Chaos excited chaos synchronizations of integral and fractional order generalized van der Pol systems

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Chaos excited chaos synchronizations of integral

and fractional order generalized van der Pol systems

Zheng-Ming Ge

*

_{, Mao-Yuan Hsu}

Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsuej Road, Hsinchu 30050, Taiwan, ROC

Accepted 30 June 2006

Communicated by Prof. Ji-Huan He

Abstract

In this paper, chaos excited chaos synchronizations of generalized van der Pol systems with integral and fractional order are studied. Synchronizations of two identiﬁed autonomous generalized van der Pol chaotic systems are obtained by replacing their corresponding exciting terms by the same function of chaotic states of a third nonautonomous or autono-mous generalized van der Pol system. Numerical simulations, such as phase portraits, Poincare´ maps and state error plots are given. It is found that chaos excited chaos synchronizations exist for the fractional order systems with the total frac-tional order both less than and more than the number of the states of the integer order generalized van der Pol system. 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Fractional calculus is an old mathematical topic from 17th century. Although it has a long history, applications are only recently focus of interest. Many systems are known to display fractional order dynamics, such as viscoelastic sys-tems[1], dielectric polarization, electrode–electrolyte polarization, and electromagnetic waves. Furthermore, some sys-tems had been found with chaotic motions in the fractional orders. There is a new topic to investigate the control and dynamics of fractional order dynamical systems recently. The behavior of nonlinear chaotic systems when their models become fractional was also investigated widely and reported[2–6].

Sensitive dependence on initial conditions is an important exhibit characteristic of chaotic systems. For this reason, chaotic systems are diﬃcult to be synchronized or controlled. Research in the area of the synchronization of dynamical systems has been widely explored in a variety of ﬁelds including physical, chemical and ecological systems, secure com-munications and so on. There are many control methods to synchronize chaotic systems such as adaptive control, sliding mode control, observer-based design methods, impulsive control and other control methods. There are more advantages in synchronization of uncoupled chaotic systems than that of coupled chaotic systems. And a new uncoupled method of synchronization is presented in this paper. Chaos excited chaos synchronizations of generalized van der Pol systems with

* _{Corresponding author. Tel.: +886 3 5712121; fax: +886 3 5720634.}

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

Chaos, Solitons and Fractals 36 (2008) 592–604

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integral and fractional order are studied. Synchronizations of two identiﬁed autonomous generalized van der Pol chaotic systems are obtained by replacing their corresponding exciting terms by the same function of chaotic states of a third non-autonomous or non-autonomous system. Numerical simulations, such as phase portraits, Poincare´ maps and state error plots are given. It is found that chaos excited chaos synchronizations exist for the fractional order systems with the total frac-tional order both less than and more than the number of the states of the integer order generalized van der Pol system. This paper is organized as follows. In Section2, the review and the approximation of the fractional order operator is presented. In Section3, the schemes of chaos excited chaos synchronization of two generalized van der Pol systems by replacing their corresponding terms by the same function of chaotic states of a third nonautonomous or autonomous generalized van der Pol system are given. In Section4, numerical simulations, such as phase portraits, Poincare´ maps error states plots, of synchronizations of various fractional order generalized van der Pol systems are presented. In Sec-tion5, conclusions are drawn.

2. The review and the approximation of fractional-order operators

There are many ways to define a fractional differential operator[7–9]. The commonly used definition for general fractional derivative is the Riemann–Liouville definition. The Riemann–Liouville definition of the fractional-order derivative is Da_{yðtÞ ¼}d n dtnD an_{yðtÞ ¼} 1 Cðn aÞ dn dtn Z t 0 yðsÞ ðt sÞanþ1ds ð1Þ

where C(Æ) is a gamma function and n is an integer such that n 1 6 a < n. This definition is different from the usual intuitive definition of derivative.

Thus, it is necessary to develop approximations to the fractional operators using the standard integer order opera-tors. Fortunately, the Laplace transform which is basic engineering tool for analyzing linear systems is still applicable and works: L d a fðtÞ dta ¼ sa_{Lff ðtÞg}X n1 k¼0 sk d a1k_f_ðtÞ dta1k t¼0 ; for all a ð2Þ

where n is an integer such that n 1 6 a < n. Upon considering the initial conditions to be zero, this formula reduces to the more expected form

L d a fðtÞ dta ¼ sa_{Lff ðtÞg} _ð3Þ

Using the algorithm in[3,10], linear transfer function of approximations of the fractional integrator is adopted. Basi-cally the idea is to approximate the system behavior based on frequency domain arguments. From[11], we get the table of approximating transfer functions for 1/sawith diﬀerent fractional orders, a = 0.1–0.9, in steps of 0.1, which give the maximum error 2 dB in calculations. These approximations will be used in the following study.

