PHYSICAL REVIEW E 69, 046206 共2004兲

Synchronized states in a ring of mutually coupled self-sustained electrical oscillators P. Woafo* Laboratoire de Me´canique, Faculte´ des Sciences, Universite´ de Yaounde´ I, Boıˆte Postale 812, Yaounde´, Cameroon

H. G. Enjieu Kadji Institut de Mathe´matiques et de Sciences Physiques, Boıˆte Postale 613, Porto-Novo, Be´nin 共Received 2 September 2003; published 29 April 2004兲 We investigate in this paper different states of synchronization in a ring of mutually coupled self-sustained electrical oscillators. The good coupling parameters leading to complete and partial synchronization or disordered states are calculated using the properties of the variational equations of stability. A stability map showing domains of synchronization to an external excitation locally injected in the ring is also obtained. In both cases, the numerical simulation validates and complements the results of the analytical investigation. DOI: 10.1103/PhysRevE.69.046206

PACS number共s兲: 05.45.Gg

I. INTRODUCTION

For many years, synchronization of coupled oscillators has been investigated by the scientific community. The main reason for this growing interest is that synchronization is frequent in nature and can help to explain many phenomena in biology, chemistry, physics, or has potential applications in engineering and communication 关1–7兴. In the majority of cases, two types of behaviors are of interest: chaotic and relaxation. For the first type, complete synchronization of chaotic oscillators has been described theoretically and observed experimentally. Zhang et al. 共see Ref. 关8兴 and references therein兲 have studied partial synchronization and spatial ordering in Ro¨ssler oscillators. They have investigated the stability of different partially synchronous spatiotemporal structures and some dynamical behaviors of these states have been discussed using numerical and analytical investigations. Recently, Chembo and Woafo 关9,10兴 studied the spatiotemporal dynamics of a ring of diffusely coupled single-well Duffing oscillators with a positive nonlinear stiffness coefficient. They used the Floquet theory to investigate various dynamical states of the ring as well as the Hopf bifurcations between them and applied the local injection method to recover the chaotic dynamics. For the second type, Somers and Kopell 关11兴 showed that a ring of identical relaxation oscillators, coupled locally in a manner that mimics fast excitatory synapes, can lead to synchronization within a couple of cycles. They also showed why relaxation oscillators can more robustly encode domains of synchrony 关12兴. Recently, Ookawara and Endo 关13兴 have investigated the effect of element value deviation on the degenerate modes in a ring of three and four coupled Van der Pol oscillators. By using the averaging method, they proved that for a ring of three coupled oscillators, two frequencies bifurcate from the degenerate mode, synchronize if they are close enough, but lose synchronization when they are separated to some extent; while for a ring of four coupled oscillators, the two frequen-

*Email address: [email protected] 1539-3755/2004/69共4兲/046206共9兲/$22.50

cies generally cannot be synchronized, even if they are close enough. The purpose of this paper is to consider the synchronized states in a ring of mutually coupled self-sustained electrical oscillators described by coupled Van der Pol equations. We first analyze the stability of the synchronization of the ring using analytical and numerical investigations. Then, we find the effects of the local injection strength on the behavior of the ring. After presenting the physical model in the next section, we analyze in Sec. III the stability of the synchronization in the ring using the Floquet theory and Whittaker method. In Sec. IV, the influence of a local injection is found. The last section is devoted to the conclusion. II. MODEL AND STATEMENT OF THE PROBLEM

The model shown in Fig. 1 is a ring of N identical mutually coupled self-excited electrical circuits described by coupled Van der Pol oscillators 共see Fig. 2兲. Each oscillator consists of a nonlinear resistor NR, an inductor L, and a condenser C, all connected in parallel. The coupling between the N identical oscillators is realized here through an inductor L c 共low-pass oscillators兲, but can also be done with a

FIG. 1. Ring of N mutually coupled self-sustained electrical oscillators. 69 046206-1

©2004 The American Physical Society

PHYSICAL REVIEW E 69, 046206 共2004兲

P. WOAFO AND H. G. ENJIEU KADJI

waves兲, and in particular the existence of an irregular degenerate mode that appears when the number N of oscillators is a multiple of 4 关22兴. III. SYNCHRONIZATION ANALYSIS IN THE RING A. Analytical treatment

FIG. 2. A self-sustained electrical oscillator.

capacitor 共high-pass oscillators兲. The volt-ampere characteristic of the nonlinear resistor for the ␯th unit is expressed by a symmetric cubic nonlinearity, which is illustrated by i ␯ ⫽⫺a 1 V ␯ ⫹a 3 V 3␯ ,

a 1 ,a 3 ⬎0;

␯ ⫽1,2,...,N.

