Commun Nonlinear Sci Numer Simulat

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1 Commun Nonlinear Sci Numer Simulat 14 (2009) Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: Dynamics of a ring network of phase-only oscillators Jacqueline Brige a, Richar Ran b, *, Si Mohame Sah c a Department of Mechanical Engineering, University of The West Inies, St. Augustine, Trinia an Tobago b Department of Theoretical an Applie Mechanics, Cornell University, Thurston Hall, Ithaca, NY 14853, Unite States c Faculty of Sciences Aïn Chock, BP 5366 Maârif, Casablanca, Morocco article info abstract Article history: Receive 25 August 2008 Accepte 26 August 2008 Available online 5 September 2008 PACS: Xt Hc We investigate the existence, stability an bifurcation of phase-locke motions in a ring network consisting of phase-only oscillators arrange in multiple simple rings (sub-rings) which are themselves arrange in a single large ring. In the case of networks with three or four sub-rings, we give approximate expressions for critical coupling coefficients which must be exceee in orer for phase-locking to occur. Ó 2008 Elsevier B.V. All rights reserve. Keywors: Synchronization Couple oscillators Networks 1. Introuction Recent research has ealt with the ynamics of a simple ring of oscillators, variously moele as phase-only oscillators, an as van er Pol oscillators in both sinusoial an relaxation limits [1,5,3,8,7,4,2]. In this work we consier the ynamics of a system compose of multiple simple rings of phase-only oscillators which themselves are arrange in a single large ring, resulting in a structure which we refer to as a ring network, see Fig. 1. Each iniviual ring, calle a sub-ring, is compose of a number of ientical phase-only oscillators which are nearest-neighbor couple by sine functions of phase ifferences, all with ientical coupling coefficients. The ith sub-ring is compose of n i oscillators with uncouple frequencies w i an coupling coefficients a i. We consier a system compose of m sub-rings which are connecte by a larger ring calle the communication ring. One oscillator on each sub-ring is ientifie as the communication oscillator, an it is these m communication oscillators which compose the communication ring, which has coupling coefficient a. We label the phase of the n i oscillators which comprise the ith sub-ring by h i,j, where j =1,...,n i. In particular the communication oscillator is labelle h i,1. Using this notation the equations of motion become: t h i;j ¼ w i þ a i ðsinðh i;jþ1 h i;j Þþsinðh i;j 1 h i;j ÞÞ þ j;1 aðsinðh iþ1;1 h i;1 Þþsinðh i 1;1 h i;1 ÞÞ; where i ¼ 1;...; m; j ¼ 1;...; n i ð1þ where j,i is the Kronecker elta, an where we use the convention that h i;ni þ1 ¼ h i;1 ; h i;0 ¼ h i;ni an h mþ1;1 ¼ h 1;1 ; h 0;1 ¼ h m;1 ð2þ * Corresponing author. Tel.: ; fax: aress: rran1@twenty.rr.com (R. Ran) /$ - see front matter Ó 2008 Elsevier B.V. All rights reserve. oi: /j.cnsns

2 3902 J. Brige et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) Fig. 1. A ring network. It is convenient to efine the phase ifferences / i;j ¼ h i;jþ1 h i;j ð3þ w i ¼ h iþ1;1 h i;1 ð4þ where / i;0 ¼ h i;1 h i;0 ¼ h i;ni þ1 h i;ni ¼ / i;ni an w 0 = h 1,1 h 0,1 = h m+1,1 h m,1 = w m whereupon the equations of motion become: t h i;j ¼ w i þ a i ðsin / i;j sin / i;j 1 Þþ j;1 aðsin w i sin w i 1 Þ; where i ¼ 1;...; m; j ¼ 1;...; n i ð5þ For example, in the case of a network consisting of three sub-rings, each of size 3, we have the following nine equations of motion (Fig. 2): t h 1;1 ¼ w 1 þ a 1 ðsin / 1;1 sin / 1;3 Þþaðsin w 1 sin w 3 Þ; t h 1;2 ¼ w 1 þ a 1 ðsin / 1;2 sin / 1;1 Þ; t h 1;3 ¼ w 1 þ a 1 ðsin / 1;3 sin / 1;2 Þ; t h 2;1 ¼ w 2 þ a 2 ðsin / 2;1 sin / 2;3 Þþaðsin w 2 sin w 1 Þ; t h 2;2 ¼ w 2 þ a 2 ðsin / 2;2 sin / 2;1 Þ; t h 2;3 ¼ w 2 þ a 2 ðsin / 2;3 sin / 2;2 Þ; t h 3;1 ¼ w 3 þ a 3 ðsin / 3;1 sin / 3;3 Þþaðsin w 3 sin w 2 Þ; t h 3;2 ¼ w 3 þ a 3 ðsin / 3;2 sin / 3;1 Þ; t h 3;3 ¼ w 3 þ a 3 ðsin / 3;3 sin / 3;2 Þ ð6þ ð7þ ð8þ ð9þ ð10þ ð11þ ð12þ ð13þ ð14þ Fig. 2. A network consisting of three sub-rings, each of size 3.