3. Schemes of chaos excited chaos synchronizations of integral and fractional order generalized van der Pol systems The generalized van der Pol system[12–18]is a nonautonomous system:

dx1 dt ¼ x2 dx2 dt ¼ x1 e 1 x 2 1 c ax2 1 x2þ b sin xt ð4Þ

where e, a, b, c are parameters, and x is the circular frequency of the external excitation b sin xt. The corresponding nonautonomous fractional order system is

dax1 dta ¼ x2 dbx2 dtb ¼ x1 e 1 x 2 1 c ax2 1 x2þ b sin xt ð5Þ

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A modiﬁed version of Eq.(5)is now proposed. The nonautonomous generalized fractional order van der Pol system

(5)with two states is transformed into an autonomous generalized fractional order van der Pol system with three states: da x1 dta ¼ x2 dbx2 dtb ¼ x1 e 1 x 2 1 c ax2 1 x2þ b sin xx3 dcx3 dtc ¼ 1 ð6Þ

where a, b, c are fractional numbers, in which the original time t in Eq.(5)is changed to a new state x3. When c = 1,

x3= t, Eq.(6)reduces to Eq.(5).

Two methods of chaos excited chaos synchronization[19–62]are proposed. In Case 1, two identical generalized frac-tional order van der Pol systems to be synchronized are

dax1 dta ¼ x2 dbx2 dtb ¼ x1 e 1 x 2 1 c ax2 1 x2þ Z ð7Þ and day1 dta ¼ y2 dby₂ dtb ¼ y1 e 1 y 2 1 c ay2 1 y₂þ Z ð8Þ

where the exciting term b sin xt in Eq.(5)is replaced in Eqs.(7) and (8)by Z which is a function of the chaotic states of a third nonautonomous chaotic system

da z1 dta ¼ z2 dbz2 dtb ¼ z1 e 1 z 2 1 c az2 1 z2þ b sinðxtÞ ð9Þ

Z is chosen as Z = bgz1, Z = bgz2, Z = gz1sin(xt), Z = gz2sin(xt), respectively, where g is a constant with diﬀerent

val-ues. Error states are deﬁned: e1= x1 y1, e2= x2 y2.

In Case 2, two identical generalized fractional order van der Pol systems to be synchronized remain unchanged, as Eqs.(7) and (8). But Z is a function of the chaotic states of a third autonomous system

daz1 dta ¼ y2 db z2 dtb ¼ z1 e 1 z 2 1 c az2 1 z2þ b sinðxzz3Þ dc z3 dtc ¼ 1 ð10Þ

Z is chosen as Z = bgz1, Z = bgz2, Z = bg exp(z1), Z = bg exp(z2), Z = g exp(z1) sin(xt), Z = g exp(z2) sin(xt),

respec-tively. g is a constant with diﬀerent values.

4. Numerical simulations for the synchronization of fractional order generalized van der Pol systems

The systems to be synchronized are systems(7) and (8)in following two cases. The parameters a = 3, b = 1.0091, c = 1.2 and d = 0.07 of system Eqs.(7)–(10)are ﬁxed. Our study of two cases consists of 10 parts:

Case 1: The third system is a nonautonomous system with two states, Eq.(9).

Part (1):

Z = bgz1 where z1 is the chaotic state of system (9), g is an adjustable constant. When (1) a = b = 1.1, x = 0.445,

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a= b = 0.7, x = 0.31812, g = 0.3, chaos synchronization is obtained. For saving space only the phase portraits and Poincare´ maps of the fractional order synchronized systems and time history of error states plot for condition (5)

are shown inFig. 1. When a, b take the fractional number less than 0.7, no chaotic synchronization is found. Part (2):

Z = bgz2 where z2 is the chaotic state of system (9). When (1) a = b = 1.1, x = 0.445, g = 0.8; (2) a = b = 1,

x= 0.12961875, g = 3; (3) a = b = 0.9, x = 0.132, g = 9; (4) a = b = 0.8, x = 0.1315, g = 13; (5) a = b = 0.7, x= 0.31812, g = 14.65, chaos synchronization is obtained. For saving space only the phase portraits and Poincare´ maps of the fractional order synchronized systems and time history of error states plot for condition(5)are shown inFig. 2. When a, b take the fractional number less than 0.7, no chaotic synchronization is found.

