Physics Letters A 371 (2007) 48–57 www.elsevier.com/locate/pla

Active control of the synchronization manifold in a ring of mutually coupled oscillators René Yamapi a,∗ , Stefano Boccaletti b,c a Department of Physics, Faculty of Science, University of Douala, PO Box 24157 Douala, Cameroon b Embassy of Italy in Tel Aviv, 25 Hamered St., Tel Aviv, Israel c CNR-Istituto dei Sistemi Complessi, Largo E. Fermi, 6, 50125 Florence, Italy

Received 6 March 2007; received in revised form 14 May 2007; accepted 16 May 2007 Available online 12 June 2007 Communicated by C.R. Doering

Abstract We investigate the active control of synchronization dynamics in a shift-invariant ring of N mutually coupled self-sustained electrical systems. Using the master stability function approach, we derive the regime of coupling parameters leading to stable and unstable synchronization phenomena in the ring. The active control technique is applied on the mutually coupled systems to suppress undesired behavior, such as the unstable synchronization manifold. We derive the range of control gain parameters which leads to a successful control and the stable control design. The effects of the control or gain parameters on the stability boundaries of the synchronization process are also studied. © 2007 Elsevier B.V. All rights reserved. PACS: 05.45.Xt Keywords: Synchronization; Master stability function; Lyapunov exponents

1. Introduction The emergence of collective and synchronized dynamics in large networks of coupled units has been investigated since the beginning of the nineties in different context and in variety of fields, ranging from biology and ecology [1–3], to semiconductor lasers [4–7], to electronic circuits [8,9]. In particular, synchronization of mutually coupled systems is one of the most intensively investigated topic in nonlinear science [10–12]. Recent attention has been also directed to the analysis of synchronization of complex networks, to understand specific collective behaviors encountered in various physical and biological systems [13,14]. A crucial issue in the study of synchronization of mutually coupled systems is the assessment of the stability of the syn* Corresponding author. Mobile phone: +237 99 32 93 76; Tel./fax: +237 33 40 75 69. E-mail addresses: [email protected] (R. Yamapi), [email protected] (S. Boccaletti).

0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.05.112

chronization process. As most of the practical implementations of such a process require to have stable synchronization regimes for all values of the coupling parameters, it is important to introduce strategies to suppress unstable synchronization. One possibility that has recently been proposed consists in coupling each unit to an external control force with the aim of stabilizing the synchronization manifold [15]. In this Letter, we consider the stability of the controlled manifold in a ring of mutually coupled systems. Our aim is to derive the range of control gain parameters which leads to control and the stability of the control design. We first concentrate on different dynamical states which appear in the mutually coupled systems without control. For this aim, we use the master stability function approach to derive the variation of the transverse Lyapunov exponents versus the coupling coefficient. The active control technique is then applied to the mutually coupled systems to suppress undesired behavior such as unstable synchronization or to find the de-synchronization phenomena. The effects of the control or gain parameters in the control process are derived and the results are shown in the stability diagram.

R. Yamapi, S. Boccaletti / Physics Letters A 371 (2007) 48–57

The Letter is organized as follows. In the next section we describe a ring of N mutually coupled self-sustained electrical systems and derive the equations of motion. Some theory for the stability of the synchronous solution is given in Section 3 (master stability function approach). Attention is focussed on numerics of the stability analysis of the synchronization process of a shift-invariant ring of mutually coupled self-sustained electrical systems, and the types of behavior which appear on the shift-invariant ring. In Section 4, after presenting the model under the active control, we analyze the stability of the control process using the master stability function approach. We analyze the effects of the control parameters on the synchronization manifold. The Letter ends with a short concluding section. 2. Description of mutually coupled systems The model considered in our Letter is shown in Fig. 1. It consists of a shift-invariant ring of N mutually coupled identical self-excited electrical systems. Each self-excited system (see Fig. 2) consists of a capacitor C, an inductor L and a nonlinear resistor NLR, all connected in series. Two types of nonlinear components are considered: the voltage of the condenser is a nonlinear function of the instantaneous electrical

