Synchronized states in a ring of four mutually coupled ...

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Communications in Nonlinear Science and Numerical Simulation 11 (2006) 186–202 www.elsevier.com/locate/cnsns

Synchronized states in a ring of four mutually coupled self-sustained electromechanical devices R. Yamapi *, P. Woafo Laboratoire de Me´canique, Faculte´ des Sciences, Universite´ de Yaounde´ I, B.P. 812, Yaounde´, Cameroun Received 4 May 2004; received in revised form 6 August 2004; accepted 6 August 2004 Available online 14 October 2004

Abstract In this paper, we study the dynamics of a ring of four mutually coupled identical self-sustained electromechanical devices both in their autonomous and nonautonomous chaotic states. The transition boundaries that can occur between instability and complete synchronization states when the coupling strength varies are derived. Numerical simulations are then performed to support the accuracy of the analytical approach. 2004 Elsevier B.V. All rights reserved. PACS: 05.45.Gg; 05.45.Ac; 04.45.Pq Keywords: Synchronization; Stability; Electromechanical system

1. Introduction The dynamics and synchronization of rings and networks of mutually coupled identical and nonidentical nonlinear oscillators is an interesting subject. The great interest devoted to such topics is due to the collective behavior in physics (Josephson junctions, granular hydrodynamics), chemistry (discrete reaction-diﬀusion systems) and biology (development of living organisms, collective dynamics of biological cells aggregates). Recent studies on these mutually coupled systems *

Corresponding author. Tel.: +237 932 9376; fax: +237 222 6275. E-mail addresses: [email protected] (R. Yamapi), [email protected] (P. Woafo).

1007-5704/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2004.08.001

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have shown various dynamical behaviors [18,11,12,1,7,4–6,16,17,14,8], such as spatiotemporal chaos, cluster and complete synchronization. The study of diﬀerent dynamical states of a series of mutually coupled excited nonlinear oscillators have been investigated recently by Chembo Kouomou and Woafo [5]. They considered a ring of diﬀusely coupled single-well Duﬃng oscillators and determined analytically various transitions that can occur between the three possible dynamical states of the ring (spatiotemporal chaos, cluster synchronization, complete synchronization) and instability when the coupling strength varied. A local injection scheme is applied to synchronize the ring with an external master oscillator. The shift-invariance symmetry is therefore broken, leading to the emergence of generalized correlated states. The transition boundaries from the correlated states to spatiotemporal chaos and complete synchronization states in a shift-invariant set of mutually coupled nonlinear oscillators are been also investigated recently by Chembo Kouomou and Woafo [6]. They found that the underlying mechanism of these transitions relies on the motion of the representative points corresponding to the systemÕs nondegenerated spatial transversal Fourier modes in the parametric Strutt diagram. Another ﬁeld where synchronization is of crucial importance is automation engineering where more devices work in a synchronization manner are required both in regular and chaotic states. In this ﬁeld, electromechanical devices with natural and created (or introduced) nonlinearities are common and can lead to various types of behavior. In the context of synchronization of two or more nonlinear electromechanical systems with one-way coupling, our recent contributions have mainly focussed on the problem of synchronizing two nonlinear electromechanical devices both to the regular and chaotic states [19]. Stability boundaries are derived through Hill equations and Floquet theory. The inﬂuence of the precision on the synchronization time is also analyzed using numerical simulation of the equations of motion. This paper considers the problem of synchronizing a ring of mutually coupled identical self-sustained electromechanical devices (a sort of network of self-sustained electromechanical devices) both to the autonomous and nonautonomous chaotic states. Each electromechanical device is described by a system consisting of an electrical Rayleigh–Duﬃng oscillator magnetically coupled to a linear mechanical oscillator. Our study uses the Hill equation and the Floquet theory [10,13] to derive the transition boundaries between diﬀerent dynamical states. The paper is organized as follows. After presenting the physical model in the next section, we identify analytically in Section 3 various transitions between diﬀerent dynamical behaviors which occur in the ring of mutually coupled identical autonomous devices. In Section 4, we extend the study to a ring of mutually coupled nonautonomous chaotic self-sustained electromechanical devices. For this aim, we present various bifurcations sequences which lead each device to the chaotic motion before analyzing various dynamical states which occur in the ring of mutually coupled self-sustained models. Section 5 is devoted to the conclusion.