Part (3):

Z = gz1sin(xt) where z1 is the chaotic state of system(9). When (1) a = b = 1.1, x = 0.445, g = 1.5; (2) a = b = 1,

x= 0.12961875, g = 3; (3) a = b = 0.9, x = 0.132, g = 0.5; (4) a = b = 0.8, x = 0.1315, g = 5; (5) a = b = 0.7, x= 0.31812, g = 10, chaos synchronization is obtained. For saving space only the phase portraits and Poincare´ maps of the fractional order synchronized systems and time history of error states plot for condition(5)are shown inFig. 3. When a, b take the fractional number less than 0.7, no chaotic synchronization is found.

Part (4):

Z = gz2sin(xt) where z2 is the chaotic state of system (9). When (1) a = b = 1.1, x = 0.445, g = 1; (2) a = b = 1,

x= 0.12961875, g = 3; (3) a = b = 0.9, x = 0.132, g = 1; (4) a = b = 0.8, x = 0.1315, g = 0.5; (5) a = b = 0.7, x= 0.31812, g = 0.5, chaos synchronization is obtained. For saving space only the phase portraits and Poincare´ maps of the fractional order synchronized systems and time history of error states plot for condition(5)are shown inFig. 4. When a, b take the fractional number less than 0.7, no chaotic synchronization is found.

Case 2: The third system is an autonomous system with three states, Eq.(10).

-2 -1 0 1 2 -5 0 5 x1 x2 -2 -1 0 1 2 -5 0 5 y1 y2 -2 -1 0 1 2 -5 0 5 z1 z2 0 500 1000 1500 2000 2500 -1 -0.5 0 0.5 1 t e1

Fig. 1. Phase portraits and Poincare´ maps of the synchronized fractional order systems and time history of states error for Z = bgz1

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-4 -2 0 2 4 -20 -10 0 10 20 x1 x2 -4 -2 0 2 4 -20 -10 0 10 20 y1 y2 -2 -1 0 1 2 -5 0 5 z1 z2 0 500 1000 1500 2000 2500 -1 -0.5 0 0.5 1 t e1

with order a = b = 0.7, x = 0.31812, g = 14.65. -2 0 2 4 -15 -10 -5 0 5 10 x1 x2 -2 0 2 4 -15 -10 -5 0 5 10 y1 y2 -2 -1 0 1 2 -5 0 5 z1 z2 0 500 1000 1500 2000 2500 -6 -4 -2 0 2x 10 -6 t e1

Fig. 3. Phase portraits and Poincare´ maps of the synchronized fractional order systems and time history of states error for Z = gz1sin(xt) with order a = b = 0.7, x = 0.31812, g = 10.

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-2 -1 0 1 2 -6 -4 -2 0 2 4 6 x1 x2 -2 -1 0 1 2 -6 -4 -2 0 2 4 6 y1 y2 -2 -1 0 1 2 -5 0 5 z1 z2 0 500 1000 1500 -6 -4 -2 0 2 4 6x 10 -5 t e1

Fig. 4. Phase portraits and Poincare´ maps of the synchronized fractional order systems and time history of states error for Z = gz2sin(xt) with order a = b = 0.7, x = 0.31812, g = 0.5.

Part (1):

Z = bgz1 where z1 is the chaotic state of system(10). When (1) a = b = c = 1.1, x = 0.34, g = 1.5; (2) a = b = 0.9,

c= 1.1, x = 0.62, g = 0.5; (3) a = b = 0.8, c = 1.1, x = 0.2807, g = 1; (4) a = b = 0.7, c = 1.1, x = 0.144, g = 3; (5) a= b = 0.6, c = 1.1, x = 0.0107, g = 6.6; (6) a = b = 0.5, c = 1.1, x = 0.001, g = 0.5; (7) a = b = 0.4, c = 1.1, x= 0.005, g = 0.5; (8) a = b = 0.3, c = 1.1, x = 0.0017, g = 1, chaos synchronization is obtained. For saving space only the phase portraits and Poincare´ maps of the fractional order synchronized systems and time history of error states plot for condition(8)are shown inFig. 5. When c < 1, no chaos exists in system(10). When a, b take the fractional number less than 0.3, no chaotic synchronization is found.