共1兲

In this case, the model has the property to exhibit selfexcited oscillations. This is due to the fact that the model incorporates through its nonlinear resistance a dissipative mechanism to damp oscillations that grow too large and a source of energy to pump up those that become too small. This form of nonlinearity was introduced by Van der Pol, who considered a lumped oscillator with two degrees of freedom to discuss simultaneous multimode oscillations 关14,15兴. As we have shown in the Appendix, the model is described by the following second-order nondimensional nonlinear differential equations:

A particular property of the Van der Pol oscillator whose final state is a sinusoidal limit cycle or relaxation oscillations 关14,15兴 is the sensibility of its phase to initial conditions. Consequently, if N identical Van der Pol oscillators are launched with different initial conditions, they will circulate on the same limit cycle but with N different phases. The aim of the synchronization is therefore to phase-lock those oscillators 共phase synchronization兲. When the oscillators are coupled as in Eqs. 共2兲, the resulting dynamical state of the system is interesting when it is stable. This requires that all the perturbed trajectories return to the original limit cycle. But for the sake of exemplification, we set N to 4. Thus, the stability of the dynamical state can be studied through the linearization of Eqs. 共2兲 around the unperturbed limit cycle 共or orbit兲 x o according to

␰¨ 1 ⫺ ␮ 共 1⫺x 2o 兲 ␰˙ 1 ⫹ 共 1⫹2K⫹2 ␮ x o x˙ o 兲 ␰ 1 ⫽K 共 ␰ 2 ⫹ ␰ 4 兲 , ␰¨ 2 ⫺ ␮ 共 1⫺x 2o 兲 ␰˙ 2 ⫹ 共 1⫹2K⫹2 ␮ x o x˙ o 兲 ␰ 2 ⫽K 共 ␰ 1 ⫹ ␰ 3 兲 , ␰¨ 3 ⫺ ␮ 共 1⫺x 2o 兲 ␰˙ 3 ⫹ 共 1⫹2K⫹2 ␮ x o x˙ o 兲 ␰ 3 ⫽K 共 ␰ 2 ⫹ ␰ 4 兲 ,

x¨ 1 ⫺ ␮ 共 1⫺x 21 兲 x˙ 1 ⫹x 1 ⫽K 共 x 2 ⫺2x 1 ⫹x N 兲 ,

␰¨ 4 ⫺ ␮ 共 1⫺x 2o 兲 ␰˙ 4 ⫹ 共 1⫹2K⫹2 ␮ x o x˙ o 兲 ␰ 4 ⫽K 共 ␰ 1 ⫹ ␰ 3 兲 ,

x¨ ␯ ⫺ ␮ 共 1⫺x ␯2 兲 x˙ ␯ ⫹x ␯ ⫽K 共 x ␯ ⫹1 ⫺2x ␯ ⫹x ␯ ⫺1 兲 , ]

]

]

]

where ␰ ␯ stands for the perturbation term. For small values of ␮, the behavior of one oscillator can be described by a pure sinusoidal trajectory of the form

]

x¨ N ⫺ ␮ 共 1⫺x N2 兲 x˙ N ⫹x N ⫽K 共 x 1 ⫺2x N ⫹x N⫺1 兲 ,

共3兲

共2兲

where ␯ ⫽2,...,N⫺1 and the dot over a quantity denotes the time derivative. x ␯ stands for the amplitude of the voltage at the ␯th oscillator, ␮ is a positive coefficient, and K is the coupling parameter. The equations of motion 共2兲 are a set of N identical coupled Van der Pol or self-sustained oscillators. Equations 共2兲 are interesting because they model several phenomena and have applications in many areas. In that channel, this model is mainly used in electronics engineering as a network of parallel microwave oscillators 关16,17兴. Such a network allows us to investigate the possibility of simultaneous multimode oscillations and accordingly the stability of several nonresonant modes of oscillations under specific boundary conditions 共two modes are nonresonant if the ratio of their frequencies is an irrational number兲 关18兴. In biology, the network of a large number of these oscillators can be used to model intestinal signal 关19兴 or colorectal myoelectrical activity in humans 关20兴. Moreover, the central pattern generator 共CPG兲 that controls the rhythmic activity in invertebrates can be modeled by the same system 关21兴. It can also be used to investigate the stability of both nondegenerate modes 共standing waves兲 and degenerate modes 共traveling

x o ⫽A cos共 ␻ t⫺ ␸ 兲 ,

共4兲

where A and ␻ are, respectively, the amplitude and the frequency of the unperturbed limit cycle in the first approximation. The values of A and ␻ are A⫽2.00 and ␻ ⫽0.999 for ␮ ⫽0.10 共obtained, for instance, by the averaging method兲. As reported in Ref. 关23兴 dealing with the synchronization of two Van der Pol oscillators, this first-order approximation gives fairly good agreement between the analytical and numerical results. If we introduce the rescaling ␶ ⫽ ␻ t⫺ ␸ and the following diagonal variables 共or Fourier modes兲 ␳ i as

␳ 1⫽ ␰ 1⫹ ␰ 2⫹ ␰ 3⫹ ␰ 4 , ␳ 2 ⫽ ␰ 4 ⫺ ␰ 2 ⫽x 4 ⫺x 2 , ␳ 3 ⫽ ␰ 3 ⫺ ␰ 1 ⫽x 3 ⫺x 1 , ␳ 4 ⫽ ␰ 4 ⫺ ␰ 3 ⫹ ␰ 2 ⫺ ␰ 1 ⫽x 4 ⫺x 3 ⫹x 2 ⫺x 1 ,