2. Description of the model and equations The electromechanical device shown in Fig. 1(a) is a ring of N mutually coupled identical selfsustained electromechanical transducers. Each self-sustained electromechanical device (Fig. 1(b)) is composed of an electrical part magnetically coupled to a mechanical part (as we have described recently, see [20,19]). The electrical part of the system consists of a capacitor C, an inductor L and

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Fig. 1. Schematic of a series of mutually coupled self-sustained electromechanical devices. (a) A ring of mutually coupled self-sustained electromechanical devices. (b) vth self-sustained electromechanical device.

a nonlinear resistor NLR, all connected in series. In the model, we have two types of nonlinear components: The voltage of the condenser is a nonlinear function of the instantaneous electrical charge qm for the mth unit and is expressed by V mC ¼

1 q þ a3 q3m ; C0 m

where C0 is the linear value of C and a3 is a nonlinear coeﬃcient depending on the type of the capacitor in use. This is typical of nonlinear reactance components such as varactor diodes widely used in many areas of electrical engineering to design for instance parametric ampliﬁers, up-converters, mixers, low-power microwave oscillators, etc [15]. The current–voltage characteristics of a resistor for the mth unit is also deﬁned as " # 3 im im ; þ V mR0 ¼ R0 i0 i0 i0

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where R0 and i0 are respectively the normalization resistance and current. The presence of such a nonlinear resistor in our electromechanical system confers to the mechanism of self-sustained oscillations. This nonlinear resistor can be realized using a block consisting of two transistors [9] and has been used recently by Chedjou et al. [3] for the dynamics of self-sustained electromechanical transducer. The mechanical part is composed of a mobile beam which can move along the ~ zm axis on both sides. The rod T which has the similar motion is bound to a mobile beam with a spring. The coupling between N identical self-sustained electromechanical transducers through electrical part is ensured by a capacitor Cm (high-pass oscillators), but can also be done with an inductor Lm (low-pass oscillators). It can easily be shown that the ring is described by the following set of coupled diﬀerential equations: " 2 # d2 qm 1 dqm dqm qm dzm 1 þ a3 q3m þ lBm ðqmþ1 2qm þ qm1 Þ; þ ¼ L 2 R 1 2 C ds ds ds C ds i0 0 m ð1Þ 2 d zm dzm dq þ kzm lBm m ¼ 0; m 2 þk ds ds ds where l is the length of the electrical wire inside the magnetic ﬁeld Bm. Let us use the dimensionless variables xm ¼

qm ; q0

ym ¼

zm ; l

t ¼ we s;

a0 ¼

w2e q20 i20

where q0 is a reference charge of qm and b¼

a3 q20 ; Lw2e

w2e ¼

l¼

R ; Lwe

k1 ¼

1 ; LC 0

w2m ¼

l2 Bm ; Lq0 we

k2 ¼

k ; m

w2 ¼

Bm q0 ; mwe

wm ; we

c¼

k : mwe

Thus the set of diﬀerential equations (1) reduces to the following set of Rayleigh–Duﬃng oscillators coupled to linear oscillators €x1 lð1 a0 x_ 21 Þ_x1 þ x1 þ bx31 þ k1 y_ 1 ¼ Kðx2 2x1 þ xN Þ; €y 1 þ cy_ 1 þ w22 y 1 k2 x_ 1 ¼ 0; .. . €xm lð1 a0 x_ 2m Þ_xm þ xm þ bx3m þ k1 y_ m ¼ Kðxmþ1 2xm þ xm1 Þ; €y m þ c_y m þ w22 y m k2 x_ m ¼ 0; .. . €xN lð1 a0 x_ 2N Þ_xN þ xN þ bx3N þ k1 y_ N ¼ Kðx1 2xN þ xN 1 Þ; €y N þ c_y N þ w22 y N k2 x_ N ¼ 0:

ð2Þ

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The dot over the quantities denotes the time derivative. l and a0 are two positive coeﬃcients, c the damping coeﬃcient, w2 the natural frequency, b the nonlinearity coeﬃcient, k1 and k2 are the coupling coeﬃcients. For mathematical convenience, we set a0 = 1 in the rest of the paper. As we have mentioned recently [20,19], each self-sustained electromechanical model is widely encountered in various branches of electromechanical engineering. But for some technological exploitation of such device, it is interesting in industry (for automation engineering) to use many identical electromechanical devices coupled mutually as shown in Fig. 1(a). For instance, it is a model of parallel operating system in manufacturing processes: a series of perforation devices or a series of macro/microelectromechanical devices uses for cutting, drilling and other machining works. One would like to take advantage of nonlinear responses of the ring of mutually coupled self-sustained electromechanical devices in manufacturing processes. By varying the coupling coeﬃcient K, one can ﬁnd various dynamical states such as clusters and complete synchronization and even instability. In the complete synchronization state, all the rods T of the mechanical oscillators vibrate in phase. For our analysis, we ﬁx N to 4. 3. Synchronization analysis Our objective in this section is to identify various dynamical states which appears in the ring of mutually coupled self-sustained electromechanical devices depending on the coupling strength K. When they are coupled mutually, the model is physically interesting so long as they are stable. In the literature, several diﬀerent tools to investigate the stability of the synchronization dynamics are proposed, such as the one (among other) developing by Butcher and Sinha [2]. But we prefer the Floquet theory [13] which is more indicated to ﬁnd the stability boundaries in the ring. Thus, the stability of the resulting dynamical states can be analyzed through the linearization of Eqs. (2) around the state (x, y) as follows €1 lð1 3_x2 Þ_1 þ ½1 þ 3bx2 1 þ k1 g_ 1 ¼ Kð2 21 þ 4 Þ; € g1 þ w22 g1 þ cg_ 1 k2 _ 1 ¼ 0; €2 lð1 3_x2 Þ_2 þ ½1 þ 3bx2 2 þ k1 g_ 2 ¼ Kð3 22 þ 1 Þ; € g2 þ w22 g2 þ cg_ 2 k2 _ 2 ¼ 0; €3 lð1 3_x2 Þ_3 þ ½1 þ 3bx2 3 þ k1 g_ 3 ¼ Kð4 23 þ 2 Þ;

ð3Þ

€ g3 þ w22 g3 þ cg_ 3 k2 _ 3 ¼ 0; €4 lð1 3_x2 Þ_4 þ ½1 þ 3bx2 _4 þ k2 g_ 4 ¼ Kð1 24 þ 3 Þ; € g4 þ w22 g4 þ cg_ 4 k2 _ 4 ¼ 0; where (k, gk) stands for the perturbations of the state (x, y) and is parametrically excited by the state x. We can replace in this ﬁrst approximation the dynamical state (x, y) of the parametric excitation by the state variable (x0, y0) which represents the dynamics of the uncoupled self-sustained electromechanical devices. Therefore, the unperturbed state variable (x0, y0) is deﬁned as x0 ¼ A cosðwt /1 Þ; y 0 ¼ B cosðwt /2 Þ;

ð4Þ

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where A and B are the amplitudes while /i (i = 1, 2) are the phases. Using the averaging method [10,13], it is found that the amplitudes A and B (see [21]) in the internal resonant state w2 = 1, are the solutions of the following nonlinear equations b6 A6 þ b4 A4 þ b2 A2 þ b0 ¼ 0;

ð5Þ

B2 ¼ MA2 ð4 3A2 Þ; with the coeﬃcients bi and M deﬁned as follows: M¼

l1 k2 ; 4ck1 w22

b6 ¼

27 Mk22 w22 b2 27M 3 k21 c2 w62 ; 16

9 b4 ¼ Mk22 w22 b2 9M 2 k22 w42 k21 18M 2 k1 k2 w42 c2 þ 81M 3 c2 k21 w62 ; 4 b2 ¼ 24M 2 k22 k21 w42 þ 16M 2 k2 k1 c2 w42 3c2 w22 Mk22 6Mk32 k1 w22 þ 48M 3 k21 c2 w62 ; b0 ¼ 4Mk22 c2 w22 þ 8Mk32 k1 w22 16M 2 k22 k21 w42 þ 64M 3 k21 c2 w62 k42 : Using the Newton–Raphson algorithm and the following set of parameters: l = 0.1, b = 0.5, k1 = 0.08, k2 = 0.4, c = 0.1 and w = 0.64 (the value of w resorted to numerical simulation), one ﬁnds that the real positive solutions are A = 0.85 and B = 1.006. In this ﬁrst approximation, we replace (x, y) in Eqs. (2) by the expressions (3) and introduce the following diagonal variables or Fourier modes (Pk, Qk) deﬁned as P 1 ¼ 1 þ 2 þ 3 þ 4 Q1 ¼ g1 þ g2 þ g3 þ g4 P 2 ¼ 4 2 ¼ x4 x2 Q2 ¼ g 4 g 2 ¼ y 4 y 2

ð6Þ

P 3 ¼ 3 1 ¼ x3 x1 Q3 ¼ g 3 g 1 ¼ y 3 y 1 ; P 4 ¼ 4 3 þ 2 1 ¼ x4 x3 þ x2 x1 ; Q4 ¼ g4 g3 þ g2 g1 ¼ y 4 y 3 þ y 2 y 1 : The variational equations (3) take the form P€ k þ ½2k þ F ðsÞP_ k þ Gk ðsÞP k þ k1 Q_ k ¼ 0; € k þ w2 Qk þ cQ_ k k2 P_ k ¼ 0; Q 2

k ¼ 1; 2; 3; 4;