Part (2):

Z = bgz2where z2is the chaotic state of system(10). When (1) a = b = c = 1.1, x = 0.34, g = 4; (2) a = b = 0.9, c = 1.1,

x= 0.62, g = 3; (3) a = b = 0.8, c = 1.1, x = 0.2807, g = 13.5; (4) a = b = 0.7, c = 1.1, x = 0.144, g = 3; (5) a= b = 0.6, c = 1.1, x = 0.0107, g = 21.3; (6) a = b = 0.5, c = 1.1, x = 0.001, g = 3; (7) a = b = 0.4, c = 1.1, x= 0.005, g = 1; (8) a = b = 0.3, c = 1.1, x = 0.0017, g = 1, chaos synchronization is obtained. For saving space only the phase portraits and Poincare´ maps of the fractional order synchronized systems and time history of error states plot for condition(8)are shown inFig. 6. When c < 1, no chaos exists in system(10). When a, b take the fractional number less than 0.3, no chaotic synchronization is found.

Part (3):

Z = bg exp(z1) where z1is the chaotic state of system(10). When (1) a = b = c = 1.1, x = 0.34, g = 0.29; (2) a = b = 0.9,

c= 1.1, x = 0.555, g = 1; (3) a = b = 0.8, c = 1.1, x = 0.2807, g = 1.8; (4) a = b = 0.7, c = 1.1, x = 0.144, g = 1; (5) a= b = 0.6, c = 1.1, x = 0.0107, g = 1; (6) a = b = 0.5, c = 1.1, x = 0.001, g = 3; (7) a = b = 0.4, c = 1.1, x = 0.005, g = 1.5; (8) a = b = 0.3, c = 1.1, x = 0.0017, g = 0.1, chaos synchronization is obtained. For saving space only the phase portraits and Poincare´ maps of the fractional order synchronized systems and time history of error states plot for condition(8)are shown inFig. 7. When c < 1, no chaos exists in system(10). When a, b take the fractional number less than 0.3, no chaotic synchronization is found.

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-1 -0.5 0 0.5 1 -5 0 5 x1 x2 -1 -0.5 0 0.5 1 -5 0 5 y1 y2 -1 -0.5 0 0.5 1 -3 -2 -1 0 1 2 3 z1 z2 0 100 200 300 400 500 -5 -4 -3 -2 -1 0 1x 10 -3 t e1

with order a = b = 0.3, c = 1.1, x = 0.0017, g = 1. -1.5 -1 -0.5 0 0.5 1 1.5 -5 0 5 x1 x2 -2 -1 0 1 2 -5 0 5 y1 y2 -1 -0.5 0 0.5 1 -3 -2 -1 0 1 2 3 z1 z2 0 1000 2000 3000 4000 5000 -20 -15 -10 -5 0 5x 10 -4 t e1

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-1 -0.5 0 0.5 1 -3 -2 -1 0 1 2 3 x1 x2 -1 -0.5 0 0.5 1 -3 -2 -1 0 1 2 3 y1 y2 -1 -0.5 0 0.5 1 -3 -2 -1 0 1 2 3 z1 z2 0 500 1000 1500 -6 -4 -2 0 2 4 x 1 0-7 t e1

Fig. 7. Phase portraits and Poincare´ maps of the synchronized fractional order systems and time history of states error for Z¼ bgez1 with order a = b = 0.3, c = 1.1, x = 0.0017, g = 0.1.

Part (4):

Z = bg exp(z2) where z2is the chaotic state of system(10). When (1) a = b = c = 1.1, x = 0.34, g = 0.2; (2) a = b = 0.9,

c= 1.1, x = 0.555, g = 0.1; (3) a = b = 0.8, c = 1.1, x = 0.2807, g = 0.1; (4) a = b = 0.7, c = 1.1, x = 0.144, g = 1.9; (5) a= b = 0.6, c = 1.1, x = 0.0107, g = 2; (6) a = b = 0.5, c = 1.1, x = 0.001, g = 1; (7) a = b = 0.4, c = 1.1, x = 0.005, g = 4; (8) a = b = 0.3, c = 1.1, x = 0.0017, g = 0.1, chaos synchronization is obtained. For saving space only the phase portraits and Poincare´ maps of the fractional order synchronized systems and time history of error states plot for con-dition(8)are shown inFig. 8. When c < 1, no chaos exists in system(10).When a, b take the fractional number less than 0.3, no chaotic synchronization is found.

