ARTICLE IN PRESS

Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx www.elsevier.com/locate/cnsns

Nonlinear dynamics and synchronization of coupled electromechanical systems with multiple functions R. Yamapi a

a,*

, F.M. Moukam Kakmeni b, J.B. Chabi Orou

c

Department of Physics, Faculty of Sciences, University of Douala, P.O. Box 24157, Douala, Cameroon b Department of Physics, Faculty of Sciences, University of Bue´a, P.O. Box 63, Bue´a, Cameroon c Institut de Mathe´matiques et de Sciences Physiques (I.M.S.P.), B. P. 613 Porto-Novo, Be´nin Received 14 February 2005; received in revised form 11 May 2005; accepted 11 May 2005

Abstract This paper deals with the nonlinear dynamics and synchronization of coupled electromechanical systems with multiple functions, described by an electrical Duffing oscillator magnetically coupled to linear mechanical oscillators. Firstly, the amplitudes of the sub- and super-harmonic oscillations for the resonant states are obtained and discussed using the multiple time scales method. The equations of motion are solved numerically using the Runge–Kutta algorithm. It is found that chaotic and periodic orbit coexist in the electromechanical system depending on the set of initial conditions. Secondly, the problem of synchronization dynamics of two coupled electromechanical systems both in the regular and chaotic states is also investigated, and estimation of the coupling coefficient under which synchronization and no-synchronization take place is made.
2005 Elsevier B.V. All rights reserved. PACS: 05.45.Xt; 05.45.Gg; 04.45.Pq Keywords: Electromechanical systems; Sub- and super-harmonic oscillations; Synchronization

*

Corresponding author. Tel.: +237 932 93 76; fax: +237 340 75 69. E-mail addresses: [email protected] (R. Yamapi), [email protected] (F.M. Moukam Kakmeni), [email protected] (J.B. Chabi Orou). 1007-5704/$ - see front matter
2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2005.05.003

ARTICLE IN PRESS

2

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx

1. Introduction In recent years, the dynamics of coupled systems including Duffing, Van der pol and Rayleigh equations have received a great attention in the scientific community. This, due to the fact that such coupled nonlinear oscillators described various physical, electrical, mechanical, electromechanical and biological systems, and can exhibit various types of behaviors [1–7]. Recent contributions on the study of the behavior of such coupled systems have been focussed on a nonlinear electromechanical system with multiple functions described by the Duffing electrical oscillator magnetically coupled to linear mechanical oscillators [7]. The amplitude of the harmonic oscillatory states and their stability boundaries have been found using respectively the harmonic balance method and the Floquet theory [1]. The chaotic state has been obtained from numerical simulation of the equations of motion. The indicators used are the bifurcation diagram and the Lyapunov exponent. The effects of the number of linear mechanical oscillators on the frequency-response and the chaotic states have been discussed and it appears that for some set of physical parameters, the hysteresis and jump phenomena disappear with the increase of the number of linear mechanical oscillators. Depending on the amplitude of the external excitation and when the frequency of the voltage source is very large or very small as compare of the values of the natural frequencies of the oscillators, the effects of the excitation will be small unless this amplitude is hard. The consequence is the generation of higher or super-harmonic and sub-harmonic oscillations where the amplitude of various orders resonant states [1] can be found and established using the multiple time scales method. This paper deals with the determination of such oscillatory states which are important in the field of nonlinear oscillations, and they frequently occur in various branches of electromechanical engineering and physical sciences [8,9]. The paper is organized as follows. In Section 2, the electromechanical system with multiple functions and its equations of motion are presented. Using the multiple time scales method, the amplitudes of sub- and super-harmonic oscillatory states of the model are found in Section 3. The effects of linear mechanical oscillators on the behavior of the model are analyzed analytically and it appears that for some set of physical parameters, the hysteresis and jump phenomena are hardly affected with the increase of the number of linear mechanical oscillators. Different dynamical states are also identified in the electromechanical system depending on the set of the initial conditions. Section 4 deals with the synchronization states of coupled electromechanical systems with 25 functions. One performers analytically and numerically the calculations to obtain for the identification of the transition boundaries of the synchronization process which appeared in the coupled models. A conclusion is given in Section 5.

2. The electromechanical system As described in Ref. [7], the electromechanical system with multiple functions is schematically represented in Fig. 1. It consists of an electrical part coupled magnetically to a mechanical part governed by n linear mechanical oscillators. The coupling between both parts is realized through the electromechanical force due to a permanent magnet. The electrical part of the system consists of a resistor R, an inductor L, the condenser C with nonlinear characteristic [7] and a sinusoidal voltage source, all connected in series, while the mechanical part is composed of n mobile beams

ARTICLE IN PRESS

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx magnet

s B

3

spring k1

C stone

N

R B

x1

magnet mobile beam (m1)

rod T1

e(t)

coupling magnet cool

rod Tn L mobile beam (mn) magnet

s B

xn

stone N s

B

spring kn

Fig. 1. Schema of the electromechanical system with multiple functions.

which can move respectively along the ~ xi ði ¼ 1; . . . ; nÞ axis on both sides. The rods Ti are bound to mobile beams with springs of constants ki. The motion of the entire electromechanical system is governed by the following n + 1 nondimensional coupled nonlinear differential equations n X €x þ c_x þ x þ bx3 þ ki x_ i ¼ E0 cos wt; i¼1

€x1 þ c1 x_ 1 þ w21 x1  k11 x_ ¼ 0; .. . €xi þ ci x_ i þ w2i xi  ki1 x_ ¼ 0; .. . €xn þ cn x_ n þ w2n xn  kn1 x_ ¼ 0;

ð1Þ

where the dots over the quantities denote differentiation with respect to time, the variables x and xi denote respectively the instantaneous electrical charge of the condenser and the displacement of the ith mobile beams of the mechanical part. The electromechanical system is described by the system consisting of an electrical Duffing oscillator magnetically coupled to linear mechanical oscillators. 3. Nonlinear dynamics 3.1. The multiple time scales method The multiple time scales method has been chosen because it is more indicated to find and establish the amplitudes of various orders of resonant states [1]. For this method, one have seen an asymptotic expansion of the solutions of Eqs. (1) in the following form:

ARTICLE IN PRESS

4

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx

x ¼ x01 ðT o ; T 2 Þ þ 3 x03 ðT o ; T 2 Þ þ    ; xi ¼ xi1 ðT o ; T 2 Þ þ 3 xi3 ðT o ; T 2 Þ þ   

ði ¼ 1; 2; 3; . . . ; nÞ;

ð2Þ

where the independent variables or time scales To = t and T2 = 2t are respectively the fast scale (associated to the unperturbed system) and the slow scale (associated to the amplitude and phase modulations induced by the global first order perturbation).  is a small dimensionless parameter. In this paper, the damping and the coupling coefficients are considered as global second order perturbations, then one writes c = 2co, ci = 2coi, ki = 2koi and ki1 = 2koi1 (with i = 1, 2, . . ., n). The amplitude E0 is taken at the order E0 = E to indicate that E0 is hard. Inserting these expansions in Eq. (1) and equating coefficients of like powers of , one obtains Order , D2o x01 þ x01 ¼ E cos wT o ; D2o x11 þ w21 x11 ¼ 0; .. .