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charge qν , which for the νth unit is expressed by 1 (1) qν + a3 qν3 , C0 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. [16]. The current-voltage characteristics of a resistor for the νth unit is also defined as     3  iν iν + , VRν0 = R0 i0 − (2) i0 i0 Vcν =

where R0 and i0 are respectively, the normalization resistance and current, iν the value of current corresponding for the limit resistor voltage. In this case, the model has the property to exhibit self-excited oscillations. This is due to the fact that the model incorporates through its nonlinear resistance a dissipative mechanism to damp oscillations that are too large and a source of energy to pump up those that become too small. The presence of such nonlinear resistor in each electrical system gives rise to a mechanism of self-sustained oscillations. The coupling between the N identical self-sustained electrical model is realized through a capacitor Cm (high-pass coupling). When the N self-sustained electrical systems are interconnected as in Fig. 1, the Kirchhoff law of the voltage and the current enables to find that the kth self-sustained electrical system is described by the following equations Vk − Vk+1 = Vself ,

(3)

dVk , Ik−1 − Ik = Iself = C dτ which leads to 1 Vk = (qk−1 − qk ). Cm Substituting Eq. (5) into Eq. (4) we have 1 1 (qk−1 − qk ) − (qk − qk+1 ). Cm Cm Therefore, the ring is described by     d 2 qν qν 1 dqν 2 dqν L 2 − R0 1 − 2 + + a3 qν3 dτ1 C0 dτ1 i0 dτ1 Vself =

Fig. 1. A ring of mutually coupled self-sustained electrical systems.

1 (qν+1 − 2qν + qν−1 ), ν = 1, 2, . . . , N, Cm with the following dimensionless variables qν t = ω e τ1 , xν = , q0 =

and α0 = Fig. 2. Schema of each self-sustained electrical system.

β=

q02 we2 C02

a3 i02 , Lωe4

,

ωe2 = K=

1 , LC0

C0 . Cm

μ=

R0 , Lωe

(4)

(5)

(6)

(7)

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R. Yamapi, S. Boccaletti / Physics Letters A 371 (2007) 48–57

Eq. (7) gives rise to the following set of coupled non-dimensional differential equations   x¨ν − μ 1 − α0 x˙ν2 x˙ν + xν + βxν3 = K(xν+1 − 2xν + xν−1 ),

ν = 1, 2, . . . , N,

(8)

where xν describes the components of the νth self-sustained system and K is the coupling parameter. We fix α0 = 1, and introduce yν = x˙ν to obtain x˙ν = yν ,   y˙ν = μ 1 − yν2 yν − xν − βxν3 + K(xν+1 − 2xν + xν−1 ).

(9)

Eqs. (9) describe a set of N mutually coupled identical Rayleigh–Duffing equations. The great interest devoted to the model shown in Fig. 1 and described by Eqs. (8) is due to the fact that this model can be used to describe several phenomena and have applications in many areas. For instance, in electrical engineering, this model is used as a model of parallel operating system of microwave oscillators [17,18]. A computer study of the mutual synchronization of a large number of these oscillators can be used to model intestinal signal [19] or colorectal myoelectrical activity in humans [20]. This model can also be a reliable description of the control pattern generator that rules the rhythmic activity in invertebrates. 3. Synchronization analysis 3.1. The master stability function approach We identify in this subsection various dynamical states which appear in the shift-invariant ring, depending on the coupling strength K and the number N of self-sustained electrical systems. As we are interesting in bifurcations from synchronous limit cycle states; these states reside on a synchronization manifold defined by M = {x1 = x2 = · · · = xN ; y1 = y2 = · · · = yN }. The central goal is the ability to predict when the synchronized state defined by xk = xs (t); yk = ys (t), is stable ∀k. The stability of such synchronization manifold M will be the object of the study. To this purpose, we make use of the master stability approach [21,22]. Let Xi be the two-dimensional vector of the dynamical variables of the ith unit, H : R2 → R2 an arbitrary function describing the coupling between each unit variables. Thus, the dynamics of the ith unit is rewritten as a function of the 2 × N column vector state Xi as N      ˙ i = F Xi + K Gij H Xj , X

i = 1, 2, . . . , N,

(10)

j

where Xi = [xi , yi ]T , F(Xi ) = [yi , μ(1 − yi2 )yi − xi − βxi3 ]T , and the function H is defined through the matrix   0 0 E= 1 0