3 J. Brige et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) Phase-locke motions The ynamics of a system like that in Fig. 2 may be expecte to be complicate, exhibiting various steay state moes of behavior in response to parameter changes. Amongst all such steay state motions perhaps the simplest is phase-locke motion in which each of the oscillators has the same constant frequency X: h i;j ¼ Xt þ k i;j ð15þ where k i,j is a phase angle, constant in time. Substituting (15) into (5) we get X ¼ w i þ a i ðsin / i;j sin / i;j 1 Þþ j;1 aðsin w i sin w i 1 Þ; where i ¼ 1;...; m; j ¼ 1;...; n i ð16þ where now / i;j ¼ h i;jþ1 h i;j ¼ k i;jþ1 k i;j ð17þ w i ¼ h iþ1;1 h i;1 ¼ k iþ1;1 k i;1 ð18þ Thus / i,j an w i are constants. Examination of the example of Fig. 2, Eqs. (6) (14), shows that by aing up all the equations we obtain (in the general case) NX ¼ X n i w i ð19þ where N ¼ P n i is the total number of oscillators. This yiels the interesting result that the phase-locke frequency X equals the average frequency of all N oscillators. A natural question to ask is what are the conitions on the parameters of the system such that phase-locke motion is possible? As an example we take the three sub-ring network of Fig. 2, Eqs. (6) (14). X ¼ w 1 þ a 1 ðsin / 1;1 sin / 1;3 Þþaðsin w 1 sin w 3 Þ; X ¼ w 1 þ a 1 ðsin / 1;2 sin / 1;1 Þ; X ¼ w 1 þ a 1 ðsin / 1;3 sin / 1;2 Þ; X ¼ w 2 þ a 2 ðsin / 2;1 sin / 2;3 Þþaðsin w 2 sin w 1 Þ; X ¼ w 2 þ a 2 ðsin / 2;2 sin / 2;1 Þ; X ¼ w 2 þ a 2 ðsin / 2;3 sin / 2;2 Þ; X ¼ w 3 þ a 3 ðsin / 3;1 sin / 3;3 Þþaðsin w 3 sin w 2 Þ; X ¼ w 3 þ a 3 ðsin / 3;2 sin / 3;1 Þ; X ¼ w 3 þ a 3 ðsin / 3;3 sin / 3;2 Þ ð20þ ð21þ ð22þ ð23þ ð24þ ð25þ ð26þ ð27þ ð28þ Aing the first three, the secon three an thir three of these equations, we obtain w 1 þ w 2 þ w 3 ¼ 3w 1 þ aðsin w 1 sin w 3 Þ; w 1 þ w 2 þ w 3 ¼ 3w 2 þ aðsin w 2 sin w 1 Þ; w 1 þ w 2 þ w 3 ¼ 3w 3 þ aðsin w 3 sin w 2 Þ ð29þ ð30þ ð31þ where we have use 3X=(w 1 + w 2 + w 3 ) from Eq. (19). We may think of Eqs. (29) (31) as replacing Eqs. (20),(23) an (26), respectively. Note that (31) is equivalent to the sum of (29) an (30). Thus we have: w 1 þ w 2 þ w 3 ¼ 3w 1 þ aðsin w 1 sinð w 2 w 1 ÞÞ; w 1 þ w 2 þ w 3 ¼ 3w 2 þ aðsin w 2 sin w 1 Þ ð32þ ð33þ where we have use w 3 = w 2 w 1. Trig-expaning, we obtain 2w 1 þ w 2 þ w 3 ¼ aðsin w 1 þ sin w 2 cos w 1 þ cos w 2 sin w 1 Þ; w 1 2w 2 þ w 3 ¼ aðsin w 2 sin w 1 Þ ð34þ ð35þ For given values of w 1, w 2, w 3 an a, we may or may not be able to solve these equations for w 1 an w 2. In orer to fin conitions on the parameters such that these equations are solvable, i.e., for a phase-locke solution to exist, we nee to o some algebraic manipulations which we escribe next. We use the computer algebra system Macsyma to accomplish these calculations. First we solve Eq. (34) for cosw 2, square the result an use the ientity (cosw 2 ) 2 =1 (sinw 2 ) 2 to eliminate cosw 2. Call the result equation A. Next we solve (35) for sinw 2 an substitute it into equation A. The resulting equation is free of w 2 but contains w 1 in the form sinw 1 an cosw 1. Using the ientity (cosw 1 ) 2 =1 (sinw 1 ) 2, we eliminate all powers of cosw 1 except the first. Then we solve for cosw 1, square the result, an use the foregoing ientity to eliminate cosw 1. Call the result equation B, which turns out to be a 6th egree polynomial on sinw 1. The general form of the equation is liste in the Appenix. We continue with a specific example efine by the frequencies:

4 3904 J. Brige et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) w 1 ¼ 1; w 2 ¼ 2:5; w 3 ¼ 3:1 ð36þ For these values, equation B takes the form: 4a 4 s 6 þ 36a 3 s 5 þ 3a 4 s 4 106:92a 2 s 4 18a 3 s 3 þ 116:64as 3 1:62a 2 s 2 41:9904s 2 þ 131:22as 147:622 ¼ 0 where we have use the abbreviation s = sinw 1. For a given value of a, Eq. (37) may or may not have a solution for w 1.By inspection, there is no solution for a = 0, an by continuity there is no solution for sufficiently small a. The critical case will correspon to a ouble root, which may be compute by requiring both Eq. (37) an its erivative with respect to s to simultaneously vanish. Eliminating s between these two equations gives the following equation on a cr : a 12 cr 112:32a10 cr þ 3145:73a8 cr 19851:3a6 cr