ð3Þ

D2o xi1 þ w2i xi1 ¼ 0; .. . D2o xn1 þ w2n xn1 ¼ 0. Order 3, D2o x03 þ x03 ¼ 2Do D2 x01  co Do x01  bx301 

n X

koi Do xi1 ;

i¼1

D2o x13 þ w21 x13 ¼ 2Do D2 x11  co1 Do x11 þ ko11 Do x01 ; .. . D2o xi3

þ

w2i xi3

ð4Þ

¼ 2Do D2 xi1  ci Do xi1 þ koi1 Do x01 ;

.. . D2o xn3 þ w2n xn3 ¼ 2Do D2 xn1  cn Do xn1 þ kon1 Do x01 ; where Do ¼ oTo o ; D2 ¼ oTo 2 . The general solutions of Eqs. (3) can be expressed as x01 ¼ AðT 2 Þ expðjT o Þ þ K expðjwT o Þ þ AðT 2 Þ expðjT o Þ þ K expðjwT o Þ; x11 ¼ A1 ðT 2 Þ expðjw2 T o Þ þ A1 ðT 2 Þ expðjw2 T o Þ; .. . xi1 ¼ Ai ðT 2 Þ expðjwiþ1 T o Þ þ Ai ðT 2 Þ expðjwiþ1 T o Þ; .. . xn1 ¼ An ðT 2 Þ expðjwnþ1 T o Þ þ An ðT 2 Þ expðjwnþ1 T o Þ;

ð5Þ

ARTICLE IN PRESS

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx

5

where the over-bar represents the complex conjugate, j2 = 1 and 1 E . 2 1  w2 Substituting x01 and xi1 into Eq. (4) yields K¼

D2o x03 þ x03 ¼ ð2jA0 þ jco A þ 3bA2 A1 þ 4bK2 AÞ expðjT o Þ þ ðjco wK þ 4bKAA þ 2K3 bÞ expðjwT o Þ  bA3 expð3jT o Þ  3bA2 K expðjð2 þ wÞT o Þ  3bAK2 expðjð2w þ 1ÞT o Þ  3bK2 A expðjð1  2wÞT o Þ  3bKA2 expðjð2  wÞT o Þ n X 3  K b expð3jwT o Þ  j wi koi Ai expðjwi T o Þ þ C.C.; i¼1

D2o x13

þ

w21 x13

¼

jw1 ð2A01

þ co1 A1 Þ expðjw1 T o Þ þ jko11 A expðjT o Þ

þ jwko11 K expðjwT o Þ þ C.C.;

ð6Þ

.. . D2o xi3 þ w2i xi3 ¼ ji ð2A0i þ coi Ai Þ expðjwi T o Þ þ jkoi1 A expðjT o Þ þ jwkoi1 K expðjwT o Þ þ C.C.; .. . D2o xn3 þ w2n xn3 ¼ jwn ð2A0n þ con An Þ expðjwn T o Þ þ jkon1 A expðjT o Þ þ jwkon1 K expðjwT o Þ þ C.C.; where the prime (on A, Ai, i = 1, 2, . . ., n) denotes the differentiation with respect to T2 and C.C. stands for the complex conjugate of the previous terms. As was mentioned before, the unknown amplitudes A and Ai may now be determined by eliminating secular terms in Eq. (6). However, due to the high dimension of the system, it is impossible to obtain a set of secular equations valid from all frequencies. According to the different values of the frequencies w and wi relatively to the natural frequency of the Duffing electrical oscillator (which is one), it comes two types of interesting resonant structures: the first one is the super-harmonic resonant state wi = 1 and 3w = 1, and the second one corresponds to the sub-harmonic resonant state w = 3 with wi = 1. Therefore, we restrict our attention in the following subsection to the case of sub- and super-harmonic resonances. 3.2. Amplitude of the super-harmonic resonant state One considers the case where the electrical Duffing oscillator enters in super-harmonic resonance with the external excitation, that is 3w = 1 + 2r and assuming that wi ¼ 1 þ 2 ri ;

ð7Þ

where r and ri are the detuning parameters indicating the accuracy of the resonances. The secular producing terms in Eqs. (6) must be eliminated and the solvability conditions are defined as

ARTICLE IN PRESS

6

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx

2jA0 þ jco A þ 4bK2 A þ 3bA2 A þ j

n X

wi koi Ai expðjri T 2 Þ þ K3 b expðjrT 2 Þ ¼ 0;

i¼1

 .. .

w1 ð2A01

 .. .

wi ð2A0i

þ co1 A1 Þ expðjr1 T 2 Þ þ ko11 A ¼ 0; ð8Þ

þ coi Ai Þ expðjri T 2 Þ þ koi1 A ¼ 0;

 wn ð2A0n þ con An Þ expðjrn T 2 Þ þ kon1 A ¼ 0. One expresses A(T2) and Ai(T2), in the polar form: 1 AðT 2 Þ ¼ aðT 2 Þ expðjbðT 2 ÞÞ; 2 ð9Þ 1 Ai ðT 2 Þ ¼ ai ðT 2 Þ expðjbi ðT 2 ÞÞ; 2 where a, ai and b, bi are respectively the amplitudes and the phases of the oscillations. Substituting Eqs. (9) into Eqs. (8), this yields after separating real and imaginary parts the following set of first order differential equations: n 3 3 1X ba  ab0 þ 2bK2 a þ bK3 cos d  wi koi ai sin di ¼ 0; 8 2 i n 1 1X co a þ a0 þ bK3 sin d þ wi koi ai cos di ¼ 0; 2 2 i   1 a1 b01 cos d1 þ a01 þ co1 a1 sin d1 ¼ 0; 2   1 1 w1 a01 þ co1 a1 cos d1  w1 b01 a1 sin d1  ko11 a ¼ 0; 2 2

.. .



