and (e

1

, e

2

, e

3

, e

4

) represent the sum and difference motions of the original coordinate system,

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respectively. We can easily see that (e

1

, e

2

, e

3

, e

4

) subspace represents the CS case, and (E

1

, E

2

, E

3

, E

4

) subspace

the AS one.

How the synchronization phenomena depend on the initial conditions will be studied. At the beginning, we

choose (x

1

, y

1

, u

1

, v

1

) ¼ (2, 5, 1, 0.3) and (x

2

, y

2

, u

2

, v

2

) ¼ (8, 9, 0, 5) as the initial conditions of systems (4)

and (5). Let the driving strengths be p ¼ 10, q ¼ 8 and p ¼ 10, q ¼ 10.

Figs. 1 and 2

show the time-series of AS

and CS phenomena for different driving strengths, respectively. The simulation results are shown in

Fig. 1

for

case (a) and in

Fig. 2

for case (b). These simulation results indicate that the ﬁnal state develops to CS or AS

depending sensitively on driving strength in spite of the identical initial conditions in both cases. For AS case

(

Figs. 1

(a) and (b)), the sums of the variables converge to zero, while the differences remain chaotic. For CS

case (

Figs. 2

(a) and (b)), on the other hand, e

1

, e

2

, e

3

, and e

4

converge to zero, while E

1

, E

2

, E

3

, and E

4

remain

chaotic.

Fig. 2. CS and AS for initial condition (x2, y2, u2, v2) ¼ (8, 9, 0, 5), and p ¼ 10, q ¼ 10. (a) e1, e2, e3, e4and (b) E1, E2, E3, E4.

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In order to know how this phenomenon depends upon the initial conditions, different initial conditions are

given for ﬁxed driving strength. The results are shown in

Figs. 3 and 4

.

Fig. 3

(b) shows that E

1

, E

2

, E

3

, and E

4

tend to zero. As shown in

Fig. 3

(a), while the e

1

, e

2

, e

3

, and e

4

do not go to zero. Comparing

Fig. 1

with

Fig. 3

,

it is found that they have contrary behavior. The only reason lies in the different initial conditions. Similar

result also exists by comparing

Fig. 2

with

Fig. 4

.

Besides, we also discover the alternative CS and AS. In

Fig. 5

, the system shows alternative switching

between these two states where the initial condition (x

1

, y

1

, u

1

, v

1

) ¼ (2, 5, 1, 0.3), (x

2

, y

2

, u

2

, v

2

) ¼ (8, 9, 0,

5), and p ¼ 12, q ¼ 12.

Fig. 4. CS and AS for initial condition (x2, y2, u2, v2) ¼ (8, 9, 0, 5), and p ¼ 10, q ¼ 13. (a) e1, e2, e3, e4and (b) E1, E2, E3, E4.

Fig. 5. Alternative CS and AS for initial condition (x1, y1, u1, v1) ¼ (2, 5, 1, 0.3), (x2, y2, u2, v2) ¼ (3, 5, 2, 9), and p ¼ 12, q ¼ 12. (a) e1,

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4. Conclusions

In this paper, parameter excited chaos synchronizations of two identical double Dufﬁng systems are studied

by adjusting the strength of the substituting variable. Numerical simulations are illustrated for CS or AS of

which the occurrence depends on initial conditions and driving strength. Besides, alternative CS and AS is also

discovered with the same initial conditions and the same driving strength.

Acknowledgments

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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