by H(Xi ) = EXi . Gij ∈ R are the elements of the N × N symmetry connectivity matrix G defined by ⎛ ⎞ −2 1 0 ... 1 ⎜ 1 −2 1 . . . 0 ⎟ ⎜ ⎟ 0 1 −2 . . . 0 ⎟ G=⎜ ⎜ . .. .. .. ⎟ .. ⎝ .. . . . . ⎠ 1 0 . . . 1 −2  and obeys to the zero row-sum condition: nj=1 Gij = 0 for all j . The synchronization manifold M is an invariant set, due to the zero row-sum condition of the coupling matrix G and due to the fact that the function H(Xi ) is the same for all shitinvariant ring units. Therefore, stability of the synchronous state reduces to take care of the system’s dynamical properties along directions in phase that are transverse to the synchronization manifold. The stability of the resulting dynamical states can be determined by letting xν = δxν + xs , yν = δyν + ys and linearizing Eqs. (10) around the state (xs , ys ). This leads to   ˙ = 1N ⊗ JF(Xs ) + KG ⊗ JH(Xs ) δX, δX (11) where ⊗ stands for the direct product between matrices, J denotes the Jacobian operator and the 2 × N column vector δXi = (δxi , δyi ) is the deviation of the ith vector state from the synchronization manifold. We remind that H(Xi ) = EXi , and J H = E. As it appears in the literature, a necessary condition for stability of the synchronization manifold [21,22] is that the set of (N − 1)∗ 2 Lyapunov exponents that corresponds to phase space directions transverse to the 2-dimensional hyperplane Xi = Xs be entirely made of negative values. Eq. (11) is used to calculate Lyapunov exponents. Since Eqs. (11) are high-dimensional, the synchronization problem is to reduce this high-dimensional set of variational equations, governing the stability of the synchronous state, to a more manageable, low-dimensional set. Let γi and vi be the set of real eigenvalues and the associated orthonormal eigenvectors of the matrix G respectively, such that Gvi = γi vi and vjT . vi = δij . The arbitrary state δX  can be written as δX = N i=1 vi ⊗ ξi (t) with ξi (t) = (ξi,1 , ξi,2 ). If one applies vjT to the left side of each of equations (11), one finally obtains the following set of N variational equations   ξ˙k = JF(Xs ) + Kγk JH(Xs ) ξk , k = 0, 1, 2, . . . , N − 1. (12) We recall that γk is the eigenvalue of G, and is given by γk = 4 sin2 (πk/N ) for the diffusive coupling [22]. One notes that each equation in (12) corresponds to a set of 2 conditional j Lyapunov exponents λk (j = 1, 2) along the eigenmode corresponding to the specific eigenvalue γk . For k = 0, we have the variational equation for the synchronization manifold (γ0 = 0) = 0 is that of the and its maximum Lyapunov exponent λmax 0 isolated dynamical unit. The remaining variations ξk , k = 1, 2, . . . , N − 1, are transverse to M, and describe the system’s response to small deviations from the synchronization manifold, and then control the stability of the synchronized state. Any deviation from the

R. Yamapi, S. Boccaletti / Physics Letters A 371 (2007) 48–57

Fig. 3. Variation of the maximum transverse Lyapunov exponents versus the coupling coefficient K for the hardening case. The parameters used are μ = 0.1; β = 0.5.

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Fig. 4. The stability diagram of the synchronization process in the plan (N, K) for the hardening case. The parameters used are μ = 0.1; β = 0.5. (US) Unstable domain and (SS) stable synchronization domain.