ð7Þ

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where

3 2 2 w A 1 ; 2 3 F ðsÞ ¼ lw2 A2 cosð2ws 2/1 Þ; 2 3 G1 ðsÞ ¼ 1 þ bA2 cosð2ws 2/1 Þ; 2 3 G2 ðsÞ ¼ G3 ðsÞ ¼ 1 þ 2K þ bA2 cosð2ws 2/1 Þ; 2 3 2 G4 ðsÞ ¼ 1 þ 4K þ bA cosð2ws 2/1 Þ: 2 Floquet theory [10,13] states that depending on the coupling coeﬃcient K (since only the eﬀect of the coupling parameter K on the dynamics of the mutually coupled self-sustained electromechanical devices is studied), (Pk, Qk) may either indeﬁnitely grow to inﬁnity or decay to zero. Setting the following rescalings: k 3 2 s þ lA w sinð2ws 2/Þ ; uk ðsÞ ¼ P k exp 2 8 ð8Þ c vk ðsÞ ¼ 2 exp s ; 2 Eq. (7) can be written in the form k¼l

€ uk þ ½dk þ 211 sinð2ws 2/Þ þ 212 cosð2ws 2/Þ þ 213 cosð4ws 4/Þuk ck1 þ k1 v_ k þ vk expðwðsÞÞ ¼ 0; 2 €vk þ d21 vk þ ððd22 þ 221 cosð2ws 2/ÞÞuk k2 u_ k Þ expðwðsÞÞ ¼ 0; where the parameters dij, w(s) and ij are given by 3 3k2 135 4 2 4 w l1 A ; d0 ¼ 1 þ bA2 0 2 4 4 d1 ¼ d0 ; d2 ¼ d3 ¼ d0 þ 2K; d4 ¼ d0 þ 4K; c2 3 k0 w2 lA2 ; 4 4 81 3 3 13 ¼ w4 l21 A4 ; 12 ¼ b1 A2 w2 l1 A2 ; 64 4 4 2 1 3 c c 11 ¼ w l1 A2 ; d21 ¼ w22 þ ; 2 4 2 3 2 k k 0 2 ; 21 ¼ w l1 k2 A2 ; d22 ¼ 8 2 1 3 wðsÞ ¼ ðc k0 Þs l1 wA2 sinð2ws 2/Þ: 2 4 d22 ¼ w22

ð9Þ

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According to the Floquet theory, the solutions of Eqs. (9) are uk ðsÞ ¼ expðh1 sÞaðsÞ ¼

n¼þ1 X

an expðan sÞ;

n¼1

vk ðsÞ ¼ expðh2 sÞbðsÞ ¼

n¼þ1 X

ð10Þ bn expðbn sÞ;

n¼1

where an = h1 + 2inw, bn = h2 + 2inw, and the functions a(s) = a(s + p) and b(s) = b(s + p) replace the Fourier series. The quantities h1 and h2 are two complex numbers, while an and bn are real constants. Inserting Eqs. (10) into Eqs. (9), and equating each of the coeﬃcients of the exponential functions to zero yields the following homogeneous equations for the am and bm coeﬃcients ðdk þ a2m Þam þ ðð12 þ i11 Þ expð2i/1 ÞÞamþ1 þ ð12 i11 Þ expð2i/1 Þam1 þ 13 expð4i/1 Þm2 ck1 þ 13 expð4i/1 Þamþ2 þ k1 bm þ expðwðsÞÞbm ¼ 0; 2 ðd21 þ b2m Þbm þ expðwðsÞÞðd22 k2 am Þam 21 expðwðsÞÞ expð2i/1 Þam1 21 expðwðsÞÞ expð2i/1 Þamþ1 ¼ 0:

ð11Þ

For the nontrivial solutions, the determinant of the matrix in Eqs. (11) must vanish. Since the determinant is inﬁnite, we divide the ﬁrst and second expressions of Eqs. (11) by (dk 4m2) and (d21 4m2) respectively for convergence considerations. When ij are small, approximate solutions can be obtained considering only the central rows and columns of the HillÕs determinant. The small Hill determinant for this case has the six rows and six columns. Thus, in the ﬁrst order, the above Eqs. (11) may have solutions if and only if the associated HillÕs determinant is set equal to zero. This condition deﬁnes the boundary dividing the parameters space in two domains: the stability and the instability ones. Thus, limiting ourselves to the central rows and columns of the HillÕs determinant, we ﬁnd that the boundary separating stability to instability domains is given by Dk ðh1 ; h2 Þ ¼ ðD12 D21 D22 D11 Þ ½ðD56 D63 þ D66 D53 ÞðD34 D45 D44 D35 Þ þ ðD56 D65 þ D66 D55 Þ ðD33 D44 D43 D34 Þ D12 D56 D31 D23 D44 D65 þ D22 D66 D31 D53 D44 D15 D22 D66 D31 D55 D13 D44 þ D22 D56 D31 D44 ðD13 D65 D15 D63 Þ D22 D66 D51 D13 ðD34 D45 D44 D35 Þ D22 D66 D51 D15 ðD33 D44 D43 D34 Þ þ D12 D66 D51 D23 ðD34 D45 D44 D35 Þ þ D12 D66 D31 D23 D44 D55 ¼ 0;

k ¼ 2; 3; 4; ð12Þ

with 2

D11 ¼ dk þ ðh1 2iwÞ ; D13 ¼ ð12 þ i11 Þ expð2i/1 Þ;

ck2 D12 ¼ expðwðsÞÞ k1 ðh2 2iwÞ þ ; 2 D15 ¼ 13 exp 4i/;

D21 ¼ expðwðsÞÞðd22 k2 ðh1 2iwÞÞ;

2

D22 ¼ d21 þ ðh2 2iwÞ ;

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D23 ¼ 21 expðwðsÞÞ exp 2i/1 ; D33 ¼ ds þ h21 ;

D31 ¼ ð12 i11 Þ expð2i/1 Þ; ck1 D34 ¼ expðwðsÞÞ k1 h2 þ ; 2

D35 ¼ expð2i/1 Þð12 þ i11 Þ; D44 ¼ d21 þ

h22 ;

D43 ¼ expðwðsÞÞðd22 k2 h1 Þ;

D45 ¼ 21 expðwðsÞÞ expð2i/1 Þ;

D51 ¼ 13 expð4i/1 Þ; 2

D55 ¼ ds þ ðh1 þ 2iwÞ ;

D53 ¼ ð12 i11 Þ expð2i/1 Þ; D56 ¼ expðwðsÞÞðk1 ðh2 þ 2iwÞ þ ck1 =2Þ;

D63 ¼ 21 expðwðsÞÞ expð2i/1 Þ;

D65 ¼ expðwðsÞÞðd22 k2 ðh1 þ 2iwÞÞ;

D66 ¼ d22 þ ðh2 þ 2iwÞ2 : The functions Pk(s) and Qk(s) are obtained as k 3 2 P k ðsÞ ¼ exp h1 s þ lwA sinð2ws 2/Þ ak ðsÞ; 2 8 c k Qk ðsÞ ¼ exp h2 s b ðtÞ: 2

ð13Þ

The Floquet theory states that the transition from stability to instability domains occurs in the two conditions: 1c k0 c • p-periodic transitions at h1 ¼ h1c 1 ¼ 2 and h2 ¼ h2 ¼ 2. 2c k0 c • 2p-periodic transitions at h1 ¼ h2c 1 ¼ i þ 2 and h2 ¼ h2 ¼ i þ 2.

Thus replacing hi by hici in Eqs. (12), we obtain for a ﬁxed value of k an equation which helps to determine the stability boundaries of the synchronization process and therefore diﬀerent dynamical states in the mutually coupled self-sustained electromechanical devices. We can now analyze through Eqs. (12) what happens in the ring when the coupling parameter K is continuously increased from 1 to +1. When K = 0, the devices are uncoupled and the diagonal variables (Pk, Qk) (k = 2, 3, 4) degenerate into (P1, Q1) which is stable (since it remains bounded as the time tends to inﬁnity). For K 5 0, the above Eq. (12) (for k = 2, 3, 4) help us to derive the range of the coupling parameter K at which the ring of mutually coupled self-sustained electromechanical devices work in a synchronizing manner. From our analytical considerations, the following results are observed. But, before identify various dynamical states which appear in the ring, let us mentioned all the stability boundaries resulting through Eqs. (12) when k takes the values 2, 3 and 4. Our investigations show that the boundaries of the stability analysis of the Fourier modes (Pk, Qk) for possible bifurcation mechanism correspond to the following selected ﬁve values of K as K a1 ¼ 1:4; K a2 ¼ 1:36; K a3 ¼ 0:36; K a4 ¼ 0:18 and K a5 ¼ 0:04 since Ds ðhic1 ; hic2 Þ vanish for these above values of K. To conﬁrm these boundaries of stability analysis deﬁned above, we have solved numerically equations (2) using the fourth-order Runge–Kutta algorithm with a time step Dt = 0.01 and the following set of initial conditions