þ 90477:9a4 cr :9a2 cr 3470:49 ¼ 0 ð38þ The smallest real positive root of Eq. (38) is a cr ¼ 1:8579 When this value of a is substitute back into Eq. (37) we fin the ouble root occurs at approximately s = sinw 1 = This observation suggests a metho for efficiently obtaining an approximation for a cr. Returning to Eqs. (34) an (35), we assert that the critical case will occur when jsinw 1 j1. So we set sinw 1 = 1, in which case cosw 1 = 0. Eqs. (34) an (35) become: 2w 1 þ w 2 þ w 3 ¼ að1 þ cos w 2 Þ; w 1 2w 2 þ w 3 ¼ aðsin w 2 1Þ For given values of w 1, w 2, w 3, we may obtain an expression for a cr, the critical value of a, by eliminating w 2 from these equations via the ientity sin 2 + cos 2 = 1. This results in a quaratic equation on a cr : ð37þ ð39þ ð40þ ð41þ a 2 cr þ 6ðw 1 w 2 Þa cr þ 2w 2 3 2w 2w 3 2w 1 w 3 þ 5w 2 2 8w 1w 2 þ 5w 2 1 ¼ 0 ð42þ which gives p a cr ¼ ffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðw 3 2w 2 þ w 1 Þðw 3 þ w 2 2w 1 Þ þ 3ðw 2 w 1 Þ In a similar way, the case for which sinw 1 = 1 gives the following caniates for a cr : p a cr ¼ ffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðw 3 2w 2 þ w 1 Þðw 3 þ w 2 2w 1 Þ 3ðw 2 w 1 Þ Eqs. (43) an (44) may be written in the following form: p a cr ¼ ffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð3x 3w 2 Þð3X 3w 1 Þ 3ðw 2 w 1 Þ where the two ± signs are to be taken inepenently, giving four caniate values. Note that Eqs. (29) (31) are invariant uner permutation of subscripts. Thus we may obtain eight more caniate values for a cr by replacing the subscripts 1,2 in Eq. (45) by 2,3, an again by 3,1. Of these 12 caniate values for a cr, our approximate metho says that the value for a cr is the minimum of those values which are both real an positive. (Note that the cases for which jsinw 2 j = 1 turn out to be the same as that obtaine by replacing subscripts 1,2 in (45) by 2,3.) Returning to the example of Eq. (36), we fin X=(w 1 + w 2 + w 3 )/3 = 2.2. Eq. (45) gives the following four caniate values for a cr : 9 pffiffiffi 9 2 ¼½1:95442; 7:04558; 1:95442; 7:04558Š ð46þ 5 2 Permuting subscripts gives eight aitional caniate values: 9 pffiffiffi 63 6 ¼½1:89092; 10:7091; 1:89092; 10:7091Š ð47þ pffiffiffi 9 6 i ¼2:20454i 1:8 ð48þ 10 5 Choosing the minimum of those real an positive values given in Eqs. (46) (48), we see that a cr ¼ 9 pffiffiffi 63 6 þ ¼ 1:89092 ð49þ 5 10 which is close to the exact value (39). So far we have obtaine a metho for etermining a cr. Now we present a similar metho for etermining the critical values of the other coupling coefficients, a icr. In orer to obtain a 1cr, we rewrite Eqs. (21) an (22) in the form ð43þ ð44þ ð45þ X ¼ w 1 þ a 1 ðsin / 1;2 sin / 1;1 Þ; X ¼ w 1 þ a 1 ðsinð / 1;2 / 1;1 Þ sin / 1;2 Þ ð50þ ð51þ

5 J. Brige et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) where we have use / 1,3 = / 1,2 / 1,1. Trig-expaning, we obtain X ¼ w 1 þ a 1 ðsin / 1;2 sin / 1;1 Þ; X ¼ w 1 þ a 1 ð sin / 1;2 cos / 1;1 cos / 1;2 sin / 1;1 sin / 1;2 Þ ð52þ ð53þ For given values of X, w 1 an a 1, we may or may not be able to solve these equations for / 1,1 an / 1,2. Note that Eqs. (52) an (53) are similar in form to Eqs. (34) an (35), except here the left han sies are the same in both equations, whereas they are in general ifferent in Eqs. (34) an (35). We solve Eqs. (52) an (53) for sin/ 1,2 an cos/ 1,2, an use the ientity sin 2 + cos 2 = 1 an some trig-simplification to obtain the following equation on / 1,1 : ðsin / 1;1 þ kþð2 cos / 1;1 sin / 1;1 þ sin / 1;1 þ 4k cos / 1;1 þ 5kÞ ¼0 where we have use k to abbreviate k=(x w 1 )/a 1. The first factor has the root sin / 1;1 ¼ k ¼ X w 1 a 1 This may or may not yiel a value for / 1,1. The critical case correspons to jsin/ 1,1 j = 1 which generates the following caniates for a 1cr :±(X w 1 ). Now let us consier the secon factor in Eq. (54). Solving this for cos/ 1,1 an using the ientity sin 2 + cos 2 = 1 gives the following polynomial on sin/ 1,1 which we abbreviate as s: 4s 4 þ 16ks 3 þð16k 2 3Þs 2 6ks þ 9k 2 ¼ 0 In orer to have a real solution / 1,1 of Eq. (56), the root s = sin/ 1,1 must be (a) real an (b) its absolute value must be less than unity. The critical case for (a) is that (56) have a ouble root, while the critical case for (b) is that s = 1. For