synchronization manifold will be reflected in the growth of one or more of these variations. We note that the stability of the synchronized state is ensured if arbitrary small transverse variations decay to zero, this means that all the transverse Lyapunov exponents are negative. For k = 0, Eqs. (12) enables to calcuof each mode k as late the maximum Lyapunov exponent λmax k the function of the coupling parameter K. In the following subsection we report the numerical computation of the Lyapunov exponents. For this purpose, a fourthorder Runge–Kutta algorithm is used to solve the equations of motion with the set of the parameters μ = 0.1, β being positive or negative. 3.2. Hardening case In this subsection, we consider the hardening case, which correspond to a sufficiently positive cubic nonlinearity (β > 0). When the coupling coefficient is turned off (i.e. K is equal to zero), the self-sustained electrical systems are uncoupled. The = 0. corresponding maximum Lyapunov exponents are λmax 0 For K = 0, and for a given value of N , the maximum Lyapunov exponents will enable us to derive the range of the coupling parameter K in which the transverse Fourier modes are stable, and then each of a group of systems of the ring of mutually self-sustained electrical systems works in a synchronizing manner. The variation of the Lyapunov exponents are derived and plotted on Fig. 3. As K increases, we find the following domains of dynamical states: The first one is the unstable domain in which there is some positive Lyapunov exponent and hardly depends on the number N as DUS (N = 4) = ]−∞; −0.447[ ∪ ]−0, 39; 0[, DUS (N > 5) = ]−∞; 0[, in which no synchronization is observed, all the modes are on the transverse manifold where variations transverse to the synchronization manifold do not decay with the time since all the > 0. In the transverse Lyapunov exponents are positive i.e. λmax k other words, the Fourier modes increase continuously or pos-

Fig. 5. Space–time-amplitude plot showing unstable synchronization in the ring for the hardening case. The parameters used are μ = 0.1; β = 0.5.

sess a bounded oscillatory behavior with increasing time. In the second one, defined as DSS (N = 4) = ]−0.447; −0.39[ ∪ ]0; +∞[, DSS (N > 5) = ]0; +∞[, the synchronization process is observed and all the Lyapunov exponents are negative. In this case, all the modes move from the unstable domain to the stable one, the ring is synchronized and stable synchronization are observed. Fig. 4 displays the two main dynamical states and the stability boundaries. We find that the number of units hardly affects the stability boundaries of the synchronization process in the ring. In particular when N > 5, there is no domain of stable synchronization for negative values of the coupling coefficient K. In order to illustrate our results, Figs. 5 and 6 show space– time-amplitude diagrams that display some behavior occurring for some values of the coupling parameter K chosen in the unstable or no-synchronization (US) and stable synchronization (SS) domains. For the unstable domain, Fig. 5 shows that for (N, K) = (30, −0.2), no synchronization is found in the ring. Considering the domain of stable synchronization, we find in Fig. 6 for (N, K) = (30, 1) and (N, K) = (30, 2) that stable synchronization is possible in the ring.

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Fig. 7. Variation of the maximum transverse Lyapunov exponents versus the coupling coefficient K for the softening case. The parameters used are μ = 0.1; β = −0.1.

Fig. 6. Space–time-amplitude plot showing stable synchronization in the ring for the hardening case. The parameters used are μ = 0.1; β = 0.5.

3.3. Softening case The softening case corresponds to sufficiently negative cubic nonlinearity (β < 0). In the following we will concentrate on the value β = −0.1. We find the following dynamical states in the shift-invariant ring, which now have some dependence on the number of units. As soon as the number of oscillators in the ring increases, one still observes the two main above mentioned dynamical states but for different ranges of K. For some value of N , the unstable domains of the synchronization process are: DUS (N = 4) = ]−∞; −0.209[ ∪ ]0; 0.094[, DUS (N = 10) = ]−∞; −0.200[ ∪ ]0; 0.500[, DUS (N = 15) = ]−∞; −0.210[ ∪ ]0; 1.090[, while the stable ones are DSS (N = 4) = [−0.209; 0[ ∪ [0.094; +∞[, DSS (N = 10) = [−0.200; 0[ ∪ [0.500; +∞[, DSS (N = 15) = ]−0.210; 0[ ∪ [1.090; +∞[. Fig. 7 shows the variation of the transverse Lyapunov expoversus the coupling parameter K and the regimes of nents λmax k coupling leading to unstable and stable synchronization can be found. Fig. 8 shows the stability diagram of the synchronization process in the plane (N, K) for the dynamics of a single sufficiently softening unit. The effects of the number of units are found on the figure. One can notice that as N increases, the