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ðx1 ð0Þ; x_ 1 ð0Þ; y 1 ð0Þ; y_ 1 ð0ÞÞ ¼ ð4:0; 4:0; 0:0; 0:0Þ; ðx2 ð0Þ; x_ 2 ð0Þ; y 2 ð0Þ; y_ 2 ð0ÞÞ ¼ ð4:3; 4:3; 0:0; 0:0Þ; ðx3 ð0Þ; x_ 3 ð0Þ; y 3 ð0Þ; y_ 3 ð0ÞÞ ¼ ð4:2; 4:2; 0:0; 0:0Þ; ðx4 ð0Þ; x_ 4 ð0Þ; y 4 ð0Þ; y_ 4 ð0ÞÞ ¼ ð4:5; 4:5; 0:0; 0:0Þ: The set of parameters used is the one deﬁned before. With numerical simulation, we ﬁnd that the stability boundaries in the ring of mutually coupled self-sustained electromechanical devices are deﬁned as K n1 ¼ 1:31; K n2 ¼ 1:3; K n3 ¼ 0:33; K n4 ¼ 0:23; K n5 ¼ 0:001 and K n6 ¼ 95:4. Although the sixth value is not predicted by the mathematical analysis, the agreement between the analytical and numerical results is quite acceptable. The fact that this sixth value of K is not obtained analytically can be explained by the fact that the analytical boundaries derived from a truncated Hill determinant (see [20,19]) and by ignoring nonlinear terms in the variational equations, approximations that can be very crude for large K. The rest of this section is to identify through the numerical simulation of Eqs. (12) diﬀerent bifurcation mechanisms which appear in the ring when the coupling coeﬃcient K varies. For this aim, let us evaluate the ﬁnal values of the Fourier modes (Pk, Qk) and thus indicate various regions of the coeﬃcient K where synchronization is achieved. We note that the ring are synchronized if each (Pk, Qk) vanish. For a ﬁxed value of the coupling coeﬃcient K in each domain, one can view synchronization or no synchronization phenomena in the ring after plotting the behavior of (Pk, Qk) (k = 2, 3, 4) versus the time in Figs. 2–4. It appears numerically that for K 2]95.4; 1.31[[]1.3; 0.33][]0.23, 0[[[0.001; +1[, the three Fourier modes (P2, Q2), (P4, Q3) and (P4, Q4) are together in the stability domain and thus tend all to the ﬁxed point (0, 0) as the time increases. In this case, the ring of mutually coupled self-sustained electromechanical devices is in the complete synchronization state since all the self-sustained electromechanical devices display the same dynamics (e.g. phase-locked) and we have (x1, y1) = (x2, y2) = (x3, y3) = (x4, y4) since (P2, Q2) = (P3, Q3) = (P4, Q4) = (0, 0). The complete synchronization appears in Fig. 2 where the temporal variation of the Fourier modes (Pk, Qk) are plotted. On the other hand, there is no synchronization in the system for the region of K deﬁned as K 2 [1.31, 1.3[[[0.33; 0.23[, because (P2, Q2) 5 (0, 0), (P3, Q3) 5 (0, 0) and (P4, Q4) 5 (0, 0). The behavior of the Fourier modes has a bounded oscillatory states as it appears in Fig. 3. In the region K 2 ]1; 95.4], we ﬁnd that the Fourier modes (Pk, Qk) will grow indefinitely leading to the instability in the ring (see Fig. 4). This means that any perturbed trajectory in this domain leads each self-sustained electromechanical device to continuously drift away from its original limit cycle.

4. Extension of a ring of four mutually coupled nonautonomous chaotic devices The aim of this section is to extend our analysis to the ring of four mutually coupled nonautonomous chaotic self-sustained electromechanical devices. For this purpose, the equations of motion of the device is now described in the following manner (we remind that a0 = 1)

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Fig. 2. Temporal variation of the Fourier modes Pk showing complete synchronization states with the parameters l = 0.1, c = 0.1, b = 0.5, k1 = 0.08, k2 = 0.4, w2 = 1.0. (a) K = 2 and (b) K = 2.0.

Fig. 3. Temporal variation of the Fourier modes Pk in the unstable domains with the parameters of Fig. 2 and K = 0.2.

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Fig. 4. Temporal variation of the Fourier modes Pk in the unstable domains with the parameters of Fig. 2 and K = 95.6.