Part (5):

Z = g exp(z1) sin(xt) where z1 is the chaotic state of system(10). When (1) a = b = c = 1.1, x = 0.445, xz= 0.34,

g = 0.43; (2) a = b = 0.9, c = 1.1, x = 0.1275, xz= 0.555, g = 0.1; (3) a = b = 0.8, c = 1.1, x = 0.1315, xz= 0.2807,

g = 0.1; (4) a = b = 0.7, c = 1.1, x = 0.31812, xz= 0.144, g = 0.1, chaos synchronization is obtained. For saving space

only the phase portraits and Poincare´ maps of the fractional order synchronized systems and time history of error states plot for condition(4)are shown inFig. 9. When c < 1, no chaos exists in system(10). When a, b take the fractional number less than 0.7, no chaotic synchronization is found.

Part (6):

Z = g exp(z2) sin(xt) where z2 is the chaotic state of system (10). When (1) a = b = c = 1.1, x = 0.445, xz= 0.34,

g = 0.1; (2) a = b = 0.9, c = 1.1, x = 0.1275, xz= 0.555, g = 0.1; (3) a = b = 0.8, c = 1.1, x = 0.1315, xz= 0.2807,

g = 0.1; (4) a = b = 0.7, c = 1.1, x = 0.31812, xz= 0.144, g = 0.1, chaos synchronization is obtained. For saving space

only the phase portraits and Poincare´ maps of the fractional order synchronized systems and time history of error states plot for condition(4)are shown inFig. 10. When c < 1, no chaos exists in system(10). When a, b take the fractional number less than 0.7, no chaotic synchronization is found.

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-1 -0.5 0 0.5 1 -4 -2 0 2 4 x1 x2 -1 -0.5 0 0.5 1 -4 -2 0 2 4 y1 y2 -1 -0.5 0 0.5 1 -3 -2 -1 0 1 2 3 z1 z2 0 500 1000 1500 -4 -2 0 2 4 x 1 0-7 t e1

Fig. 8. Phase portraits and Poincare´ maps of the synchronized fractional order systems and time history of states error for Z¼ bgez2 with order a = b = 0.3, c = 1.1, x = 0.0017, g = 0.1. -2 -1 0 1 2 -5 0 5 x1 x2 -2 -1 0 1 2 -5 0 5 y1 y2 -2 -1 0 1 2 -5 0 5 z1 z2 0 1000 2000 3000 4000 -4 -2 0 2 4x 10 -5 t e1

Fig. 9. Phase portraits and Poincare´ maps of the synchronized fractional order systems and time history of states error for Z¼ gez1_{sinðxtÞ with order a = b = 0.7, c = 1.1, x = 0.31812, x}

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Table 1

The ranges of g for various chaos synchronization cases

Part (1) Part (2) Part (3) Part (4)

Z = bgz1 Z = bgz2 Z = gz1sin(xt) Z = gz2sin(xt)

Case 1: The third system is a nonautonomous system with two states, Eq.(9)

a= b = 1.1 0.23–1.64 0.18–4.20 0.51–2.27 0.32–6.51

a= b = 1 0.09–3.43 0.14–7.6 0.15–3.7 0.7–9.8

a= b = 0.9 0.11–6.13 0.05–9.59 0.6–14.9 0.5–6.1

a= b = 0.8 0.18–8.23 0.1–13.53 0.3–8.5 0.5–14.4

a= b = 0.7 0.07–12.27 0.3–14.65 0.09–14.9 0.12–26.6

Case 2: The third system is an autonomous system with three states, Eq.(10)