共5兲

we get, after some algebraic manipulation, the following variational equations:

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␳¨ i ⫹ 关 2␭⫹F 共 ␶ 兲兴 ␳˙ i ⫹G i 共 ␶ 兲 ␳ i ⫽0,

i⫽1,2,3,4,

共6兲

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SYNCHRONIZED STATES IN A RING OF MUTUALLY . . .

with ␭⫽⫺

␮A2 cos 2 ␶ , 2␻

1 共 1⫹2K⫺ ␮ A 2 ␻ sin 2 ␶ 兲 , ␻2

1 共 1⫹4K⫺ ␮ A 2 ␻ sin 2 ␶ 兲 . ␻2

␩ i ⫽e ␣ i ␶ sin共 n ␶ ⫺ ␴ 兲 ,

From the expression of G 2 ( ␶ ) and G 3 ( ␶ ), we find that if K苸]⫺⬁,⫺0.50关 , ␳ 2 and ␳ 3 will grow indefinitely, leading to the instability, in the ring. The same phenomenon also occurs for ␳ 4 from the expression of G 4 ( ␶ ) when K 苸]⫺⬁,⫺0.25关 共兴a,b关 means the internal from a to b but with a and b excluded兲. Taking the union of the two domains, this means that any perturbed trajectory in the region of K 苸]⫺⬁,⫺0.25关 leads the oscillators to continuously drift away from their original limit cycles because the restoring force turns out to be repelling and the cycle loses its attraction character of the disturbed trajectory. To discuss further the stability of the synchronization process, let us rewrite Eqs. 共6兲 in a standard form. For this objective, we use the transformation

冉 冕

␳ i ⫽n i exp共 ⫺␭ ␶ 兲 exp ⫺

1 2

F共 ␶⬘兲d␶⬘

冊

共7兲

and obtain that ␩ i satisfies the following set of independent Hill equations 关24,25兴: n¨ i ⫹ 共 a 0i ⫹2a 1s sin 2 ␶ ⫹2a 1c cos 2 ␶ ⫹2a 2c cos 4 ␶ 兲 n i ⫽0, 共8兲

i⫽1,2,3,4, where a 01⫽

冉

冋 冋

␮2 1 A2 1⫺ 1⫺ ␻2 4 2

冉

冊

2

⫺␮2

␮2 1 A2 1⫺ a 02⫽a 03⫽ 2 1⫹2K⫺ ␻ 4 2 a 04⫽

冋

冉

␮2 1 A2 1⫹4K⫺ 1⫺ ␻2 4 2 a 1s ⫽⫺ a 1c ⫽

冉

冊

冊

2

册

A4 , 32

册

A4 ⫺␮ , 32

2

⫺␮2

␮ 2A 4 . 64␻ 2

From Eq. 共8兲, the stability boundaries of the synchronization are to be found around the two main parametric resonances defined at a 0 ⫽n 2 (n⫽1,2). Floquet theory 关24,25兴 states that ␩ i may decay to zero or grow to infinity, and therefore decide the behavior of the independent Fourier modes ␳ i 关23,26兴. Consequently, the stability of each ␳ i depends on the coupling coefficient K and we need to determine the range of K for the synchronization process to be achieved. Thereby, we use the Whittaker method 关24兴 to discuss the unstable solutions. Thus we assume that at the nth unstable region, each solution of Eqs. 共8兲 has the form

1 共 1⫺ ␮ A 2 ␻ sin 2 ␶ 兲 , ␻2

G 2 共 ␶ 兲 ⫽G 3 共 ␶ 兲 ⫽ G 4共 ␶ 兲 ⫽

冊

␮ A2 1⫺ , 2␻ 2

F共 ␶ 兲⫽ G 1共 ␶ 兲 ⫽

冉

a 2c ⫽⫺

2

册

共9兲

with ␣ i being the characteristic exponents and ␴ a parameter. Substituting Eqs. 共9兲 into Eqs. 共8兲 and equating the coefficient of cos n␶ and sin n␶ separately to zero, we find that the characteristic exponents have the following expressions:

␣ 2i ⫽⫺ 共 a 0i ⫹n 2 兲 ⫹ 冑4n 2 a 0i ⫹a 2n ,

共10兲

2 2 ⫹a nc . The synchronization process is stable with a 2n ⫽a ns when the Fourier modes ␳ i go to zero with increasing time, so that the real part of ⫺␭⫾ ␣ i should be negative. Since ␭ is real and positive, the stability condition is reduced to ␭ 2 ⬎ ␣ 2i . Consequently, from the relations 共7兲, the synchronization process is stable under the conditions