€x1 lð1 x_ 21 Þ_x1 þ x1 þ bx31 þ k1 y_ 1 ¼ E0 cos qt þ Kðx2 2x1 þ x4 Þ; €y 1 þ c_y 1 þ w22 y 1 k2 x_ 1 ¼ 0; €x2 lð1 x_ 22 Þ_x2 þ x2 þ bx32 þ k1 y_ 2 ¼ E0 cos qt þ Kðx3 2x2 þ x1 Þ; €y 2 þ c_y 2 þ w22 y 2 k2 x_ 2 ¼ 0; €x3 lð1 x_ 23 Þ_x3 þ x3 þ bx33 þ k1 y_ 3 ¼ E0 cos qt þ Kðx4 2x3 þ x2 Þ;

ð14Þ

€y 3 þ c_y 3 þ w22 y 3 k2 x_ 3 ¼ 0; €x4 lð1 x_ 24 Þ_x4 þ x4 þ bx34 þ k1 y_ 4 ¼ E0 cos qt þ Kðx1 2x4 þ x3 Þ; €y 4 þ c_y 4 þ w22 y 4 k2 x_ 4 ¼ 0; where E0 and q are respectively the amplitude and frequency of the external excitation. The objective of this extension is to ﬁnd if our analytical procedure can help to identify the synchronization states which appear in the mutually coupled chaotic devices. This chaotic state is particularly of interest when the ring of mutually coupled self-sustained electromechanical systems is used for industrial operations such as the mixing of diﬀerent liquids, chemicals or powders. But, before proceeding to the synchronization analysis, one ﬁrst consider the behavior of each nonautonomous self-sustained electromechanical device as the amplitude of the external excitation E0 varies. Chaos appears in the forced self-sustained electromechanical model with the following set of the physical parameters l = 4, c = 0.1, b = 1.32, k1 = 0.01, k2 = 0.06, w2 = 1.2, q = 5, E0 = 2.72. Figs. 5 and 6 show respectively the phase portrait of the forced chaotic self-sustained electromechanical model, a representative bifurcation diagram and the variation of the Lyapunov exponent (Lya) versus E0. The following results are observed. When E0 increases from zero to E0 = 5.0, the self-sustained electromechanical device exhibits a chaotic behavior in the region of E0 deﬁned

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Fig. 5. A chaotic phase portrait of each self-sustained electromechanical device with the parameters l = 4, E0 = 2.72, k1 = 0.01, k2 = 0.06, c = 0.1, q = 5, w2 = 1.2.

Fig. 6. Bifurcation diagram (a) and variation of the Lyapunuv exponent Lya (b) versus E0 with the parameters of Fig. 5.

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as E0 2]2.6; 2.69[[]2.71; 2.76[[]2.85; 2.9[[]2.96; 2.99[[]3.07; 3.15[ and E0 = 4.12. In other regions, we have quasi-periodic and periodic oscillations. To use the analytical procedure, we are obviously limited by the fact that a chaotic orbit is aperiodic and is composed of an inﬁnite number of Fourier component. However, we can assume that in the chaotic states, the virtual orbit is that obtained using the harmonic balance method and therefore deﬁned by Eqs. (4). Then, inserting the solutions (4) into the equations of motion of each nonautonomous self-sustained electromechanical device, and equating the coeﬃcients of sin qt and cos qt separately (assuming that the terms due to higher frequencies can be neglected), it is found that the amplitude of the virtual orbit is the solution of the following equations: 9 2 6 3 ðl q þ b2 ÞA6c þ ðbF þ lGq3 ÞA4c þ ðF 2 þ G2 ÞA2c E20 ¼ 0; 16 2 2 2 k A B2c ¼ 2 c ; D

Fig. 7. Idem as in Fig. 2 in the chaotic states with K = 8 for (a) and K = 5 for (b).