Z = bg exp(z1) Z = bg exp(z2) a= b = c = 1.1 0.8–1.64 0.1–4.16 0.29–0.015 0.12–0.015 a= b = 0.9, c = 1.1 0.16–0.52 0.2–0.132 0.0001–1.36 0.001–0.18 a= b = 0.8, c = 1.1 0.1–11.6 0.07–13.5 0.0001–1.9 0.001–0.5 a= b = 0.7, c = 1.1 0.04–10.4 0.02–19.1 0.0001–3.6 0.001–1.9 a= b = 0.6, c = 1.1 0.06–6.6 0.05–21.3 0.0001–2.3 0.001–2.0 a= b = 0.5, c = 1.1 0.05–8.7 0.04–34.2 0.0001–3.2 0.001–3.6 a= b = 0.4, c = 1.1 0.06–10.1 0.1–38.2 0.02–3.7 0.001–4 a= b = 0.3, c = 1.1 0.06–11.9 0.04–50 0.01–4.7 0.01–5.5 Part (5) Part (6)

Z = g exp(z1) sin(xt) Z = g exp(z2) sin(xt)

a= b = c = 1.1 0.44–0.08 0.13–0.01 a= b = 0.9, c = 1.1 0.03–2.14 0.003–0.33 a= b = 0.8, c = 1.1 0.06–3.48 0.02–0.63 a= b = 0.7, c = 1.1 0.05–5.22 0.007–1.93 -2 -1 0 1 2 -6 -4 -2 0 2 4 6 x1 x2 -2 -1 0 1 2 -6 -4 -2 0 2 4 6 y1 y2 -2 -1 0 1 2 -5 0 5 z1 z2 0 200 400 600 -2 -1 0 1 2 3x 10 -5 t e1

Fig. 10. Phase portraits and Poincare´ maps of the synchronized fractional order systems and time history of states error for Z¼ gez2_{sinðxtÞ with order a = b = 0.7, c = 1.1, x = 0.31812, x}

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By the results of simulation, it is found that the chaos excited chaos synchronizations are obtained in Case 1 for lowest total fractional order 0.7· 2 = 1.4, while synchronizations can be achieved in Case 2 for lowest total fractional order 0.3· 2 = 0.6.

5. Conclusions

Chaos excited chaos synchronization of a generalized van der Pol system with integral and fractional order is stud-ied. Synchronizations of two identical generalized van der Pol chaotic systems are obtained by replacing their corre-sponding exciting terms by the same function of chaotic states of a third nonautonomous or autonomous system. Numerical simulations, such as phase portraits, Poincare´ maps and state error plots are given. It is found that chaos excited chaos synchronizations exist for the fractional order systems with the total fractional order both less than and more than the number of the states of the integer order generalized van der Pol system. Synchronizations are obtained in Case 1 for lowest total fractional order 0.7· 2 = 1.4, while synchronizations can be achieved in Case 2 for lowest total fractional order 0.3· 2 = 0.6.

Acknowledgments

This research was supported by the National Science Council, Republic of China, under grant number NSC94-2212-E-009-013.

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[23] Ge Z-M, Chang C-M. Chaos synchronization and parameter identiﬁcation of single time scale brushless DC motors. Chaos, Solitons & Fractals 2004;20:883–903.

[24] Ge Zheng-Ming, Cheng Jui-Wen, Chen Yen-Sheng. Chaos anticontrol and synchronization of three time scales brushless DC motor system. Chaos, Solitons & Fractals 2004;22:1165–82.

[25] Ge Z-M, Cheng J-W. Chaos synchronization and parameter identiﬁcation of three time scales brushless DC motor system. Chaos, Solitons & Fractals 2005;24:597–616.

[26] Ge Zheng-Ming, Lee Ching-I. Control, anticontrol and synchronization of chaos for an autonomous rotational machine system with time-delay. Chaos, Solitons & Fractals 2005;23:1855–64.

[27] Ge Zheng-Ming, Chen Yen-Sheng. Adaptive synchronization of unidirectional and mutual coupled chaotic systems. Chaos, Solitons & Fractals 2005;26:881–8.

[28] Jiang G, Zheng W, Chen G. Global chaos synchronization with channel time-delay. Chaos, Solitons & Fractals 2004;20:267–75. [29] Chen G, Liu S. On generalized synchronization of spatial chaos. Chaos, Solitons & Fractals 2003;15:311–8.

[30] Li Z, Xu D. A secure communication scheme using projective chaos synchronization. Chaos, Solitons & Fractals 2004;22:477–81. [31] Jiang Guo-Ping, Tang Wallace Kit-Sang, Chen Guanrong. A simple global synchronization criterion for coupled chaotic systems.

Chaos, Solitons & Fractals 2003;15:925–35.