H ni ⫽ 共 a 0i ⫺n 2 兲 2 ⫹2 共 a 0i ⫹n 2 兲 ␭ 2 ⫹␭ 4 ⫺a 2n ⬎0,

n⫽1,2. 共11兲

We note that in the second main parametric resonance 共i.e., for n⫽2), the conditions 共11兲 are satisfied for all value of i. Thus the stability is analyzed in the first main parametric resonance 共i.e., for n⫽1) and H 1i helps us to determine the synchronization domain and the stability boundaries. We can now analyze through H 1i what happens in the ring when the coupling strength K increases from ⫺0.25 to infinity. It should be noticed that H 11 does not depend on the coupling strength. When K⫽0, the system is uncoupled and the Fourier modes ␳ 2 , ␳ 3 , and ␳ 4 degenerate into ␳ 1 , which is stable 共since it remains bounded as t tends to infinity兲. Then, the model belongs to the stability area. As K increases, our investigation shows that both H 12 and H 13 are positive for K 苸]⫺0.25;⫺0.0011]艛 关 0.004;⫹⬁ 关 while H 14 is positive for the range defined as K苸]⫺0.25;⫺0.0006]艛 关 0.002;⫹⬁ 关 . We can thus discern three domains as follows:

A4 , 32

I 1a ⫽]⫺0.25;⫺0.0011]艛 关 0.004;⫹⬁ 关 , I 2a ⫽]⫺0.0011;⫺0.0006]艛 关 0.0020;0.004关 ,

␮A2 , 4␻

冊

A2 2 ␮2 A , 2 1⫺ 8␻ 2

I 3a ⫽]⫺0.0006;0 关 艛 兴 0;0.0020关 . When K苸I 1a , the three modes ␳ 2 , ␳ 3 , and ␳ 4 are together in the stability domain and thus tend all to zero as the time 046206-3

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P. WOAFO AND H. G. ENJIEU KADJI

increases. Thus the ring is in the complete synchronization state where we have the constraint x 1 ⫽x 2 ⫽x 3 ⫽x 4 .

共12兲

In this case, all four oscillators display the same dynamics 共e.g., are phase-locked兲. For K苸I 2a , only the fastest mode ␳ 4 reaches the stability domain and the ring satisfies the constraint x 4 ⫺x 3 ⫹x 2 ⫺x 1 ⬅0,

共13兲

while we have x 1 ⫽x 3 , x 2 ⫽x 4

共14兲

since ␳ 2 and ␳ 3 remain in the unstable domain. This corresponds to what can be called a standard correlated state 共SCS兲. For K苸I 3a , the modes ␳ 2 , ␳ 3 , and ␳ 4 enter into the instability domain. This means that the ring satisfies the following constraint: x 1 ⫽x 3 , x 2 ⫽x 4 ,

共15兲

x 4 ⫺x 3 ⫹x 2 ⫺x 1 ⫽0.

共16兲

and

In this case, there is no synchronization in the ring. B. Results of the numerical simulation

We use the numerical simulation to check the validity and complement the analytical results obtained from Eqs. 共11兲. The numerical simulation uses the fourth-order Runge-Kutta algorithm with a time step ⌬t⫽0.01 and the initial conditions „x 1 (0);x˙ 1 (0)…⫽(1.0;1.0), „x 2 (0);x˙ 2 (0)…⫽(1.5;1.5), „x 3 (0);x˙ 3 (0)…⫽(2.0;2.0), and „x 4 (0);x˙ 4 (0)…⫽(3.0;3.0). Let us evaluate the final values of ␳ i and thus indicate various areas of K where synchronization is achieved. The ring is considered synchronized if each ␳ i vanishes with the precision 10⫺4 . For a fixed value of K in each area I ja ( j ⫽1,2,3), we have plotted the behavior of ␳ k (k⫽2,3,4) versus the time in Fig. 3 to show how they look when there is synchronization, when there is no synchronization, and when there is instability. From the numerical simulation of Eq. 共2兲, complete synchronization occurs for K苸I 1n ⫽ 关 ⫺0.2363; ⫺0.0017兴 艛 关 0.0037;⫹⬁ 关 . The system is in the SCS for K 苸I 2n ⫽]⫺0.0011;⫺0.0009]艛 关 0.0029;0.0037关 . This is due to the fact that ␳ 4 ⫽0 while ␳ 2 ⫽0 and ␳ 3 ⫽0. For the region of K defined as K苸I 3n ⫽]⫺0.0009;0 关 艛 兴 0;0.0029关 , there is no synchronization in the ring because ␳ 2 ⫽0, ␳ 3 ⫽0, and ␳ 4 ⫽0. Two clusters also come from this numerical analysis which do not appear from the analytical investigations. The first one is defined for K苸 关 ⫺0.25;⫺0.2364兴 , where ␳ 2 ⫽ ␳ 3 ⫽0, ␳ 4 ⫽0 corresponding to the state x 4 ⫽x 2 and x 3

FIG. 3. Temporal variation of the Fourier modes ␳ i with ␮ ⫽0.10: 共a兲 K⫽⫺0.10, 共b兲 K⫽⫺0.0008, 共c兲 K⫽0.0010, and 共d兲 K⫽⫺0.26. ␳ 2 共lines兲, ␳ 3 共line points兲, and ␳ 4 共dashed lines兲.