 1 0 0 ai bi cos di þ ai þ coi ai sin di ¼ 0; 2   1 1 wi a0i þ coi ai cos di  wi b0i ai sin di  koi1 a ¼ 0; 2 2 .. . an b0n  wn

 cos dn þ a0n

a0n

 1 þ con an sin dn ¼ 0; 2

 1 1 þ con an cos dn  wn b0n an sin dn  kon1 a ¼ 0; 2 2

ð10Þ

ARTICLE IN PRESS

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx

7

where d = rT2  b and di = riT2 + bi  b. Since it is particularly interested in studying the steadystate responses, one must have a0 ¼ a0i ¼ 0 and d0 ¼ d0i ¼ 0. Thus b 0 = r and b0i ¼ r  ri . Eliminating d, di from Eqs. (10), we obtain the following set of nonlinear equations: 9 2 6 3 b asp þ bF n a4sp þ ðF 2n þ G2n Þa2sp  b2 K6 ¼ 0; 64 4 a2iðspÞ

¼

k2oi1 a2sp 2

w2i ðc2oi þ 4ðr  ri Þ Þ

ð11Þ

;

where n 1X koi koi1 ðr  ri Þ ; F n ¼ 2bK  r  2 i¼1 4ðr  ri Þ2 þ c2oi 2

n X 1 koi koi1 coi . Gn ¼ co  2 2 2 i¼1 4ðr  ri Þ þ coi

When the linear mechanical oscillators are identical, Fn and Gn take the following expressions: F n ¼ F idn ¼ 2b2 K2  r 

1 nko1 ko11 ðr  ri Þ ; 2 4ðr  r1 Þ2 þ c2o1

1 nko1 ko11 coi Gn ¼ Gidn ¼ co þ . 2 4ðr  r1 Þ2 þ c2o1 Thus, in the case of super-harmonic resonances, the motion of the n + 1 oscillators are coupled and described by xðtÞ ¼ asp cosð3wt þ dÞ þ K cos wt þ Oð3 Þ; xi ðtÞ ¼ aiðspÞ cosð3wt þ di  dÞ þ Oð3 Þ.

ð12Þ

Using the Newton–Raphson algorithm, the amplitudes a and ai are plotted as a function of the detuning parameter r, since ri = 0 (exact internal resonance) will be considered throughout the paper. With the appropriate set of parameters E = 0.5, co = 0.01; co1 = 1.2, ko1 = 0.12, ko11 = 0.2, b = 0.6, ri = 0, the super-harmonic frequency–response curves obtained are presented in Fig. 2(i, ii) for several value of n and show the well-known hysteresis phenomena. The effects of the number of linear mechanical oscillators on the behavior of the model are observed (see Fig. 2(i, ii)) and it appears that with the above set of parameters, the hysteresis phenomenon hardly affects with the increasing of linear mechanical oscillators. In Fig. 2(iii, iv), one finds the effects of the amplitude E on the frequency–response curves and it appears that the hysteresis phenomenon disappears in the considering r-band when E increases. With the form of solutions (12), one finds through analytical investigations that the super-harmonic response is the periodic soluis the period of the external excitation). This is confirmed by the tions with T3 -period (T ¼ 2p w numerical simulation as it shown in Fig. 3 where the temporal evolution of the linear mechanical

ARTICLE IN PRESS

8

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx 0.2

1.6 1.4

n=1 n=10 n=20

(i)

1.2

n=1 n=10 n=20

(ii) 0.15

1

a 0.8

0.1

ai

0.6 0.4

0.05

0.2 0

0 –0.4

–0.2

0

0.2

σ

0.4

0.6

–0.4

–0.2

0

σ

0.2

0.4

0.6

0.25 E=0.5 E=1.5 E=5.0

3.5 3

(iii)

E=0.5 E=1.5 E=5.0

(iv) 0.2

2.5

0.15

2

ai

a

0.1

1.5 1

0.05

0.5 0 –2

0 –1

0

1

σ

2

3

–2

–1

0

σ

1

2

3

Fig. 2. (i, ii) Effects of the number of linear mechanical oscillators on the super-harmonic frequency–response curves a(r) and ai(r) (i = 1, . . ., n) with E = 0.5. (iii, iv) Effects of the amplitude E on the super-harmonic frequency–response curves a(r) and ai(r). The parameters used are:  = 0.001; ko1 = 0.12; ko11 = 0.2; co = 0.01; co1 = 1.2; b = 0.6; ri = 0 and n = 1.

0.03

T/3

0.02

x(t)

0.01 0

–0.01 –0.02 –0.03 5000

5005

5010

5015

5020

5025

5030

Time (s)

Fig. 3. Super-harmonic response curves xi(t) showing the T/3-periodic motion with the parameters of Fig. 2 and r = 2 and n = 25.

solution is plotted. This temporal variation xk(t) is the T3 -harmonic oscillations. All numerical simulations of the equations of motion in this paper are carried out using the fourth-order Runge– Kutta algorithm.

ARTICLE IN PRESS

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx

9

The stability of the super-harmonic steady-state motions can be determined by investigating the nature of the super-harmonic steady-state solutions of Eqs. (10). To accomplish this, we let a ¼ as þ ap ; ai ¼ ais þ aip ;

d ¼ ds þ dp ;

ð13Þ

di ¼ dis þ dip ;

where as and ais are the amplitudes of the super-harmonic steady-state solutions and ds and dis its phases. Substituting (13) into (10), expending for small ap, aip, dp and dip, and keeping linear terms in ap, dp, aip and dip, we obtain the following n + 2 set of first order differential equations a_ p ¼ P1 ap þ P2 dp þ d_ p ¼ P2 ap þ P4 dp þ

n X i¼1 n X

ðP2i dip þ P1i aip Þ; ðP4i dip þ P3i aip Þ;

i¼1

a_ 1p ¼ C11 a1p þ C21 d1p þ C31 ap ; d_ 1p ¼ C41 a1p þ C51 d1p þ C61 ap þ C71 dp ; .. . a_ ip ¼ C1i aip þ C2i dip þ C3i ap ; d_ ip ¼ C4i aip þ C5i dip þ C6i ap þ C7i dp þ Co5i

ð14Þ n X ðP4j djp þ P3j ajp Þ; j6¼i

.. . a_ np ¼ C1n anp þ C2n dnp þ C3n ap ; d_ np ¼ C4n anp þ C5n dnp þ C6n ap þ C7n dp þ Co5n

n X

ðP4j djp þ P3j ajp Þ;

j6¼n

where the parameters Pi, Pij and Cij are given in Appendix A. Then the stability of the super-harmonic steady-state solutions depends on the eigenvalues S of the coefficient matrix on the righthand sides of (14). But due to the order of this matrix (2n + 2 · 2n + 2), its difficult to find the eigenvalue equation, we restrict our analyze to the case of one function (n = 1) and Eq. (14) become n X ðP2i dip þ P1i aip Þ; a_ p ¼ P1 ap þ P2 dp þ i¼1

d_ p ¼ P2 ap þ P4 dp þ

n X

ðP4i dip þ P3i aip Þ;

i¼1

a_ 1p ¼ C11 a1p þ C21 d1p þ C31 ap ; d_ 1p ¼ C41 a1p þ C51 d1p þ C61 ap þ C71 dp

ð15Þ

ARTICLE IN PRESS

10

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx

Using the above equations, one can obtain the following eigenvalue equation S 4 þ Q1 S 3 þ Q2 S 2 þ Q4 ¼ 0;