a ouble root we ifferentiate (56) with respect to s an eliminate s, giving a polynomial on k: k 2 ð4k 1Þ 2 ð4k þ 1Þ 2 ðk 2 þ 3Þ ¼0 which has the real solution k = ± 1/4, giving the following caniates for a 1cr : ±4(X w 1 ). For (b) the critical case correspons to s = ± 1, which together with (56) gives k = ± 1/5, giving the following caniates for a 1cr : ±5(X w 1 ). Thus Eq. (54) has yiele six caniates for a 1cr, namely ±(X w 1 ), ±4(X w 1 ) an ± 5(X w 1 ). Of these six caniate values, the true value of a 1cr is the minimum of those values which are both real an positive. Thus we conclue that Similarly, a 1cr ¼jX w 1 j ð58þ a 2cr ¼jX w 2 j; a 3cr ¼jX w 3 j ð59þ Continuing with the previous example, we again suppose that w 1 =1,w 2 = 2.5 an w 3 = 3.1, giving X = 2.2. We previously foun that (see Eq. (49)) a cr = We now obtain a 1cr ¼ 1:2; a 2cr ¼ 0:3; a 3cr ¼ 0:9 ð60þ Numerically simulating the network in Fig. 2 for w 1 =1,w 2 = 2.5 an w 3 = 3.1 by numerically integrating Eqs. (6) (14), we fin that phase-locking occurs as long as each of a, a 1, a 2, a 3 is larger than the respective critical value obtaine above. 3. Generalization The formulas for a cr, a 1cr, a 2cr, a 3cr erive above apply to the example of Fig. 2 which consists of m=3 sub-rings, each having n 1 = n 2 = n 3 = 3 oscillators. These formulas have been extene to a general system with m = 3 sub-rings, each sub-ring with n i oscillators. We fin that for each sub-ring, a icr ¼ n i 1 jx w i j; i ¼ 1; 2; 3 ð61þ 2 An for the communication ring, an approximate expression for a cr is obtaine by taking the minimum of those values which are both real an positive from amongst the following 12: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2C i C j ðc i C j Þ; ði; jþ ¼ð1; 2Þ; ð2; 3Þ; ð3; 1Þ ð62þ where the two ± signs are to be taken inepenently, giving four caniate values, an where C i ¼ n i ðx w i Þ As an example, we take a system in which n 1 =3,n 2 = 4 an n 3 = 5, with frequencies w 1 =1,w 2 = 2.5 an w 3 = 3.1. From Eq. (19) we obtain the phase-locke frequency X ¼ð3w 1 þ 4w 2 þ 5w 3 Þ=12 ¼ 2:375 ð64þ ð54þ ð55þ ð56þ ð57þ ð63þ

6 3906 J. Brige et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) Eq. (61) then gives us the following values for the critical coupling coefficients of the sub-rings: a 1cr ¼ 1:375; a 2cr ¼ 0:1875; a 3cr ¼ 1:45 ð65þ To obtain an approximate value for the critical coupling coefficient of the communication ring, a cr, we start by computing the C i s from Eq. (63): C 1 ¼ 4:125; C 2 ¼ 0:5; C 3 ¼ 3:625 ð66þ Then from Eq. (62), a cr is the minimum of those values which are both real an positive from amongst the following list: ½6:65602; 2:59399; 1:90395i 3:125; 2:28134; 13:2187Š ð67þ Therefore we obtain a cr ¼ 2:28134 Numerical simulation of the 12 ODE s which escribe the system of Fig. 3 gives a value of a cr = 2.22 which is close to the value (68) given by our approximate metho. ð68þ 4. Systems with four Sub-rings In this section we consier conitions for phase-locking in systems which have m = 4 sub-rings, each with n i oscillators. As in the previous section, we fin that for each sub-ring, a icr ¼ n i 1 jx w i j; i ¼ 1; 2; 3; 4 ð69þ 2 where X =(w 1 + w 2 + w 3 + w 4 )/4. For the communication ring we offer an approximate formula for a cr which involves taking the minimum of those values which are positive an which are greater than maxfa icr g, i.e., the largest of the a icr, from amongst the following four permutations: 2C ic j ðc j þ C k Þ ; ði; j; kþ ¼ð1; 2; 3Þ; ð2; 3; 4Þ; ð3; 4; 1Þ; ð4; 1; 2Þ ð70þ 2 4C i C j þðc i þ C k Þ where C i ¼ n i ðx w i Þ ð71þ As an example, we take the system in Fig. 4, in which each sub-ring has n i =4 oscillators with frequencies We fin: w 1 ¼ 1; w 2 ¼ 2:5; w 3 ¼ 3:1; w 4 ¼ 1:7 ð72þ C 1 ¼ 4:3; C 2 ¼ 1:7; C 3 ¼ 4:1; C 4 ¼ 1:5 ð73þ a 1cr ¼ 1:6125; a 2cr ¼ 0:6375; a 3cr ¼ 1:5375; a 4cr ¼ 0:5625 ð74þ An approximate value for the critical coupling coefficient of the communication ring, a cr, is given by the minimum of those values which are positive an which are greater than maxfa icr g¼1:6125 from the following list (Eq. (70)): ½2:904; 1:298; 2:904; 1:298Š ð75þ Therefore we obtain a cr ¼ 2:904 ð76þ Fig. 3. A system with three sub-rings with frequencies w 1 =1,w 2 = 2.5 an w 3 = 3.1.