ð15Þ

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where D ¼ ðw22 q2 Þ2 þ c2 q2 ; k1 k2 q2 ðw22 q2 Þ F ¼ 1 q2 ; D k1 k2 cq3 : G ¼ lq þ D With the parameters deﬁned in Fig. 5, one can ﬁnd through the Newton–Raphson algorithm that the virtual orbit has the amplitude Ac = 0.092. Following this approximation, Eqs. (12) indicates that Ds ðhic1 ; dic2 Þ are vanish at the values of K deﬁned as K a1c ¼ 10:0; K a2c ¼ 9:45; K a3c ¼ 6:53; K a4c ¼ 5:51; K a5c ¼ 2:17; K 6c ¼ 4:3 and K a7c ¼ 0:75, which correspond to the stability boundaries of the Fourier modes (Pk, Qk). From the numerical simulation of Eq. (14), we ﬁnd the following stability boundaries K n1c ¼ 11:09; K n2c ¼ 9:19; K 3cn ¼ 7:59; K n4c ¼ 6:19; K 5cn ¼ 4:59; K n6c ¼ 4:1; K n7c ¼ 3:39; K n8c ¼ 0:2; K n9c ¼ 0:5; K n10c ¼ 37:8: The last three values are obtained only from numerical simulation and do not match with the analytical formalism. This can be explained as in Section 3, but also because of the approximation made by replacing a chaotic orbit by a mono frequency virtual periodic orbit.

Fig. 8. Idem as in Fig. 3 in the chaotic states with K = 10 for (a) and K = 7 for (b).

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Let us ﬁnd numerically what happens in the mutually coupled chaotic self-sustained electromechanical devices as the coupling parameters K varies. With the ﬁnal states of the Fourier modes (Pk, Qk), represented in Figs. 7 and 8, our investigations show that the complete synchronization in the series of mutually coupled self-sustained electromechanical devices requires that K 2]37.8; 11.09[[]9.19; 7.59[[]6.19; 4.59[[]4.59; 3.39][]3.39; 0.2[[]0.5; + 1[. The ring of mutually coupled self-sustained electromechanical devices is in the unstable area with K 2 [11.09, 9.19][[7.59, 6.19][[[0.2; 0.5]. In these regions, the Fourier modes (Pk, Qk) never go to zero but have a bounded oscillatory behavior as it appears in Fig. 8. We ﬁnd through this ﬁgure that thereÕs some range of the K parameter at which (P2, Q2) = (P3, Q3) 5 0 and (P4, Q4) 5 0. Indeed, for other types of coupled chaotic nonlinear model as the case of soft Duﬀing oscillators, one ﬁnds good agreement between the analytical and numerical results [4,5]. 5. Conclusion In summary, we have studied the dynamics and bifurcation behaviors in the ring of mutually coupled self-sustained electromechanical devices both in the autonomous and nonautonomous chaotic states. Each self-sustained electromechanical device is described by the system consisting of an electrical Rayleigh–Duﬃng oscillator magnetically coupled to a linear mechanical oscillator. The transition boundaries amongst dynamical states (instability and complete synchronization states) in the ring of mutually coupled self-sustained electromechanical devices have been derived using the Hill equations and the Floquet theory. The agreement between the analytical and numerical results is good. References [1] Belykh VN, Belykh IV, Mosekilde E. Cluster synchronization modes in an ensemble of coupled chaotic oscillators. Phys Rev E 2001;63:036216. [2] Butcher EA, Sinha SC. Symbolic computation of local stability and bifurcation surfaces for nonlinear timeperiodic systems. Nonlinear Dynam 1998;17:1–121. [3] Chedjou JC, Woafo P, Domngang S. Shilnikov chaos and dynamics of a self-sustained electromechanical transducer. J Vibrat Acoust 2001;123:170–4. [4] Chembo Kouomou Y, Woafo P. Stability and optimization of chaos synchronization through feedback coupling with delay. Phys Lett A 2002;298:18–28. [5] Chembo Kouomou Y, Woafo P. Generalized correlated states in a ring of coupled nonlinear oscillators with a local injection. Phys Rev E 2002;66:066201. [6] Chembo Kouomou Y, Woafo P. Transitions from spatiotemporal chaos to cluster and complete synchronization states in a shift-invariant set of coupled nonlinear oscillations. Phys Rev E 2003;67:046205. [7] De SE, Pinto S, Lopes SR, Viarra RL. Collective behavior in a chain of van der Pol oscillators with power-law coupling. Physica A 2002;303:339–56. [8] Endo T, Mori S. Mode analysis of a ring of a large number of mutually coupled Van der Pol oscillators. IEEE Trans Circuits Syst 1978;CAS-25(1):7. [9] Hasler MJ. Electrical circuit with chaotic behavior. Proc IEEE 1987;75:1009. [10] Hayashi C. Nonlinear Oscillations in Physical Systems. NY: McGraw-Hill; 1964. [11] Heagy JF, Caroll TL, Pecora LM. Synchronous chaos in coupled oscillator systems. Phys Rev E 1994;50:1874–95. [12] Heagy JF, Caroll TL, Pecora LM. Short wavelength bifurcations and size instabilities in coupled oscillator systems. Phys Rev Lett 1995;74:4185–8.

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