ð16Þ

where the coefficients Qi are given in Appendix B. The super-harmonic steady-state motions are stable under the following conditions: Ca ¼ Q4 > 0;

ð17Þ

Cb ¼ Q1 ðQ2 Q3  Q1 Q4 Þ  Q23 > 0

and unstable otherwise. The stability boundary curves of the super-harmonic oscillatory solutions in the as  r plane are drawn in Fig. 4. One sees that for a given value of the amplitude of the external excitation E, a point M(as, r) will fixed on the stability chart and can be moved along the five following regions (I), (II), (III), (IV) and (V) as it appears in Fig. 4. Hence, by virtue of the stability conditions (17), the super-harmonic oscillatory solutions becomes unstable if the point M lies in the unstable regions. The unstable portions are the regions (I), (II), (III), and (IV) in Fig. 4. We remind that the super-harmonic oscillatory solutions are stable if both Ca and Cb are positive, and shall now consider each unstable region in detail. When the point M lies in the regions (I) and (III), Ca is negative and Cb positive, then the stability conditions are not satisfied. In the region (II), we have Ca < 0 and Cb < 0 while in the region (IV), Ca is positive and Cb negative. Our investigations shown that the stability portion is the region (V), since both Ca and Cb are positive. To summarize the above consideration, we mention that the stability portion of the super-harmonic oscillatory solutions is the region in which the super-harmonic oscillation is sustained. 3.3. Amplitude of the sub-harmonic resonant state To analyze the sub-harmonic resonances, we set w = 3 + 2ro. ro is another detuning parameter indicating the accuracy of the sub-harmonic resonances. Eliminating in Eq. (6) the terms that produce secular terms in xo3, xi3 and considering the expressions given by Eq. (7), we have

5 4.5 (III)

4 3.5

asp

3

(II)

(I)

2.5 2 (V): Stable portions

1.5 1 (IV )

0.5 0 –1

0

1

2

σ

3

4

5

Fig. 4. Stability boundaries of super-harmonic response curves with the parameters of Fig. 2.

ARTICLE IN PRESS

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx

2jA0 þ jco A þ 4bK2 A þ 3bA2 A þ j

n X

11

 2 expðjro T 2 Þ ¼ 0; wi koi Ai expðjri T 2 Þ þ 3KbA

i¼1

 w1 ð2A01 þ co1 A1 Þ expðjr1 T 2 Þ þ ko11 A ¼ 0; .. .  wi ð2A0i þ coi Ai Þ expðjri T 2 Þ þ koi1 A ¼ 0; .. .  wn ð2A0n þ con An Þ expðjrn T 2 Þ þ kon1 A ¼ 0.

ð18Þ

Again, introducing the polar notations for A, Ai and separating real and imaginary parts, we obtain the following first order differential equations: n 3 3 3 1X wi koi ai sin di ¼ 0; ba  ab0 þ 2bK2 a þ bKa2 cos do  8 4 2 i¼1 n 1 3 1X wi koi ai cos di ¼ 0; co a þ a0 þ bKa2 sin do þ 2 4 2 i¼1   1 0 0 a1 b1 cos d1 þ a1 þ co1 a1 sin d1 ¼ 0; 2   1 1 w1 a01 þ co1 a1 cos d1  w1 b01 a1 sin d1  ko11 a ¼ 0; 2 2 .. . ð19Þ   1 0 0 ai bi cos di þ ai þ coi ai sin di ¼ 0; 2   1 1 wi a0i þ coi ai cos di  wi b0i ai sin di  koi1 a ¼ 0; 2 2 .. .   1 0 0 an bn cos dn þ an þ con an sin dn ¼ 0; 2   1 1 wn a0n þ con an cos dn  wn b0n an sin dn  kon1 a ¼ 0. 2 2 where do = roT2  3b and di = riT2 + bi  b. For the steady-state responses, one has b0 ¼ r3o and b0i ¼ r3o  ri . Eliminating do and di from Eq. (19), one obtains the following set of nonlinear equations:   9 2 6 3 9 2 2 4 b asb þ bM n  b K asb þ ðM 2n þ N 2n Þa2sb ¼ 0; 64 4 16 2 ð20Þ k a2 i a2iðsbÞ ¼ h  oi1 sb2 w2i 4 r3o  ri þ c2oi

ARTICLE IN PRESS

12

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx

with   n koi koi1 r3o  ri ro X M n ¼ 2bK   ; ro 2 3 þ c2oi i¼1 4 3  ri 2

n 1 1X koi koi1 coi . N n ¼ co  ro  2 2 i¼1 4  ri 2 þ c2oi 3

In the case of identical linear mechanical oscillators, Mn and Nn take the following form:   ro nko1 ko11 r3o  r1 2 id M n ¼ M n ¼ 2bK    ;  3 4 ro  r1 2 þ c2 o1

3

1 1 nko1 ko11 co1 . N n ¼ N idn ¼ co    2 2 4 ro  r1 2 þ c2 o1 3 Eqs. (20) show that either asb = 0 or   9 2 4 3 9 2 2 2 b asb þ bM n  b K asb þ M 2n þ N 2n ¼ 0; 64 4 16 which is quadratic in asb. Its solutions are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bo b2o  4ao co 2 asb ¼ 2ao

ð21Þ

ð22Þ

with ao ¼

9b2 ; 64

3 9 bo ¼ bM n  b2 K2 ; 4 16

co ¼ M 2n þ N 2n .

One finds that nontrivial sub-harmonic oscillations occur only when b2o P 4ao co . This condition demands that ac K4 þ bc K2 þ cn P 0;

ð23Þ

where 81 4 b; ac ¼ 256

27 bc ¼ b4 ; 16

"  # n koi koi1 r3o  ri 27 3 ro X 9  b2 N 2n . cn ¼  b   ro 2 32 16 3 þ c2oi i¼1 4 3  ri

It follows from Eqs. (23) that, for a given ro, the boundary of the region where nontrivial sub-harmonic solutions in the Kro-plane can exist is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bc b2c  4ac cn . ð24Þ K2 ¼ 2ac

ARTICLE IN PRESS

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx

13

In the sub-harmonic resonant state, the motion of the electromechanical system is described by   1 wt þ do þ K cos wt þ Oð3 Þ; xðtÞ ¼ asb cos 3   ð25Þ 1 3 wt þ di  do þ Oð Þ. xi ðtÞ ¼ aiðsbÞ cos 3 In the exact internal resonances, one finds in Fig. 5 the behaviors of the amplitudes asb and aisb when the detuning parameter ro and the amplitude E are varied. The effects of the number of the linear mechanical oscillators on the sub-harmonic frequency–response curves (resp. amplitude–response curves) and the boundary of the region where nontrivial solutions can exist are observed. It is found that the behaviors of the electromechanical system are hardly affected and this boundary increases when the number of linear mechanical oscillators n varies from (n = 1), roc = 0.9 to (n = 20), roc = 2.4176 while for the case a(E) and ai(E), this boundary decreases for the increase of n from Ec = 11.4 (n = 1) to Ec = 3.21 (n = 20). With the form of solutions (25), one also finds through analytical investigations that the sub-harmonic response is the periodic solutions with 3T-period. This is confirmed by the numerical simulation as it shown in Fig. 6 where the temporal evolution of the linear mechanical solutions is plotted, this response xk(t) is

Fig. 5. (i, ii) Effects of the number of linear mechanical oscillators on the sub-harmonic frequency–response curves a(r) and ai(r) (i = 1, . . ., n), E = 0.5. (iii, iv) Effects of the number of linear mechanical oscillators on the sub-harmonic amplitude–response curves a(E) and ai(E) with r = 2.5. The parameters used are:  = 0.001; ko1 = 0.3; ko11 = 0.25; co = 0.2; co1 = 0.5; b = 0.6; ri = 0.