Fig. 8. The stability diagram of the synchronization process in the plan (N, K) for the softening case. The parameters used are μ = 0.1; β = −0.1. (US) Unstable domain and (SS) stable synchronization domain

domain of stable synchronization reduces drastically in the region of positive K. Figs. 9 and 10 show space–time-amplitude diagrams that display stable and unstable synchronization respectively. 4. Active control and its stability 4.1. Mathematical formulation The main outcome of the previous analysis is that the synchronization process is stable or unstable, depending upon the coupled coefficient K. It is then interesting to study how unstable synchronization can be controlled, and devise an active control strategy to solve the problem. To illustrate that, we consider the following equations modelling the controlled system   x¨ν − μ 1 − x˙ν2 x˙ν + xν + βxν3 + λ1 u˙ ν = K(xν+1 − 2xν + xν−1 ), u¨ ν + u˙ ν + uν − λ2 x˙ν = 0,

ν = 1, 2, . . . , N,

(13)

where uν is the control force of the νth self-sustained electrical system, λi the control gain parameters. Practically, this type of control can be carried out in engineering by making use of a mechanical system, putting together to the original self-sustained

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we remind that we are interesting in bifurcations from synchronous states; which reside on a synchronization manifold defined by M = {x1 = x2 = · · · = xN ; y1 = y2 = · · · = yN ; u1 = u2 = · · · = uN ; v1 = v2 = · · · = vN }. The master stability approach [21,22] can be also used to find the appropriate range of the control gain parameters. For this purpose, let Yi be the four-dimensional vector of the dynamical variables of the ith unit. Eqs. (14) can be rewritten as N      ˙ i = F Yi + K Gij H Yj , Y

i = 1, 2, . . . , N,

(15)

j

where Yi = [xi , yi , ui , vi ]T , F(Yi ) = [yi , μ(1 − yi2 )yi − xi − βxi3 − λ1 vi , vi , −vi − ui + λ2 yi ]T . As in the above section, the stability of the control reduces to take care of the system’s dynamical properties along directions in phase that are transverse to the synchronization manifold. Thus, the stability of the resulting dynamical states can be determined by letting xν = δxν + xs , yν = δyν + ys , uν = δuν + us , vν = δvν + vs and linearizing Eqs. (15) around the state (xs , ys , us , vs ). This leads to   ˙ = 1N ⊗ JF(Ys ) + KG ⊗ JH(Ys ) δY, δY (16)

Fig. 9. Space–time-amplitude plot showing unstable synchronization in the ring for the softening case. The parameters used are μ = 0.1; β = −0.1.

Fig. 10. Space–time-amplitude plot showing stable synchronization in the ring for the softening case. The parameters used are μ = 0.1; β = −0.1.

electrical system. The original self-sustained electrical system and the mechanical system are coupled to the electromagnetic force to form a self-sustained electromechanical system such as those presented in Ref. [23]. Introducing vk = u˙ k , Eqs. (13) yield to x˙ν = yν ,   y˙ν = μ 1 − yν2 yν − xν − βxν3 − λ1 vν + K(xν+1 − 2xν + xν−1 ), u˙ ν = vk , v˙ν = −vν − uν + λ2 yν ,

ν = 1, 2, . . . , N,

(14)

where the 4 × N column vectors δYi = (δxi , δyi , δui , δvi ) is the deviation of the ith vector state from the controlled synchronization manifold. We remind that H is also defined through the matrix E, and J H = E with ⎛ ⎞ 0 0 0 0 ⎜1 0 0 0⎟ E=⎝ ⎠. 0 0 0 0 0 0 0 0  The arbitrary state δY can be written as δY = N i=1 vi ⊗ ηi (t) with ηi (t) = (ηi,1 , ηi,2 , ηi,3 , ηi,4 ). If one applies vjT to the left side of each of equations (16), one finally obtains the following set of N variational equations   η˙ k = JF(Ys ) + Kγk JH(Ys ) ηk , k = 0, 1, 2, . . . , N − 1. (17) As γk is the eigenvalue of G, it is given by γk = 4 sin2 (πk/N ) for the diffusive coupling [22], and each equation in (17) corj responds to a set of 4 conditional Lyapunov exponents λk (j = 1, 2, 3, 4) along the eigenmode corresponding to the specific eigenvalue γk . For λi = 0, Eqs. (17) enables to calculate of each mode k as the the maximum Lyapunov exponent λmax k function of the coupling gain parameter λi . 4.2. Effects of the control We here analyze the effects of the control on the synchronization manifold. It is important to note that the choice of K can been find in the unstable or stable region for the uncontrolled ring of self-sustained electrical systems. Several values of K will be used. It is also important to notice that, when the value of K is chosen in the unstable synchronization domain, the role of the active control is to ensure that all arbitrary small

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Fig. 12. The stability boundary of the control process in the plan (λ1 , λ2 ) for the hardening case. The parameters used are μ = 0.1; β = 0.5, K = −0.2. SC: Domain of stable control and UC: Domain of unstable control.