[32] Ge Zheng-Ming, Yang Cheng-Hsiung. Generalized synchronization of quantum-CNN chaotic oscillator with diﬀerent order systems. Chaos, Solitons & Fractals, accepted for publication.

[33] Ge Z-M, Chang C-M, Chen Y-S. Anti-control of chaos of single time scale brushless DC motor and chaos synchronization of diﬀerent order systems. Chaos, Solitons & Fractals 2006;27:1298–315.

[34] Bi Qinsheng. Dynamical analysis of two coupled parametrically excited van der Pol oscillators. Int J Nonlinear Mech 2004;39:33–54.

[35] dos Santos AM, Lopes SR, Viana RL. Rhythm synchronization and chaotic modulation of coupled van der Pol oscillators in a model for the heartbeat. Physica A 2004;338:335–55.

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[42] Kim CM, Rim S, Kye WH, Ryu JW, Park YJ. Anti-synchronization of chaotic oscillators. Phys Lett A 2003;320:39–46. [43] Yang SP, Niu HY, Tian G, et al. Synchronizing chaos by driving parameter. Acta Phys Sin 2001;50:619–23.

[44] Dai D, Ma XK. Chaos synchronization by using intermittent parametric adaptive control method. Phys Lett A 2001;288:23–8. [45] Chen HK. Synchronization of two diﬀerent chaotic systems: a new system and each of the dynamical systems Lorenz, Chen and

Lu¨. Chaos, Solitons & Fractals 2005;25:1049–56.

[46] Chen HK, Lin TN. Synchronization of chaotic symmetric Gyros by one-way coupling conditions. ImechE Part C: J Mech Eng Sci 2003;217:331–40.

[47] Chen HK. Chaos and chaos synchronization of a symmetric gyro with linear-plus-cubic damping. J Sound Vibrat 2002;255:719–40.

[48] Ge ZM, Yu TC, Chen YS. Chaos synchronization of a horizontal platform system. J Sound Vibrat 2003:731–49.

[49] Ge ZM, Lin TN. Chaos, chaos control and synchronization of electro-mechanical gyrostat system. J Sound Vibrat 2003;259(3). [50] Ge ZM, Lin CC, Chen YS. Chaos, chaos control and synchronization of vibrometer system. J Mech Eng Sci 2004;218:1001–20. [51] Chen HK, Lin TN, Chen JH. The stability of chaos synchronization of the Japanese attractors and its application. Jpn J Appl

Phys 2003;42(12):7603–10.

[52] Ge ZM, Shiue. Non-linear dynamics and control of chaos for tachometer. J Sound Vibrat 2002;253(4).

[53] Ge ZM, Lee CI. Non-linear dynamics and control of chaos for a rotational machine with a hexagonal centrifugal governor with a spring. J Sound Vibrat 2003;262:845–64.

[54] Ge ZM, Hsiao CM, Chen YS. Non-linear dynamics and chaos control for a time delay duﬃng system. Int J Nonlinear Sci Numer 2005;6(2):187–99.

[55] Ge ZM, Tzen PC, Lee SC. Parametric analysis and fractal-like basins of attraction by modiﬁed interpolates cell mapping. J Sound Vibrat 2002;253(3).

[56] Ge ZM, Lee SC. Parameter used and accuracies obtain in MICM global analyses. J Sound Vibrat 2004;272:1079–85.

[57] Ge ZM, Lee JK. Chaos synchronization and parameter identiﬁcation for gyroscope system. Appl Math Comput 2004;63:667–82. [58] Chen HK. Global chaos synchronization of new chaotic systems via nonlinear control. Chaos, Solitons & Fractals

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[59] Chen HK, Lee CI. Anti-control of chaos in rigid body motion. Chaos, Solitons & Fractals 2004;21:957–65.

[60] Ge ZM, Wu HW. Chaos synchronization and chaos anticontrol of a suspended track with moving loads. J Sound Vibrat 2004;270:685–712.

[61] Ge ZM, Yu CY, Chen YS. Chaos synchronization and chaos anticontrol of a rotational supported simple pendulum. JSME Int J Ser C 2004;47(1):233–41.

[62] Ge ZM, Lee CI. Anticontrol and synchronization of chaos for an autonomous rotational machine system with a hexagonal centrifugal governor. Chaos, Solitons & Fractals 2005;282:635–48.