ARTICLE IN PRESS

14

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx 0.03

3T

0.02

x(t)

0.01

0

–0.01 –0.02

–0.03 5000

5005

5010

5015

5020

5025

5030

Time(s)

Fig. 6. Sub-harmonic response curves xi(t) showing the 3T-periodic motions with the parameters defined in Fig. 4 and n = 25 and r = 2.

25

20 n=20

E

15

n=10

10

5 n=1 0 0

0.2

0.4

0.6

0.8

1

σ0

1.2

1.4

1.6

1.8

2

Fig. 7. Boundary of the existence of sub-harmonic oscillations with the parameters defined in Fig. 4 in the (E, r)-plane.

the 3T-harmonic oscillations. These investigations enable us to find the variation of the boundary of the existence of sub-harmonic oscillations (below the curve) in the E–r plane and the results are provided in Fig. 7. 3.4. Coexistence of chaotic and periodic orbits One finds in this subsection different dynamical states which appear in the nonlinear electromechanical system depending on the set of initial conditions. For this aim, we numerically solve the equations of motion (1) and plot the resulting Poincare´ cross-section and phase portrait with the following set of parameters: c = ci = 0.2; ki = 0.01; ki1 = 0.25; wi = 1; b = 1.0; E0 = 25 and w = 0.8 where i = 1, 2, . . ., 25 since one considers the electromechanical system with 25 functions.

ARTICLE IN PRESS

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx 10

15

1.5

(i)

1

(ii)

5

dxi/dt

dx/dt

0.5 0

0

–0.5 –5 –1 –10 –5

0

5

x

–1.5 –2

0

1

2

0.3

0.4

0.5

xi

1.4

7

(iii)

(iv)

6

1.2

5

dxi/dt

dx/dt

–1

1

4 0.8 3 2 –3.5

–3

x

–2.5

–2

0.6 0.1

0.2

xi

Fig. 8. (i, ii) Phase portrait and (iii, iv) Poincare´ cross-section showing the periodic orbit with the parameters c = 0.2; c1 = 0.2; k1 = 0.01; k11 = 0.25; w1 = 1.0; b = 1.0; E = 25; w = 0.8; n = 25 and the initial conditions Ip: ðxð0Þ; x_ ð0ÞÞ ¼ ð0.0; 0.01Þ and ðxi ð0Þ; x_ i ð0ÞÞ ¼ ð0.0; 0.0Þ (i = 1, . . ., 25).

For the judicious choice of the initial conditions, one finds that the nonlinear electromechanical system with 25 functions is able to exhibit two degenerate attractors with the above mentioned set of parameters. The first one shown in Fig. 8 is obtained with the initial conditions Ip: ðxð0Þ; x_ ð0ÞÞ ¼ ð0.0; 0.01Þ and ðxi ð0Þ; x_ i ð0ÞÞ ¼ ð0.0; 0.0Þ which is the periodic orbit. The second one, obtained with the initial conditions Ic: ðxð0Þ; x_ ð0ÞÞ ¼ ð5.5; 2.0Þ and ðxi ð0Þ; x_ i ð0ÞÞ ¼ ð0.01; 0.01Þ is the chaotic attractor as it appears in Fig. 9. With the above two sets of initial conditions, a bifurcation diagram is drawn (see Fig. 10) when the amplitude E0 varies, and one finds around E0 = 25, two types of degenerate attractors have appeared depending on the set on initial conditions.

4. Synchronization of two coupled nonlinear electromechanical systems Today great and extensive interest in synchronization problems is displayed, spanning quite different fields of science. 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 behavior, which is a characteristic of synchronization, seems to be a manifestation of the natural tendency of self-organization existing in nature [10]. Considerable attention paid to such

ARTICLE IN PRESS

16

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx 10

1.5 (i)

(ii)

1

5

dxi /dt

dx/dt

0.5 0

0 –0.5

–5 –1 –10 –5

0

–1.5 –2

5

–1

0

(iii)

1

dxi /dt

6

5.5

0.9 0.8

5

0.7

4.5 4 –3.4

(iv)

1.1

6.5

dx/dt

2

1.2

7.5 7

1

xi

x

–3.2

–3

x

–2.8

–2.6

0.6 0.1

–2.4

0.15

0.2

0.25

xi

0.3

0.35

Fig. 9. (i, ii) Phase portrait and (iii, iv) Poincare´ cross-section showing the chaotic orbit with the parameters defined in Fig. 7 and the initial conditions Ic: ðxð0Þ; x_ ð0ÞÞ ¼ ð5.5; 2.0Þ and ðxi ð0Þ; x_ i ð0ÞÞ ¼ ð0.01; 0.01Þ (i = 1, . . ., 25). 4.5

(i)

x

4 3.5

4.5

3

3.5

4

3 24.5

2.5 18

20

22

E0 24

26

25

25.5

28

30

4.5

(ii)

x

4 4.5 4

3.5

3.5

3

3 24.5

18

20

22

E0

24

26

25

25.5

28

30

Fig. 10. Bifurcation diagrams showing the coordinate x versus E0 with the parameters defined in Fig. 7 and the following initial conditions: Ip for (i) and Ic for (ii).