Fig. 11. Effects of the gain parameter λ2 on the variation of the transverse Lyapunov exponents versus the coefficient λ1 for the hardening case. The parameters used are μ = 0.1; β = 0.5, K = −0.2.

transverse variations decay to zero, and consequently the controlled system developed a stable synchronization. On the other hand, for values of K in the stable domain (the stable synchronization is observed in the uncontrolled system), we need to verify whether the active control has a positive or a negative effect on the synchronization manifold, or find if the gain parameter introduces de-synchronization phenomena in the origin system. Fig. 11 shown the variation of the maximum transverse Lyapunov exponents as function of the coupling parameter λ1 for a fixed value of λ2 , while Fig. 12, the stability boundary of the control process in the plan (λ1 , λ2 ) for the hardening case. We find on these figures that as λ1 increases, the transverse Lyapunov exponents are more negative, and the active control process of the synchronization manifold is better.

Fig. 13. Space–time-amplitude plot showing unstable control for the hardening case. The parameters used are μ = 0.1; β = 0.5; K = −0.2.

To illustrate our results of the control strategy, Figs. 13 and 14 show space–time-amplitude diagrams that display some behaviors found for various values of the coupling parameter K chosen in the unstable or no-control (UC) and stable control (SC) domains. For the unstable domain, Fig. 13 shows that for (N, K, λ1 λ2 ) = (30, −0.2, 2, 0.1) and (N, K, λ1 λ2 ) = (30, −0.2, 0.1, 0.5), no control is found in the ring. Considering the domain of stable control, we find in Fig. 14 for (N, K, λ1 λ2 ) = (30, −0.2, 1, 6) and (N, K, λ1 λ2 ) = (30, −0.2, 2, 4) that stable control is possible in the ring. In the case in which K is chosen in the unstable domain of synchronization, for example (K, N ) = (−0.2, 10), let us find the variation of the maximum Lyapunov exponents λmax k

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Fig. 14. Space–time-amplitude plot showing stable control for the hardening case. The parameters used are μ = 0.1; β = 0.5; K = −0.2.

as function of control gain parameters. Fig. 15 shows the variaversus λ1 for tion of the maximum Lyapunov exponent λmax k several different values of the coefficient λ2 . It appears that starts at a positive value as the coefficient λ1 increases, λmax k when λ1 = 0, since the uncontrolled shift-invariant ring of mutually coupled self-sustained electrical systems are in the unstable synchronization states ((K, N ) = (−0.2, 10)), the control decreases from there as process is unstable. The value of λmax k λ1 increases and leads to the stable control process when all the are negative. The boundmaximum Lyapunov exponents λmax k aries of stable domain of the control process depend on the gain parameter λ2 as DSC (N = 10, λ2 = 0.1) = [12.8; +∞[, DSC (N = 10, λ2 = 1.0) = [0.74; +∞[, DSC (N = 10, λ2 = 2.0) = [0.30; +∞[, DSC (N = 10, λ2 = 3.0) = [0.21; +∞[. It appears that when the control is applied with a control gain in the DSC region, all the maximum Lyapunov exponents λmax are k negative. Therefore, the shift-invariant ring of mutually coupled self-sustained electrical systems bifurcates from the unstable synchronization states to the stable synchronization states because the controlled system is stable. Analyzing the effects of the gain parameter λ2 on the maximum Lyapunov exponents λmax k (λ1 ), we find that the stable control domain increases as the gain parameter λ2 increases. The results are provided on Fig. 12 in the stability boundary of the control process in the plan (λ1 , λ2 ) for the hardening case.

Fig. 15. Effects of the gain parameter λ2 on the variation of the transverse Lyapunov exponents versus the coefficient λ1 for the hardening case. The parameters used are μ = 0.1; β = 0.5, K = 1.

For K chosen in the stable synchronization domain, for example K = 1, Fig. 15 also shows the variation of the maximum Lyapunov exponents λmax k (λ1 ) for several different values starts of λ2 . It appears that as the coefficient λ1 increases, λmax k at a negative value when λ1 = 0, since the uncontrolled shiftinvariant ring of mutually coupled self-sustained electrical systems are in the stable synchronization states ((K, N ) = (1, 10)), the uncontrolled system is stable as we reported in the above section. As λ1 increases all modes at first become more stable and the control process is stable. As λ1 increases further, we have a transition from the stable region to the unstable one which is defined as follows DUC (N = 10, λ2 = 0.1) = ]0.50; 4.60[, DUC (N = 10, λ2 = 1.0) = ]0.06; 0.44[,

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Fig. 16. The stability boundary of the control process in the plan (λ1 , λ2 ) for the hardening case. The parameters used are μ = 0.1; β = 0.5, K = 1. SC: domain of stable control and UC: domain of unstable control.

DUC (N = 10, λ2 = 2.0) = ]0.03; 0.22[, DUC (N = 10, λ2 = 3.0) = ]0.02; 0.15[. After the region DUC , the controlled system passes to the stable one DSC , which also depend to gain parameters and defined as DSC (N = 10, λ2 = 0.1) = [0; 0.50] ∪ [4.60; +∞[, DSC (N = 10, λ2 = 1.0) = [0; 0.06] ∪ [0.44, +∞[, DSC (N = 10, λ2 = 2.0) = [0; 0.03] ∪ [0.22; +∞[,

Fig. 17. Space–time-amplitude plot showing unstable control for the hardening case. The parameters used are μ = 0.1; β = 0.5; K = 1.

DSC (N = 10, λ2 = 3.0) = [0; 0.02] ∪ [0.15; +∞[. In these above domain DSC , all the maximum Lyapunov exare negative and the control process is stable. The ponents λmax k results of this investigation appears in Fig. 16 where the stability diagram of a shift-invariant ring of 10 controlled mutually coupled self-sustained electrical systems is plotted in the (λ1 , λ2 ) plane. One find on the figure two domains: the first one is the domain of the stable control process DSC , while in the second one DUC , the control process is not stable, since all the maxiare not negative. Figs. 17 and mum Lyapunov exponents λmax k 18 also show space–time-amplitude diagrams displaying stable and unstable control respectively. In general, the results of the control are to enable the unstable synchronization manifold to be stable, and then the oscillators display the same dynamics as the original uncoupled oscillator. But on Fig. 18, it appears that the effects of control strategy to the unstable synchronization manifold may lead to the suppression of oscillation since all the eigenvalues become negative. This situation can be explained by the fact that Eqs. (13) describe the behaviors of the self-sustained electromechanical system [23], which for the set of parameters used here, correspond to a complete quenching phenomena of self-excited vibrations already found in the reference [23]. 5. Conclusions We have studied in this Letter the possibility of controlling the unstable synchronization manifold on a ring of mutually coupled self-sustained electrical systems. Stability boundaries

Fig. 18. Space–time-amplitude plot showing stable control for the hardening case. The parameters used are μ = 0.1; β = 0.5; K = 1.

R. Yamapi, S. Boccaletti / Physics Letters A 371 (2007) 48–57

for the synchronization process in a shift-invariant ring of selfsustained electrical systems have first been derived follows the master stability function approach, and the transverse Lyapunov exponents. A generic feature is that there is a threshold for synchronization for the K > 0 attractive parameter. For K < 0, the repulsive case, the ring can be synchronized only if the number N of units is small and K lies in a special interval of the coupling. Secondly, we have used the active control strategy to stabilize the controlled system under variations of the gain parameters. The stability of the control process have been done using the master stability function approach in the hardening case, and the results have been derived in the stability diagrams in the (λ1 , λ2 ) plan. These observations can guide practical implementations of stable synchronous behavior in all the mentioned situations where the model represents a suitable description of the dynamical behavior. References [1] [2] [3] [4]

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