7 J. Brige et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) Fig. 4. A system with four sub-rings, each with the same number of oscillators, n i =4,i =1,2,3,4. Numerical integration of the governing ifferential equations for the system of Fig. 4 with frequencies (72) in the case that all coupling parameters are taken to be equal, a = a 1 = a 2 = a 3 = a 4, gives that a cr = 2.903, in close agreement with the foregoing. In the case that four of the five coupling parameters are fixe at values greater than their critical values, e.g., a =3,a 1 = 1.7, a 3 = 1.6, a 4 = 0.6, an a 2 is allowe to vary, we fin that locking occurs at a 2 = , in agreement with the foregoing. As another example, we take the system in Fig. 5, in which n 1 =3,n 2 =4,n 3 = 5 an n 4 = 6, with frequencies Eq. (72). The phase-locke frequency is X ¼ð3w 1 þ 4w 2 þ 5w 3 þ 6w 4 Þ=18 ¼ 2:15 An from Eq. (69) we obtain the following values for the critical coupling coefficients of sub-rings: a 1cr ¼ 1:15; a 2cr ¼ 0:525; a 3cr ¼ 1:9; a 4cr ¼ 1:125 ð78þ In orer to fin an approximate value for the critical coupling coefficient of the communication ring, a cr, we first compute the C i s: C 1 ¼ 3:45; C 2 ¼ 1:4; C 3 ¼ 4:75; C 4 ¼ 2:7 ð79þ Then from Eq. (70), a cr is the minimum of those values which are positive an which are greater than max fa icr g¼1:9 from the following list: ½3:370; 0:964; 3:180; 0:981Š Therefore we obtain a cr ¼ 3:180 Numerical integration of the 18 ODE s which escribe the system with frequencies Eq. (72) in the case that all sub-ring coupling parameters are larger than their respective critical values, e.g., a 1 = 1.2, a 2 = 0.6, a 3 =2,a 4 = 1.2, gives that a cr = 3.15 which is close to the above approximate value. For a thir example, we again take the system in Fig. 5 but with frequencies: We fin w 1 ¼ 3:3; w 2 ¼ 1:7; w 3 ¼ 2:1; w 4 ¼ 1 ð82þ a 1cr ¼ 1:456; a 2cr ¼ 0:217; a 3cr ¼ 0:511; a 4cr ¼ 2:111 ð83þ ð77þ ð80þ ð81þ Fig. 5. A system with four sub-rings, with n 1 =3,n 2 =4,n 3 = 5 an n 4 =6.

8 3908 J. Brige et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) The list of caniates for max fa icr g is ½0:162; 0:194; 1:520; 2:960Š ð84þ giving the approximate result: a cr ¼ 2:960 ð85þ Numerical integration with all sub-ring coupling parameters taken larger than their respective critical values gives a cr = In the first example, which involves Fig. 4 with frequencies (72), there are only 2 positive caniates for a cr. One of these is less than maxfa icr g. In the secon example, which involves Fig. 5 with frequencies (72), there are four positive caniates for a cr. Two of these are less than maxfa icr g. In the thir example, which involves Fig. 5 with frequencies (82), there are 4 positive caniates for a cr. Three of these are less than maxfa icr g. In all three cases the rule of choosing the minimum of those positive caniate values which are greater than maxfa icr g gives a reasonable approximation of the numerically compute value for a cr. 5. Symmetric solutions We have thus far focuse on conitions for phase-locking, but we have not consiere the phases themselves in a phaselocke motion. So let us return to Eq. (55) which we use to obtain a conition on a 1 for locking. Now however we will use it to stuy the phases. Substituting Eq. (55) into (52), we see that sin / 1;2 ¼ 0 ð86þ Subtracting (22) from (21) an using (86), we obtain sin / 1;3 ¼ sin / 1;1 ) sinðh 1;3 h 1;1 Þ¼sinðh 1;2 h 1;1 Þ ð87þ One solution of this equation is h 1,2 = h 1,3, in which case the sub-ring is sai to be symmetric. A phase-locke solution will be sai to be symmetric if all its sub-rings are symmetric. Eq. (55) gives a value for sin/ 1,1 : sin / 1;1 ¼ sinðh 1;2 h 1;1 Þ¼ X w 1 a 1 ð88þ Since arcsine is a multivalue function, Eq. (88) will give two values for / 1,1. One of these will have cos/ 1,1 > 0, an the other will have cos/ 1,1 <0. As an example we consier the system of Fig. 2 with Eqs. (6) (14) an frequencies (36) for parameters a = a 1 = a 2 = a 3 =2. Note that these values lie above the previously compute critical values, cf. Eqs. (49) an (60), so a phase-locke motion is expecte to