ARTICLE IN PRESS

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx

17

topics is due to the potential applications of synchronization in communication engineering (using chaos to mask the information bearing signal) [11–14], in biology, chemistry and medicine the phenomenon of synchronization is of interest to those studying rhythms, electric rhythms of the brain in particular, cardiac rhythms, wave reactions in chemistry. In industry, synchronization is also used in the power generation to ensure exact coincidence of frequencies of several alternating current generator operating in parallel mode for common loading. In medecine, the effect of synchronization has found wide application in the construction of various vibro-technical devices. In this section, one considers the problem of the synchronization dynamics of two coupled electromechanical systems with a large number of functions, both on the regular and chaotic states. 4.1. Statement of the problem and variational equations 4.1.1. Statement of the problem In the regular regime, for some sets of physical parameter and due to the nonlinearity, the response of the system to the external excitation shows the well-known hysteresis phenomena with two stable harmonic oscillations and different amplitudes. Each harmonic state has its own basin of attraction in the space of initial conditions. Consequently, if two systems are launched with different initial conditions belonging to different basins of attraction, they will finally circulate on different orbits. The goal of the synchronization in this case is to call one of the system (slave) from its orbit to that of the other system (master) as it appears in Fig. 11. The study of the transition boundaries and the derivation of the characteristics of the synchronization process of two coupled electromechanical systems with 25 functions will be considered here. The master system is described by the components x and xi while the slave system has the corresponding components u and ui. The coupling between the master and the slave systems is carried out as

8 6 4

dx/dt

2

. Transition slave system

*

0

master system

–2 –4 –6 –8 –3

–2

–1

0 x

1

2

3

Fig. 11. The two basins of attractions with the initial conditions: I aA : ðxð0Þ; x_ ð0ÞÞ ¼ ð0.0; 0.01Þ and ðxi ð0Þ; x_ i ð0ÞÞ ¼ ð0.0; 0.0Þ (i = 1, . . ., 25) with tin-line and I cA : ðxð0Þ; x_ ð0ÞÞ ¼ ð5.0; 5.01Þ and ðxi ð0Þ; x_ i ð0ÞÞ ¼ ð0.0; 0.0Þ (i = 1, . . ., 25) with fatline. The parameters used are: c = 0.01; c1 = 0.1; k1 = 0.2; k11 = 0.4; w1 = 1; b = 0.95; w = 3; E = 1.05.

ARTICLE IN PRESS

18

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx

€x þ c_x þ x þ bx3 þ

n X

ki x_ i ¼ E0 cos wt;

i¼1

€x1 þ c1 x_ 1 þ w21 x1  k11 x_ ¼ 0; .. . €xn þ cn x_ n þ w2n xn  kn1 x_ ¼ 0; n X 3 € ki u_ i ¼ E0 cos wt  Kðu  xÞH ðt  T 0 Þ; u þ cu_ þ u þ bu þ

ð26Þ

i¼1

€ u1 þ c1 u_ 1 þ w21 u1  k11 u_ ¼ 0; .. . € un þ cn u_ n þ w2n un  kn1 u_ ¼ 0; where K is the feedback coupling synchronization coefficient, T0, the onset time of the synchronization and H(a) is the Heaviside function defined by H(a) = 0 for a < 0 and H(a) = 1 for a P 0. The schematic circuit of the two coupled identical nonlinear electromechanical systems with a unidirectionally homogenous coupling element is realized as it is shown in Ref. [5]. The two electromechanical systems, namely the master and slave systems, are coupled by a linear condenser and a buffer. The buffer acts on a signal-driving element that isolates the master system variable from the slave system variable, thereby providing a one-way coupling or unidirectional coupling. In the absence of the buffer, the system represents two identical self-sustained models coupled by a common condenser, when both the master and the slave systems will mutually affect each other. 4.1.2. Nonlinear variational equations From the instant t > T0, the system of coupled electromechanical systems changes its configuration and becomes physically interesting only as long as the dynamics of the slave electromechanical system is stable. The determination of the range of K for which the synchronization process is achieved is equivalent to the boundedness of  and i defined as eðtÞ ¼ uðtÞ  xðtÞ; ei ðtÞ ¼ ui ðtÞ  xi ðtÞ;

ð27Þ

i ¼ 1; 2; . . . ; 25.

The variables e and ei are the measure of the relative nearness of the slave to the master and obey the following nonlinear variational equations €eðtÞ þ c_eðtÞ þ ð1 þ K þ 3bx2s ÞeðtÞ þ 3bxs eðtÞ2 þ beðtÞ3 þ

n X i¼1

€ei ðtÞ þ ci e_ i ðtÞ þ

x2i ei ðtÞ

 ki1 e_ ðtÞ ¼ 0

ki e_ i ðtÞ ¼ 0;

ð28Þ

for i ¼ 1; 2; . . . ; 25.

Synchronization is achieved when the variations of the variables  and i go to zero as the time t increases or is less than a given precision, on the other hand, the synchronization is not achieved

ARTICLE IN PRESS

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx

19

and the variations  and i diverge. The behaviors of  and i depend on K, and the form of the master (xs, x1s, . . ., xns). The master time evolution in the harmonic states can be described as xs ðtÞ ¼ A cosðxt  /Þ; xis ðtÞ ¼ Ai cosðxt  /Þ;

ð29Þ

i ¼ 1; 2; . . . ; 25;

where A and Ai are the amplitudes while / and /i are the phases. As reported in Ref. [7], the amplitudes A and Ai are the solutions of the following nonlinear equations 9 2 6 3 b A þ bF m A4 þ ðF 2m þ Gm ÞA2  E20 ¼ 0; 16 2 where / ¼ tan1

) Gm ; F m þ 34 bA2

wki1 Ai ¼ pffiffiffiffiffi A; Di

i ¼ 1; 2; . . . ; 25;

ð30Þ

(

F m ¼ 1  x2 

Di ¼ ðx2i  x2 Þ þ x2 c2i ;

n X ki ki1 xðx2i  x2 Þ ; Di i¼1

Gm ¼ cx þ

n X ki ki1 ci x3 . Di i¼1

Using the Newton–Raphson algorithm, it appears that the corresponding two real positive physical solutions of Eqs. (30) with the parameters c = 0.01; c1 = 0.1; k1 = 0.2; k11 = 0.4; w1 = 1; b = 0.95; w = 3; E0 = 1.05, are Aa = 0.186 and Ac = 2.97. With the expressions (24), Eqs. (29) yield to the following parametric equations: n X 2 €e þ c_e þ ðX1 þ g cosð2wt  2/ÞÞe þ ki e_ i þ 3bA cosðwt  /Þe2 þ be3 ¼ 0; ð31Þ i¼1 €ei þ ci e_ i þ w2i i  kk1 e_ ¼ 0;

i ¼ 1; 2; . . . ; n;

where 3 g ¼ bA2 . 2 2 From the expression of X1 , one finds that if K < 1  32 bA2 , e(t) and ei(t) will grow indefinitely leading the slave electromechanical system to continuously drift away from it original basins of attraction. In this case the feedback coupling is dangerous. Since it continuously adds energy to the slave electromechanical system. The appropriate tool to investigate the stability of the nonlinear parametric equations (31) is the Floquet theory [1] or the method of multiple time scales [1], but due to the complexity in solving analytically this Eq. (31), we will use in the following section the numerical simulation to find the range of K for the stability of the synchronization process of the two coupled nonlinear electromechanical systems. X21 ¼ 1 þ K þ g;