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⫽x1 with x 4 ⫺x 3 ⫹x 2 ⫺x 1 ⫽0. The second cluster is for K 苸 关 ⫺0.0016;⫺0.0011兴 since ␳ 2 ⫽0 and ␳ 3 ⫽ ␳ 4 ⫽0. It corresponds to x 1 ⫽x 3 and x 4 ⫹x 2 ⫽x 1 ⫹x 3 .

x¨ 3 ⫺ ␮ 共 1⫺x 23 兲 x˙ 3 ⫹x 3 ⫽K 共 x 4 ⫺2x 3 ⫹x 2 兲 ,

IV. INFLUENCE OF THE LOCAL INJECTION

x¨ 4 ⫺ ␮ 共 1⫺x 24 兲 x˙ 4 ⫹x 4 ⫽K 共 x 1 ⫺2x 4 ⫹x 3 兲 ,

As we have mentioned before, different dynamical states are observed in several identical coupled oscillators such as clustering or complete synchronization. However, sometimes, due to environmental constraints or because of its potential application, the system can be coupled to an external independent oscillator or excitation. This is commonly achieved through the local injection technique consisting of a unidirectional coupling between the external command oscillator and a fixed representative of the nonlinear coupled system 关27兴. This local injection scheme is sometimes indispensable for the description of undesirable parasite couplings or external perturbations. For example, in the case where external perturbation is the noise, it plays a dual role if applied to a synchronized system. Depending on the system’s parameter, the noise can disrupt synchronization or produce a new ordered state whose coherence depends resonantly on the noise intensity. For instance, there is an optimal value of noise intensity which produces maximally regular biperiodic oscillations, and thus coherence resonance 关28兴. Local injection can also be willingly introduced to force the nonlinear system to replicate the dynamics of the external master oscillator. For instance, in the chaotic oscillators, the local injection method can enable us to recover a particular chaotic orbit when the unidirectional command coupling is suitably designed 关9兴. When the local injection is taken into account, Eqs. 共2兲 become x¨ 1 ⫺ ␮ 共 1⫺x 21 兲 x˙ 1 ⫹x 1 ⫽K 共 x 2 ⫺2x 1 ⫹x 4 兲 ⫺⌫ 共 x 1 ⫺x s 兲 ,

x¨ 2 ⫺ ␮ 共 1⫺x 22 兲 x˙ 2 ⫹x 2 ⫽K 共 x 3 ⫺2x 2 ⫹x 1 兲 ,

共17兲

where x s represents the dynamics of the external oscillator and also plays the role of the command signal, and ⌫ is the local injection strength. Generally, the literature places emphasis upon the control of the coupled system to the trivial equilibrium state (x s ⫽0). Even for this simple target, further simplifications are often imposed for the analytical results to be derived. For example, to be sure that the first oscillator can be pinned to the target state x s ⫽0, ⌫ should be directly set to infinity. Throughout our study, we take x s as the periodic solution of a Van der Pol equation. Then we have x¨ s ⫺ ␮ 共 1⫺x s2 兲 x˙ s ⫹x s ⫽0.

共18兲

Let us rewrite the first-order perturbation equations 共3兲 as follows:

␰¨ 1 ⫺ ␮ 共 1⫺x s2 兲 ␰˙ 1 ⫹ 共 1⫹2 ␮ x s x˙ s 兲 ␰ 1 ⫽K 共 ␰ 2 ⫺2 ␰ 1 ⫹ ␰ 4 兲 ⫺⌫ ␰ 1 ,

␰¨ 2 ⫺ ␮ 共 1⫺x s2 兲 ␰˙ 2 ⫹ 共 1⫹2 ␮ x s x˙ s 兲 ␰ 2 ⫽K 共 ␰ 3 ⫺2 ␰ 2 ⫹ ␰ 1 兲 , ␰¨ 3 ⫺ ␮ 共 1⫺x s2 兲 ␰˙ 3 ⫹ 共 1⫹2 ␮ x s x˙ s 兲 ␰ 3 ⫽K 共 ␰ 4 ⫺2 ␰ 3 ⫹ ␰ 2 兲 , ␰¨ 4 ⫺ ␮ 共 1⫺x s2 兲 ␰˙ 4 ⫹ 共 1⫹2 ␮ x s x˙ s 兲 ␰ 4 ⫽K 共 ␰ 1 ⫺2 ␰ 4 ⫹ ␰ 3 兲 共19兲 with the deviation ␰ ␯ ⫽x ␯ ⫺x s . Following the analysis of Sec. III, Eqs. 共19兲 may now be written under the form of a set of coupled Hill’s equations:

␩¨ 1 ⫹ 共 a 01⫹2a 1s sin 2 ␶ ⫹2a 1c cos 2 ␶ ⫹2a 2c cos 4 ␶ 兲 ␩ 1 ⫽

1 关 K 共 ␩ 2 ⫺2 ␩ 1 ⫹ ␩ 4 兲 ⫺⌫ ␩ 1 兴 , ␻2

␩¨ 2 ⫹ 共 a 01⫹2a 1s sin 2 ␶ ⫹2a 1c cos 2 ␶ ⫹2a 2c cos 4 ␶ 兲 ␩ 2 ⫽

1 关 K 共 ␩ 3 ⫺2 ␩ 2 ⫹ ␩ 1 兲兴 , ␻2

␩¨ 3 ⫹ 共 a 01⫹2a 1s sin 2 ␶ ⫹2a 1c cos 2 ␶ ⫹2a 2c cos 4 ␶ 兲 ␩ 3 ⫽

1 关 K 共 ␩ 4 ⫺2 ␩ 3 ⫹ ␩ 2 兲兴 , ␻2

␩¨ 4 ⫹ 共 a 01⫹2a 1s sin 2 ␶ ⫹2a 1c cos 2 ␶ ⫹2a 2c cos 4 ␶ 兲 ␩ 4 ⫽

1 关 K 共 ␩ 1 ⫺2 ␩ 4 ⫹ ␩ 3 兲兴 , ␻2

共20兲

where

冉 冕

␰ ␯ ⫽ ␩ ␯ exp共 ⫺␭ ␶ 兲 exp ⫺

1 2

冊

F共 ␶⬘兲d␶⬘ ,

␯ ⫽1,2,3,4.

Let us investigate the stability of the synchronization process in the ring. We assume that each solution of Eqs. 共20兲 has the expression

␩ ␯ ⫽C ␯ e S ␶ sin共 n ␶ ⫺ ␴ 兲 , 046206-5

共21兲

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where S is the characteristic exponent and C ␯ are arbitrary constants. Substituting the solutions ␩ ␯ into Eqs. 共20兲 and equating the coefficients of sin n␶ and cos n␶ separately to zero gives us the following set of algebraic equations in C ␯ : 关共 S 2 ⫺n 2 ⫹ ␦ ⫺a nc 兲 cos ␴ ⫹ 共 2nS⫺a ns 兲 sin ␴ 兴 C 1 ⫺

K K 2 cos ␴ C 2 ⫺ 2 cos ␴ C 4 ⫽0, ␻ ␻

关共 2nS⫹a ns 兲 cos ␴ ⫺ 共 S 2 ⫺n 2 ⫹ ␦ ⫹a nc 兲 sin ␴ 兴 C 1 ⫹

K K sin ␴ C 2 ⫹ 2 sin ␴ C 4 ⫽0, ␻2 ␻

⫺

K K 2 2 2 cos ␴ C 1 ⫹ 关共 S ⫺n ⫹a 02⫺a nc 兲 cos ␴ ⫹ 共 2nS⫺a ns 兲 sin ␴ 兴 C 2 ⫺ 2 cos ␴ C 3 ⫽0, ␻ ␻

K K 2 2 2 sin ␴ C 1 ⫹ 关共 2nS⫹a ns 兲 cos ␴ ⫺ 共 S ⫺n ⫹a 02⫹a nc 兲 sin ␴ 兴 C 2 ⫹ 2 sin ␴ C 3 ⫽0, ␻ ␻ ⫺

K K cos ␴ C 2 ⫹ 关共 S 2 ⫺n 2 ⫹a 02⫺a nc 兲 cos ␴ ⫹ 共 2nS⫺a ns 兲 sin ␴ 兴 C 3 ⫺ 2 cos ␴ C 4 ⫽0, ␻2 ␻

K K 2 2 2 sin ␴ C 2 ⫹ 关共 2nS⫹a ns 兲 cos ␴ ⫺ 共 S ⫺n ⫹a 02⫹a nc 兲 sin ␴ 兴 C 3 ⫹ 2 sin ␴ C 4 ⫽0, ␻ ␻ ⫺

K K 2 2 2 cos ␴ C 1 ⫺ 2 cos ␴ C 3 ⫹ 关共 S ⫺n ⫹a 02⫺a ac 兲 cos ␴ ⫹ 共 2nS⫺a ns 兲 sin ␴ 兴 C 4 ⫽0, ␻ ␻

K K 2 2 2 sin ␴ C 1 ⫹ 2 sin ␴ C 3 ⫹ 关共 2nS⫹a ns 兲 cos ␴ ⫺ 共 S ⫺n ⫹a 02⫹a nc 兲 sin ␴ 兴 C 4 ⫽0. ␻ ␻

冏

Upon elimination of C 1 , C 2 , C 3 , C 4 , and ␴ in Eqs. 共22兲, we have

⌬ n共 S 兲 ⬅

⌬ 11 ⌬ 12

冏

⌬ 13

0

0

0

⌬ 17

0

⌬ 24

0

0

0

⌬ 28

0

0

0

⌬ 46

0

0

⌬ 57

0

⌬ 66

0

⌬ 68

⌬ 77

⌬ 78

⌬ 21

⌬ 22

0

⌬ 31

0

⌬ 33

0

⌬ 42

0

0

⌬ 53

0

0

0

⌬ 71

0

0

0

⌬ 75

0

0

⌬ 82

0

0

0

⌬ 86

⌬ 34 ⌬ 35

⌬ 43 ⌬ 44 0

0

⌬ 55 ⌬ 56

⌬ 64 ⌬ 65

⫽ 0,

⌬ 87 ⌬ 88

with n⫽1 or n⫽2 and the parameters ⌬ lm (l,m⫽1,2,3,4,5,6,7,8) are given by the following expressions: ⌬ 11⫽S 2 ⫹⌰ n ,

⌬ 22⫽⫺ 共 S 2 ⫹⌶ n 兲 ,

⌬ 44⫽⌬ 66⫽⌬ 88⫽⫺ 共 S 2 ⫹⌿ n 兲 ,

⌬ 33⫽⌬ 55⫽⌬ 77⫽S 2 ⫹⌼ n ,

⌬ 12⫽⌬ 34⫽⌬ 56⫽⌬ 78⫽2nS⫺a ns ,

⌬ 21⫽⌬ 43⫽⌬ 65⫽⌬ 87⫽2nS⫹a ns , ⌬ 13⫽⌬ 31⫽⌬ 17⫽⌬ 71⫽⌬ 35⫽⌬ 53⫽⌬ 57⫽⌬ 75⫽⫺

⌬ 24⫽⌬ 42⫽⌬ 28⫽⌬ 82⫽⌬ 46⫽⌬ 64⫽⌬ 68⫽⌬ 86⫽ 046206-6

K , ␻2

K , ␻2

共22兲

共23兲

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SYNCHRONIZED STATES IN A RING OF MUTUALLY . . .

FIG. 4. Stability map.

the validity of our analytic investigation, we have solved numerically Eqs. 共17兲 with the fourth-order Runge-Kutta algorithm. Synchronization between two oscillators p and q occurs with a criterion that the distance of the phase trajectories be

where ⌰ n ⫽ ␦ ⫺n 2 ⫺a nc ,

⌶ n ⫽ ␦ ⫺n 2 ⫹a nc ,

⌼ n ⫽a 02⫺n 2 ⫺a nc , ⌿ n ⫽a 02⫺n 2 ⫹a nc ,

␦⫽

d pq ⫽ 兩 x p ⫺x q 兩 ⬍h,

2K⫹⌫ ⫹a 01 , ␻2

with n⫽1 for ⌬ 1 (S) and n⫽2 for ⌬ 2 (S) 关see Eq. 共23兲兴. The characteristic exponent S is given by Eq. 共23兲, that is, ⌬ n (S)⫽0. Since the stability condition is given by ␭ 2 ⫺S 2 ⬎0 when assuming that ␭⬎0, we have ⌬ n 共 ␭ 兲 ⫽0

共25兲

⫺3

共24兲

at the boundary of the nth unstable domain. In the second main resonance, ⌬ 2 (␭) is positive and does not change its sign. Accordingly, the stability analysis is reduced once more around the first unstable region. Thus, when ⌫⫽0, our analytical investigation shows that the synchronization process is stable for the range of K defined as ]⫺0.25;⫺0.001关 艛 兴 ⫺0.003;0 关 艛 兴 0;0.0004关 艛 兴 0.0039; ⫹⬁ 关 , which is comparable with the interval I 1a . Analyzing the effects of the local injection strength on the stability boundary of the ring, we find two ranges as ⌫ varies. The first range is defined as 0⬍⌫⭐1.5, where the stability of the ring depends on the local injection strength ⌫. For example, when ⌫⫽0.06, the synchronization process is achieved if K苸]⫺0.25;⫺0.0012关 艛 兴 ⫺0.0011; ⫺0.0006关 艛 兴 0.0039;⫹⬁ 关 and becomes兴⫺0.25;⫺0.0035关艛 兴⫺0.0011;⫺0.0006关艛兴0.0023;0.0040关艛关0.0110;⫹⬁关 when ⌫ is 0.6. In the second range, i.e., ⌫苸]1.5;⫹⬁关, we find that the stability domain of the synchronization does not change with the variation of the local injection strength and is defined as K苸]⫺0.25;⫺0.0036关 艛 兴 ⫺0.0011; ⫺0.0006关 艛 兴 0.0023;0.0040关 艛 兴 0.0133;⫹⬁ 关 . To confirm

where h⫽10 is the precision. Synchronization among all the oscillators occurs if the total separation of all pairs of trajectories is smaller than an accuracy, namely d⫽

兺

pairs共 pq 兲

d pq ⬍h.

共26兲

For higher accuracy 共with a smaller h兲, computational time has been extended to 105 . In Fig. 4, we show the stability map by applying the numerical simulation of the equation of motion 共17兲 and the preceding analytical investigation. The resulting synchronized states in the (K,⌫) plane are drawn for a fixed value of the injection strength ⌫ when the coupling parameter K varies. The following results are observed. The map shows four different areas: (D 1 ), (D 2 ), (D 3 ), and (D 4 ) 共see Fig. 4兲. The intersection between both analytical and numerical instability areas corresponds to (D 1 ), while (D 4 ) is the intersection between the analytical and the numerical stability areas. As for (D 2 ), it shows the instability domain that is not predicted analytically while (D 3 ) is the stability domain forecasted analytically but not numerically. As ⌫ increases, both analytical and numerical instability areas become closer. For example, when ⌫⫽0.06, the numerical simulation gives that the synchronization is unstable for K苸 关 ⫺0.25;⫺0.241兴 艛 关 ⫺0.003;0 关 艛 兴 0;0.021兴 and becomes 关 ⫺0.004;0 关 艛 兴 0;0.013兴 when ⌫⫽0.60. The existence of clusters in the numerical instability domain (D 2 ) should be noted since, to obtain complete synchronization, all the clusters should be synchronized between them. This phenomenon ( ␳ 2 ⫽0) can be displayed, for example, when

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PHYSICAL REVIEW E 69, 046206 共2004兲

P. WOAFO AND H. G. ENJIEU KADJI

⌫⫽0.03 in the interval 关⫺0.25; ⫺0.239兴. When ⌫⬎1.44, the map configuration remains unchanged. It is important to remark that (D 1 ) and (D 4 ) are two regions where the agreement between analytical and numerical results is quite good. It is also clear throughout our analytic investigation that, in opposition to the case where the coupled system is studied around the trivial equilibrium state x s ⫽0 and for which we need to set directly ⌫ to infinity to assure the synchronization, it is not necessary when x s ⫽0.

V ␯ ⫺V ␯ ⫹1 ⫽L c

I ␯ ⫺1 ⫺I ␯ ⫽i L ⫹i C ⫹i ␯ ⫽

冕

V ␯ d ␶ ⫺C

共A1兲

dV ␯ ⫺a 1 V ␯ ⫹a 3 V ␯3 . d␶ 共A2兲

The first time derivative of Eq. 共A2兲 leads us to dV ␯ dV ␯ d 2V ␯ dI ␯ ⫺1 dI ␯ 1 ⫺ ⫽ V ␯ ⫺C ⫺a 1 ⫹3a 3 V 2␯ . d␶ d␶ L d␶2 d␶ d␶ 共A3兲

V. CONCLUSION

In this paper, we have studied the stability of the synchronization in a ring of mutually coupled self-sustained oscillators with and without a local injection. The Whittaker method has permitted us to obtain the boundaries of the synchronization process when the local injection is not present. When we take into account the local injection effect, the same analytical method helps us to obtain a stability map for complete synchronization to the external excitation. As noted in the Introduction, the model analyzed in this paper is a representative of many systems. We think that following the preliminary results obtained here, a close inspection of the realistic models in the context of physics, biology, and electronics is still an interesting task. Indeed, coming back to the electronic system shown in Fig. 1, it should be stressed that when K belongs to I 1n , all four microwave oscillators are phase-locked. Thus the wave signal emitted appears to be more powerful. The state where x 1 ⫽x 3 and x 2 ⫽x 4 is also interesting since it corresponds to the situation where two microwave oscillators are phase-locked one after the other with possible implications in automation engineering.

1 L

dI ␯ , d␶

Then using Eq. 共A1兲, we obtain that the voltage in the capacitor of the ␯th oscillator obeys the equation 1 1 共 V ␯ ⫺1 ⫺V ␯ 兲 ⫺ 共 V ␯ ⫺V ␯ ⫹1 兲 Lc Lc ⫽

冉

冊

dV ␯ 1 d 2V ␯ a3 ⫺a 1 1⫺3 V 2␯ . V ␯ ⫹C L d␶2 a1 d␶

共A4兲

This latter equation can be rewritten as follows:

冉

冊

dV ␯ a3 1 d 2V ␯ a 1 V 1⫺3 V 2␯ ⫹ 2 ⫺ d␶ C a1 d␶ LC ␯ ⫽

1 ⫺2V ␯ ⫹V ␯ ⫹1 兲 . 共V L c C ␯ ⫺1

共A5兲

The substitution of the quantities w 2e ⫽

ACKNOWLEDGMENT

1 , LC

t⫽w e ␶ ,

V ␯⫽

冑

The authors thank T. Endo for enriching contributions.

a1 x 3a 3 ␯

gives the set of Eqs. 共2兲 with APPENDIX

␮ ⫽a 1

When the N electrical oscillators are interconnected, the ␯th oscillator is described by the following equations:

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冑

L , C

K⫽

L . Lc

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SYNCHRONIZED STATES IN A RING OF MUTUALLY . . . 34, 943 共1986兲. 关18兴 T. Endo and S. Mori, IEEE Trans. Circuits Syst. 23, 100 共1976兲. 关19兴 B. Robertson-Dunn and D. A. Linkens, J. Med. Biol. Eng. 12, 750 共1974兲. 关20兴 D. A. Linkens, IEEE Trans. Biomed. Eng. 23, 101 共1976兲. 关21兴 J. D. Murray, Mathematical Biology 共Springer, New York, 1989兲. 关22兴 T. Endo and S. Mori, IEEE Trans. Circuits Syst. 25, 7 共1978兲. 关23兴 P. Woafo and R. A. Kraenkel, Phys. Rev. E 65, 036225 共2002兲.

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