exist. Numerical integration of Eqs. (6) (14) gives the following values for the phase-locke steay state: h 1;1 ¼ 0 h 1;2 ¼ 5:6397 h 1;3 ¼ 5:6397 h 2;1 ¼ 0:9683 h 2;2 ¼ 1:1189 h 2;3 ¼ 1:1189 h 3;1 ¼ 1:3516 h 3;2 ¼ 1:8183 h 3;3 ¼ 1:8183 ð89þ ð90þ ð91þ ð92þ ð93þ ð94þ ð95þ ð96þ ð97þ Since the phase of this perioic motion is arbitrary, we have chosen h 1,1 as zero in the above list. These give the following values for / i,j an w i : / 1;1 ¼ 5:6397; sin / 1;1 ¼ 0:6; cos / 1;1 ¼ 0:8 ð98þ / 1;2 ¼ 0; sin / 1;2 ¼ 0; cos / 1;2 ¼ 1 ð99þ / 1;3 ¼ 5:6397; sin / 1;3 ¼ 0:6; cos / 1;3 ¼ 0:8 ð100þ / 2;1 ¼ 0:1506; sin / 2;1 ¼ 0:15; cos / 2;1 ¼ 0:9887 ð101þ / 2;2 ¼ 0; sin / 2;2 ¼ 0; cos / 2;2 ¼ 1 ð102þ / 2;3 ¼ 0:1506; sin / 2;3 ¼ 0:15; cos / 2;3 ¼ 0:9887 ð103þ / 3;1 ¼ 0:4667; sin / 3;1 ¼ 0:45; cos / 3;1 ¼ 0:8930 ð104þ / 3;2 ¼ 0; sin / 3;2 ¼ 0; cos / 3;2 ¼ 1 ð105þ / 3;3 ¼ 0:4667; sin / 3;3 ¼ 0:45; cos / 3;3 ¼ 0:8930 ð106þ w 1 ¼ 0:9683; sin w 1 ¼ 0:8239; cos w 1 ¼ 0:5667 ð107þ

9 J. Brige et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) w 2 ¼ 0:3833; sin w 2 ¼ 0:3740; cos w 2 ¼ 0:9274 ð108þ w 3 ¼ 1:3516; sin w 3 ¼ 0:9761; cos w 3 ¼ 0:2174 All of these agree with the erive formula for symmetric solutions, cf. Eq. (88). Note that all have positive values for the cosines of / i,j an w i. This raises the following question: since we have seen in Eq. (87) that there are other solutions besies symmetric solutions, an since we have seen in Eq. (88) that in the case of symmetric solutions, they exist with both positive an negative cos/ i,j, we may ask why the numerical result showe only symmetric solutions, an those with only positive cos/ i,j? The answer lies in the stability of motion. ð109þ 6. Stability consierations So far we have stuie the existence of phase-locke motions in ring networks. We have ientifie critical values of the coupling coefficients a an a i which must be exceee for phase-locke motions to exist. In this section we consier the question of the stability of phase-locke motions [6]. We have efine phase-locke motions by Eq. (15): h i;j ¼ Xt þ k i;j ð110þ In orer to stuy the stability of such a motion, we set h i;j ¼ Xt þ k i;j þ u i;j ð111þ an linearize in u i,j. This will result in a linear system with constant coefficients, the eigenvalues of which will etermine stability. As an example we consier the system of Fig. 2 with Eqs. (6) (14). Substituting (111) in (6) (14) an linearizing, we obtain: t u 1;1 ¼ a 1 ½cos / 1;1 ðu 1;2 u 1;1 Þ cos / 1;3 ðu 1;1 u 1;3 ÞŠ þ a½cos w 1 ðu 2;1 u 1;1 Þ cos w 3 ðu 1;1 u 3;1 ÞŠ; t u 1;2 ¼ a 1 ½cos / 1;2 ðu 1;3 u 1;2 Þ cos / 1;1 ðu 1;2 u 1;1 ÞŠ; t u 1;3 ¼ a 1 ½cos / 1;3 ðu 1;1 u 1;3 Þ cos / 1;2 ðu 1;3 u 1;2 ÞŠ; t u 2;1 ¼ a 2 ½cos / 2;1 ðu 2;2 u 2;1 Þ cos / 2;3 ðu 2;1 u 2;3 ÞŠ þ a½cos w 2 ðu 3;1 u 2;1 Þ cos w 1 ðu 2;1 u 1;1 ÞŠ; t u 2;2 ¼ a 2 ½cos / 2;2 ðu 2;3 u 2;2 Þ cos / 2;1 ðu 2;2 u 2;1 ÞŠ; t u 2;3 ¼ a 2 ½cos / 2;3 ðu 2;1 u 2;3 Þ cos / 2;2 ðu 2;3 u 2;2 ÞŠ; t u 3;1 ¼ a 3 ½cos / 3;1 ðu 3;2 u 3;1 Þ cos / 3;3 ðu 3;1 u 3;3 ÞŠ þ aðcos w 3 ðu 1;1 u 3;1 Þ cos w 2 ðu 3;1 u 2;1 ÞÞ; t u 3;2 ¼ a 3 ½cos / 3;2 ðu 3;3 u 3;2 Þ cos / 3;1 ðu 3;2 u 3;1 ÞŠ; t u 3;3 ¼ a 3 ½cos / 3;3 ðu 3;1 u 3;3 Þ cos / 3;2 ðu 3;3 u 3;2 ÞŠ where the arguments of the trig terms are to be evaluate on the solution whose stability is being sought. As in the previous section, we take as an example this system with frequencies (36) an parameters a = a 1 = a 2 = a 3 =2. We substitute the values for cos/ i,j an cosw i erive in Eqs. (98) thru (109) into Eqs. (112) thru (120), an obtain the following 9 9 matrix: 0 1 4:76 1:60 1:60 1: : :60 3:60 2: :60 2:0 3: : :94 1:97 1:97 1: :97 3:97 2: ð121þ :97 2:0 3: : : :86 1:79 1:79 B :79 3:79 2:0 A :79 2:0 3:79 which turns out to have the following nine eigenvalues: ð112þ ð113þ ð114þ ð115þ ð116þ ð117þ ð118þ ð119þ ð120þ k ¼ 0; 0:55; 0:94; 4:66; 5:42 1:00i; 6:93 0:73i; 9:45 ð122þ