4.2. Results of the numerical simulations 4.2.1. Synchronization of the regular states As one has mentioned before, our aim now is to use the numerical simulation for determining the range of the appropriate coupling coefficient which enables to call one of the electromechan-

ARTICLE IN PRESS

20

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx

ical system from its orbit to that of the other electromechanical system. The coupled equations (26) are solved numerically and determined the range of K in which the variables e and ei tend to zero, since the synchronization process is achieved. It can be noticed that the stability of the synchronization process means that the deviations e(t) and ei(t) tend to zero as the time increases, and the synchronization process is achieved. In the unstable region of K, e(t) and ei(t) show a bounded oscillatory behavior (the slave system is stable, but the synchronization process is not achieved) or grow indefinitely (no synchronization phenomena). One has found through a direct numerical simulation of Eq. (26) the appropriate range of K and the following results are observed. The master and slave electromechanical systems are initially launched with the initial con_ ¼ ditions ðxð0Þ; x_ ð0ÞÞ ¼ ð0.0; 0.01Þ and ðxi ð0Þ; x_ i ð0ÞÞ ¼ ð0.0; 0.0Þ (i = 1, . . ., 25), and ðuð0Þ; uð0ÞÞ ð5.0; 5.01Þ and ðui ð0Þ; u_ i ð0ÞÞ ¼ ð0.0; 0.0Þ, (i = 1, . . ., 25) respectively. These set of initial conditions lead to periodic oscillation with amplitudes approximately equal to Aa and Ac respectively. Here the slave system is forced to come from the orbit Aa to the orbit Ac. We just inverse the initial conditions for the case Ac to Aa. In this analysis, the synchronization is launched at T0 = 500 and the coefficient K is varied until synchronization is achieved. It appears through this numerical simulation of Eqs. (26) that the slave transition from Ac to Aa requires that K 2 [3.1; 3.9] [ [4.4; 4.8] [ [5.1; +1[, while for the slave transition Aa to Ac, one has K 2 [0.92; 3,3] [ [4.05; +1[. The synchronization time Ts, defined as Ts = ts  T0 is computed following the time trajectory of the slave system relative to that of the master, ts is the time instant at which the two trajectories are close enough to be considered as synchronized. Let us look now for the behavior of the condition of the synchronization process between the master and slave electromechanical systems. Synchronization is achieved with the following synchronization condition: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 25 25 u X X ð32Þ ðui  xi Þ2 þ ðu_ i  x_ i Þ2 < h; jdðtÞj ¼ tðu  xÞ2 þ ðu_  x_ Þ2 þ i¼1

i¼1

where h is the synchronization precision or tolerance. For these investigations, the value of the precision is h = 1010 for the slave transition from Ac to Aa, while in the reverse case, the synchronization process with this precision was not possible. In fact, with the tolerance h = 104, one finds that the synchronization process is achieved with the range of K defined before. This is due to the fact that it is easy to move from the orbit with small amplitude to the one with high amplitude, than in the reverse case. This situation can be understood by the fact that we have Aa < Ac. We remind that this synchronization has enabled us to identify different bifurcation mechanisms which appear in the coupled system. To illustrate the above results, let us evaluate the temporal variation of the quantity     25 25 X X  2 2 2 2 ðui  xi Þ þ ðu_ i  x_ i Þ  pðtÞ ¼ ðu  xÞ þ ðu_  x_ Þ þ   i¼1 i¼1 with the value of the coupling coefficient K both chosen in the unstable and stable regions of the synchronization process. The results are shown in Fig. 12 for the variation of p(t) and in Fig. 13 for the variation of Ts versus K, and one finds that in the stable domain of K, p(t) decreases and tend to zero when t evolves (see Fig. 13(iii, iv)), while in the unstable region of K, p(t) presents the

ARTICLE IN PRESS

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx 50

50

(iii)

p(t)

p(t)

(i)

0

–50 400

500

600

700

0

–50 400

800

500

Time (s)

600

700

50

(ii)

(iv)

p(t)

p(t)

800

Time (s)

50

0

–50 400

21

500

600

700

0

–50 400

800

500

600

700

800

Time (s)

Time (s)

Fig. 12. Temporal variation of p(t) versus time for the slave transition from periodic orbit Aa to the periodic orbit Ac with the value of K chosen both in the unstable region ((i) K = 1 and (ii) K = 3.5) and stable region ((iii) K = 2 and (iv) K = 5).

3500

3000

Ts

2500

2000

1500

1000 0

1

2

3

4

5

6

7

8

9

K

Fig. 13. Synchronization time Ts versus the coupling coefficient K for the slave transitions from the orbit Aa to the orbit A c.

bounded oscillatory behavior (see Fig. 13) or grows to infinity (see Fig. 13(i, ii)) or exhibits the chaotic behavior (see Fig. 13(i)).

ARTICLE IN PRESS

22

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx

4.2.2. Synchronizing chaotic orbit (respectively periodic orbit) on the periodic orbit (respectively chaotic orbit) Now one considers the two electromechanical systems with the following parameters c = 0.2; c1 = 0.2; k1 = 0.01; k11 = 0.25; w1 = 1.0; b = 1.0; E = 25; w = 0.8. This set of parameters leads to the following approximated amplitude A = 3.18. In this case, the model can exhibit the chaotic and periodic behavior, depending on the initial conditions as we have shown in Figs. 7 and 8. If two electromechanical systems with the above set of parameters are launched with the following initial conditions Ip and Ic, they will finally circulate on the periodic and chaotic orbits, respectively. The objective of the synchronization is to call the slave system from the periodic orbit (respectively chaotic orbit) to follow the master system to the chaotic orbit (periodic orbit). The numerical simulation of the nonlinear equations (26) indicates that, the slave transition from the periodic orbit to the chaotic orbit occurs for K 2 [2.85; 2.75] [ [2.55; 2.15] [ [1.55; 2.25] [ [4.4; +1[, while to the reverse transition, one needs K 2 [2.15; 1.4] [ [0.75; 1.40] [ [3.9; +1[. In Fig. 14, the temporal variation of p(t) is presented for the slave transition from the chaotic orbit to the periodic one with the value of K chosen both in the unstable and stable

40

60

(i)

40

(iii) 20

P(t)

P(t)

20 0

0

–20 –20 –40 –60 400

450

500

550

–40 400

600

500

600

700

Time (s)

Time (s) 50

30

(ii)

(iv)

20

P(t)

P(t)

10 0

0 –10 –20

–50 400

450

500

Time (s)

550

600

–30 400

500

600

700

Time (s)

Fig. 14. Temporal variation of p(t) versus time for the slave transition from chaotic orbit to the periodic orbit with the value of K chosen both in the unstable region ((i) K = 3.2 and (ii) K = 3.5) and stable region ((iii) K = 2 and (iv) K = 4.6).

ARTICLE IN PRESS

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx

23

2500

2000

Ts

1500

1000

500

0 –4

–2

0

2

4

6

8

10

12

14

16

K

Fig. 15. Synchronization time Ts versus the coupling coefficient K for the slave transitions from the chaotic orbit to the periodic orbit.

regions of K. The temporal variations of p(t) versus time for the slave transitions from the chaotic orbit to the periodic orbit with the values of K chosen both in the unstable region (K = 3.2; 3.5) and the stable domain (K = 2.1; 4.6) is shown in Fig. 14, and we find that p(t) in the stable region of K versus the time increases while in the unstable region, p(t) shows the chaotic behaviors. Fig. 15 shows the variation of the synchronization time Ts versus the coupling coefficient K for the slave transition from the chaotic orbit to the periodic orbit.

5. Conclusion In this paper, the dynamics and synchronization of coupled electromechanical systems with multiple functions in series is studied. Using the multiple time scale method, attention is focussed on the amplitude of sub- and super-harmonic resonances. It is observed that the number of linear mechanical oscillators can be increased in order to destroy the hysteresis phenomena in the system. This is very important in electromechanical system where the hysteresis phenomena must be destroyed in order to avoid distortions. On the other hand, uni-directional coupling have been applied to study the synchronization of the master and slave system both in the regular and chaotic states. These phenomena of synchronization or nonsynchronization is proved by direct numerical simulations of the coupled system. The threshold value of the coupling parameter K for the phenomena (synchronization or nonsynchronization) to be observed is determined from the nonlinear variational equation defining the synchronization error dynamics.

Appendix A The coefficients of Eqs. (14) are

ARTICLE IN PRESS

24

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx

Di ¼ xi ais sin2 dis  xi ais cos dis ; xi ais sin dis ; C01i ¼ Di ais cos dis C02i ¼ ; Di xi C03i ¼ ; Di sin dis C04i ¼ ; Di xi ais sin2 dis þ xi ais cos dis C05i ¼ ; as Di 1 C11i ¼ ðr  ri Þ cos dis þ c0i sin dis ; 2 1 1 C2i ¼ ais ðr  ri Þ sin dis þ c0i ais cos dis ; 2 1 1 C3i ¼ xi ðr  ri Þ sin dis þ c0i cos dis ; 2 1 1 C4i ¼ xi ais ðr  ri Þ cos dis  c0i sin dis ; 2 C1i ¼ C01i C11i þ C02i C13i ; C2i ¼ C01i C12i þ C02i C14i ; 1 C3i ¼  C02i k0i1 ; 2 1 0 1 C4i ¼ C3i C1i þ C04i C13i  C05i xi k0i sin dis ; 2 1 0 1 0 1 C5i ¼ C3i C2i þ C4i C4i  C05i xi k0i ais cos dis ; 2  1 0 2 0 9 2 C6i ¼  C4i k0i1 þ C5i bas  r þ 2bK ; 2 8 3 0 C7i ¼ C5i bK sin ds ; 1 P1 ¼  c0 ; 2 P2 ¼ bK3 cos ds ;   1 9 2 bas  r þ 2bK2 ; P3 ¼  as 8 1 P4 ¼ bK3 sin ds ; as P1i ¼ xi k0i cos dis ; P2i ¼ xi k0i ais sin dis ; 1 P3i ¼  xi k0i sin dis ; 2 1 P4i ¼  xi k0i ais cos dis ; 2

ð33Þ

ARTICLE IN PRESS

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx

Appendix B.

25

The coefficients Qi of Eqs. (16) are

Q1 ¼ P4  P1  C11  C51 ; Q2 ¼ P2 P3 þ P4 ðP1 þ C11 þ C51 Þ þ P1 C11 þ C51 ðP1 þ C11 Þ  C61 P21  C41 C21  C31 P11 þ C71 P41 ; Q3 ¼ P2 P3 ðC11 þ C51 Þ  P2 C61 P41  P2 C31 P31  P4 P1 C11  P4 C51 ðC11 þ P1 Þ  P1 C11 C51 þ P4 C61 P21 þ P4 C41 C21 þ P4 C31 P11  C31 C41 P21  C61 P11 C21 þ C61 C11 P21 þ P1 C41 C21 þ C31 P11 C51  C71 P31 C21  C71 P3 P21 þ C71 P41 ðP1 þ C11 Þ; Q4 ¼ P2 P3 C11 C51  P41 P2 C31 C41  P2 C61 P31 C21 þ P2 C61 C11 P41 þ P2 P3 C41 C21 þ P2 C31 P31 C51 þ P4 P1 C11 C51 þ P4 C31 C41 P21 þ P4 C61 C21 P11  P4 C61 P21 C11  P4 C61 C11 P1  P4 C31 P11 C51 þ C71 P1 P31 C21 þ C71 P3 P21 C11 þ C71 C31 P41 P11  C71 C31 P31 P21  C71 C11 P41 P1  C71 C21 P3 P11 .

References [1] Nayfeh AH, Mook DT. Nonlinear oscillations. New York: Wiley-Interscience; 1979. [2] Awrejcewicz J. Bifurcation and chaos in coupled oscillators. Singapore: World Scientific; 1991. [3] El-Bassiony AF, Eissa M. Response of three-degree-of-freedom with cubic non-linearities to harmonic excitations. Phys Scr 1999;59:183–94. [4] Kozlov AK, Sushchik MM, Molkov YaI, Kuznetsov AS. Bistable phase synchronization and chaos in a system of coupled Van der Pol–Duffing oscillators. Int J Bifurc Chaos 1999;9:2271–7. [5] Yamapi R, Woafo P. Dynamics and synchronization of coupled self-sutained electromechanical devices. J Sound Vibr 2005;285(4):1151–70. [6] Yamapi R, Bowong S. Dynamics and chaos control of the self-sustained electromechanical device with and without discontinuity. Commun Nonlinear Sci Numer Simul, in press. [7] Woafo P, Yamapi R, Chabi Orou JB. Dynamics of a nonlinear electromechanical system with multiple functions in series. Commun Nonlinear Sci Numer Simul 2005;10:229–51. [8] Stocker JJ. Nonlinear vibration. New York: Interscience Publishers; 1950. [9] Hayashi C. J Appl Phys 1953;24:521. [10] Blekhman II. Synchronization in nature and techniques. Moskou: Nanka; 1981. [11] Pecora LM, Caroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:812–24. [12] Oppenheim AV, Wornell GW, Isabelle SH, Cuoma K. Signal processing in the context of chaotic signals. Proceedings of the international conference on acoustic, speech and signal processing, vol. 4. New York: IEEE; 1992. p. 117–20. [13] Kovarev LJ, Halle KS, Eckert K, Parlitz U, Chua LO. Experimental demonstration of secure communications via chaotic synchronization. Int J Bifurc Chaos 1992;2:709–13. [14] Perez G, Cerdeira HA. Extracting messages masked by chaos. Phys Rev Lett 1995;74:1970–3.