10 3910 J. Brige et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) The presence of the zero eigenvalue is associate with the non-uniqueness of h 1,1, corresponing to a one imensional continuum of equilibria. The other eight eigenvalues all have negative real parts, confirming the stability of the phase-locke motion. This example suggests the result that if coupling parameters a an a i are larger than their critical values, then there will exist a phase-locke solution which is symmetric an stable. This may be prove by following an approach taken by Rogge an Aeyels [8] (for simple rings) base on Gershgorin s theorem: Let A be a square matrix. Imagine a isk in the complex plane centere at a point corresponing to a ii, being an element on the main iagonal. Now let the raius of this isk be the sum of the absolute value of all the other terms in that row P j i ja ijj. Then every eigenvalue of A lies in one of these isks. In orer to use this theorem we nee to have an expression for the general stability matrix, this being a generalization of the example matrix (121). The row corresponing to a generic non-communication oscillator h i,j contains three non-zero terms: a i cos / i;j 1 a i ðcos / i;j þ cos / i;j 1 Þ a i cos / i;j ð123þ where the mile term lies on the main iagonal an the other two terms lie on other iagonals. In the case of a communication oscillator, the corresponing row contains three aitional terms: a cos w i... aðcos w i þ cos w i 1 Þ...a cos w i 1 ð124þ where again the mile term lies on the main iagonal an the other two terms lie on other iagonals. By choosing the coupling coefficients a i an a larger than their critical values we guarantee that sin/ i,j an sinw i will give real values for / i,j an w i. However since sine is multivalue, each of / i,j an w i can take on two values mo 2p, leaing to each of the cosine terms in (121) being either positive or negative. If we choose the solution for which the cosine terms are positive, then we may use Gershgorin s theorem to prove that the phase-locke solution is stable. Note that the row sum of each row is zero. Also note that the terms on the main iagonal are negative an all other terms are positive. Thus the Gershgorin isks lie in the left half-plane an are tangent to the imaginary axis at the origin. Then Gershgorin s theorem tells us that all eigenvalues lie in the left half-plane or are 0, which proves stability of the phase-locke motion. 7. Bifurcations In this section, the number of phase-locke solutions is investigate for varying coupling coefficients a i an a. We again use the example of the system in Fig. 2 with parameters (x 1, x 2, x 3 ) = (1, 2.5, 3.1) an a = a 1 = a 2 = a 3 = 2. We saw previously that for this case (a) a stable symmetric solution exists, (b) that there are other unstable solutions which exist, an (c) the critical coupling coefficient within each sub-ring (a i,cr ) is inepenent of the corresponing values in the other sub-rings an the communication ring. Consequently, we may inepenently investigate the effect of varying each coupling coefficient The communication ring The Appenix gives the generalize formula for the existence of phase-locke solutions in the communication ring. Note that if a = 0, there are no phase-locke solutions but as a? 1, there are six solutions for the phase-locke equation. We therefore expect the appearance of new roots at critical values of a, i.e. where there is a ouble root. Thus, for the x parameters iscusse above, Eq. (38) yiels critical communication ring coupling coefficients at a = ±1.8579, ±5.7564, ±8.5119, with the corresponing number of real solutions to sinw 1 increasing from 0 to 2, from 2 to 4 an from 4 to 6, respectively. Each of these is accompanie by an appropriate sinw 2 an sinw 3 which satisfies Eq. (126). We note that there are two possible caniates for w i for each term; however the appropriate phase-locke solution must further satisfy the consistency conition, w 1 þ w 2 þ w 3 ¼ 0 moð2pþ: Imposing this conition yiels a single possible solution (w 1, w 2, w 3 ) per root of Eq. (127). Fig. 6 shows the corresponing bifurcation iagram The sub-rings From Eqs. (54) an (57), the critical coupling coefficients for the sub-rings are (a) a i =(X x i ) an (b) a i =4(X x i ). Conition (a) yiels sin / 1;2 ¼ 0 from which / 1;2 ¼ 0; p: Eq. (88) gives the actual phase ifference.

11 J. Brige et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) Fig. 6. Bifurcation iagram for the communication ring ynamics. Fig. 7. Bifurcation iagram for the sub-ring ynamics. Combining this with the consistency conition, / i;1 þ / i;2 þ / i;3 ¼ 0 mo2p the solutions corresponing to / 1,2 = 0 are equivalent to h 1,2 = h 1,3, i.e. the solution is symmetric. For one of these symmetric solutions all the cos/ i,j terms are positive while the other has two negative cos/ i,j terms. The other combination / 1,2 = p an the consistency conition, yiels two solutions of the form h 1,2 = p h 1,3. These solutions also have two negative cos/ i,j terms. Thus on exceeing the lower critical coupling coefficient, four phase-locke solution caniates appear. Now consier (b). Eq. (57) shows that there are two ouble roots of the sub-ring coupling coefficient equation at a =±4 (X x i ). Thus four new phase-locke solutions emerge at the secon critical coefficient, two corresponing to sin/ 1,1 =1 an two with sin/ 1,1 = 1. Imposing the consistency conition yiels Fig. 7, the bifurcation iagram for sub-ring 1; similar curves are obtaine for the other sub-rings. 8. Conclusions We have stuie the existence, stability an bifurcation of phase-locke motions in ring networks which consist of a large communication ring, each noe of which contains an oscillator attache to a sub-ring. In the case of systems with three or four sub-rings we have given approximate expressions for critical coupling coefficients which must be exceee for phase-

12 3912 J. Brige et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) locking to occur. We have ientifie a type of phase-locke motion calle symmetric, an we have shown that if the coupling coefficients are all greater than critical, a stable symmetric phase-locke motion will exist. An important conclusion of this work is that phase-locking in each sub-ring can occur inepenently of the motion in the other sub-rings, cf. Eq. (61). On the other han phase-locking in each sub-ring is not sufficient for phase-locking of the entire structure. For the latter an aitional conition must be satisfie, cf. Eqs. (62) an (70). This work represents a first step towars unerstaning the ynamics of networks of oscillators that are more complicate than simple rings. Future research is expecte to inclue comparable stuies of yet more complicate networks, an of motions within such networks which are more complicate than phase-locke motions. Acknowlegements Authors JB an SS gratefully acknowlege the hospitality of the Department of Theoretical an Applie Mechanics, Cornell University. Support receive from the Moroccan American Commission for Eucational an Cultural Exchange through the Fulbright Program is gratefully acknowlege by author SS. This work was partially supporte uner the CMS NSF Grant CMS Nonlinear Dynamics of Couple MEMS Oscillators. Appenix. The equations for the phase ifferences within the communication ring in the general three sub-ring system may be written as (cf. Eqs. (33) an (32)): sin w 2 ¼ sin w 1 þ C 2 ð125þ a sinðw 1 þ w 2 Þ¼sin w 1 þ C 2 a þ C 3 ð126þ a where C i is efine by Eq. (63). Expaning the trigonometric term an substituting for sinw 2 in Eq. (126) cos w 2 ¼ 1þcot w 1 sin w 1 þ C 2 þ C 2 þ C 3 a a sin w 1 a sin w 1 Squaring, aing to (125) square, an finally substituting for cosw 1, yiels the following relationship for sinw 1 (cf. Eq. (37)): F a ðsin w 1 Þ¼a 6 sin 6 w 1 þ a 5 sin 5 w 1 þ a 4 sin 4 w 1 þ a 3 sin 3 w 1 þ a 2 sin 2 w 1 þ a 1 sin w 1 þ a 0 ¼ 0 where ð127þ a 6 ¼ 4a 4 a 5 ¼ 8a 3 ðc 2 þ 2C 3 Þ a 4 ¼ 4a 2 ðc 2 3 þ 6C 2C 3 þ 6C 2 2 Þ 3a4 a 3 ¼ 8aC 2 ðc 2 þ C 3 Þð2C 2 þ C 3 Þ 4a 3 ðc 3 þ 2C 2 Þ a 2 ¼ 4C 2 2 ðc 2 þ C 3 Þ 2 þ 2a 2 ðc 2 3 2C 3C 2 2C 2 2 Þ a 1 ¼ 4aC 2 3 ðc 3 þ 2C 2 Þ a 0 ¼ C 2 3 ðc 3 þ 2C 2 Þ 2 When a = 0, Eq. (127) is reuce to F 0 ðsin w 1 Þ¼4C 2 2 ðc 2 þ C 3 Þ 2 sin 2 w 1 þ C 2 3 ðc 3 þ 2C 2 Þ 2 ¼ 0: Since both terms are semi-positive efinite, this equation has no real roots unless C 3 =0orC 3 +2C 2 = 0. Meanwhile as a? 1, the equation is reuce to F 1 ðsin w 1 Þ¼4a 4 sin 6 w 1 3a 4 sin 4 w 1 ¼ 0 p which yiels six real roots (two simple roots at ffiffiffi 3 =2 an a quaruple root at 0) for sinw1. Therefore, as a is increase the number of real roots of the equation increases from 0 to 6. There is therefore a minimum value of a for which a real solution will exist. If we let sinw 1 = s, then the number of roots change when both F(s) an F/s = 0, i.e., when there is a ouble root. References [1] Ashwin P, King GP, Swift JW. Three ientical oscillators with symmetric coupling. Nonlinearity 1990;3: [2] Brige J, Menelowitz L, Ran R, Sah S, Verugo A. Dynamics of a ring of three couple relaxation oscillators. Commun Nonlinear Sci Numer Simulat. oi: /j.cnsns

13 J. Brige et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) [3] Collins JJ, Stewart I. A group-theoretic approach to rings of couple biological oscillators. Biol Cybern 1994;71: [4] Menelowitz L, Verugo A, Ran R. Dynamics of three couple limit cycle oscillators with application to artificial intelligence. Commun Nonlinear Sci Numer Simulat 2009;14: [5] Nishiyama N, Eto K. Experimental stuy on three chemical oscillators couple with time elay. J Chem Phys 1994;100: [6] Ran RH. Lect Notes Nonlinear Vib (version 52). available on-line at < (2005). [7] Ran R, Wong J. Dynamics of four couple phase-only oscillators. Commun Nonlinear Sci Numer Simulat 2008;13: [8] Rogge JA, Aeyels D. Stability of phase locking in a ring of uniirectionally couple oscillators. J Phys A: Math Gen 2004;37: