On Synchronization of Kuramoto Oscillators

Transcription

1 Proceedngs of the 44th IEEE Conference on Decson and Control, and the European Control Conference 5 Sevlle, Span, December 1-15, 5 TuC14. On Synchronzaton of uramoto Oscllators khl Chopra and Mark W. Spong Abstract Synchronzaton s a key concept to the understandng of self-organzaton phenomena occurrng n coupled oscllators of the dsspatve type. In ths paper we study one of the most representatve models of coupled phase oscllators, the uramoto model. The tradtonal uramoto model all-toall connectvty s sad to synchronze f the angular frequences of all oscllators converge to the mean frequency of the group and the oscllators get phase locked. Recently, Jadbabae et. al. calculated a lower bound on the couplng gan whch s necessary for the onset of synchronzaton n the tradtonal uramoto model. It was also shown that there exsts a large enough couplng gan so that the phase dfferences are locally asymptotcally stable. Furthermore, the authors demonstrated that the convergence s exponental when all oscllators have the same natural frequency. In ths paper we assume that the natural frequences of all oscllators are arbtrarly chosen from the set of reals. We develop a tghter lower bound on the couplng gan, as compared to the one proposed by Jadbabae et. al., whch s necessary for the onset of synchronzaton n the tradtonal uramoto model. Our man result says that t s possble to fnd a couplng gan such that the angular frequences of all oscllators locally exponentally synchronze to the mean frequency of the group. To the best of our knowledge, ths s the frst result whch demonstrates that n the tradtonal uramoto model, wth all-to-all couplng and dfferent natural frequences, the oscllators locally exponentally synchronze. Smulatons are also presented to valdate the proposed results. I. ITRODUCTIO Collectve synchronzaton phenomena have been observed n bologcal, chemcal, physcal and socal systems for centures. The concept of synchronzaton mples that multple perodc processes wth dfferent natural frequences come to acqure a common natural frequency as a result of ther mutual or one-sded nteracton. Ths phenomenon s observed when system of oscllators lock on to a common frequency despte dfferences n the natural frequency of the ndvdual oscllators. Bologcal examples nclude groups of synchronously flashng frefles [1], crckets that chrp n unson [13] etc. Examples n physcs nclude the superconductng Josephson juncton [16]. The mportance of synchronzaton n nature may be realzed from the fact that what looks lke a sngle perodc process on a macroscopc scale often turns out to be collectve oscllaton resultng from the mutual synchronzaton among large number of consttuent oscllators. The human heartbeat may be taken as an example of ths phenomenon. As the consttuent Research partally supported by the Offce of aval Research under grant , by the atonal Scence Foundaton under grants ECS- 141, HS-33314, and CCR-9.Chopra and M.W.Spong are wth Coordnated Scence Laboratory, Unversty of Illnos at Urbana-Champagn, 138 W. Man St., Urbana, IL 6181 USA nchopra,mspong@uuc.edu oscllators n nature rarely posses dentcal frequences, mutual synchronzaton appears to be a unque mechansm for producng and mantanng macroscopc rhythmcty. Collectve synchronzaton was frst studed by Wener [15], who speculated that t s nvolved n the generaton of alpha rhythms n the bran. It was then taken up by Wnfree [17] who used t to study crcadan rhythms n lvng organsms. He contended that theoretcal understandng of the orgn of collectve rhythmcty would be best studed by studyng ts onset, that s, by treatng t as a knd phase transton or bfurcaton. Hs attempt to study mutual synchronzaton n mult-oscllator systems was based on a phase descrpton [18]. Wnfree s model was sgnfcantly extended by uramoto n [5], [6] where he developed results what s now popularly known as the uramoto model. uramoto s work, and the later attempts to answer the questons that were rased by hs formulatons, have been elegantly summarzed n [11]. Recently, control theoretc methods have been used n [14], [1], [4], [] to address the synchronzaton phenomenon. Phase models of coupled oscllators were used to derve control laws for stablzng collectve moton of a group of self-propelled partcles n [1]. In [7] consensus problems were dscussed for a network of dynamc agents wth fxed and swtchng topologes. Stablty analyss was carred out for a undrectonal rng of oscllators wth uramoto type dynamcs n [9]. In [8] t was shown that only phase lockng solutons correspondng to π, π can be locally asymptotcally stable, and a condton for guaranteeng local asymptotc stablty was also derved. However, the condton was dependent on the parameter r, along wth the couplng gan and natural frequences of the oscllators. Recently n [3], control and graph theoretc methods were used to analyze uramoto oscllators for an arbtrary bdrectonal graph topology. The authors derved the value of the couplng gan L for the onset of synchronzaton n the tradtonal uramoto Model all-to-all connectvty. It was also shown that there exsts a large enough couplng gan gan so that the phase dfferences are locally π, π asymptotcally stable. Furthermore, the authors also demonstrated that the convergence s exponental, when all oscllators have the same natural frequency. In ths paper we study the case of a large but fnte number of uramoto oscllators where every oscllator s connected to every other oscllator the orgnal uramoto model. We assume n ths note that the natural frequences ω of all oscllators are arbtrarly chosen from the set of reals and we do not mpose any partcular dstrbuton on them. We construct a tghter lower bound on the couplng /5/$. 5 IEEE 3916

2 gan, as compared to the one developed by Jadbabae et. al., whch s necessary for the onset of synchronzaton n the tradtonal uramoto model. Our man result says that t s possble to fnd a couplng gan = nv such that all trajectores whch start wthn the set defned by D = {θ,θ j R θ θ j π ɛ} where ɛ< π 4 s an arbtrary postve number, exponentally synchronze. II. SUMMARY OF URAMOTO S RESULTS In ths secton we descrbe the orgnal uramoto model and summarze the man fndngs. The uramoto model conssts of a populaton of oscllators who dynamcs are governed by the followng equatons θ = ω + snθ j θ, =1,,..., 1 j=1 where θ S 1 s the phase of the th oscllator, ω R s ts natural frequency and > s the couplng gan. The natural frequences are dstrbuted wth probablty densty gω, where gω s assumed to be unmodal and symmetrc about the mean frequency Ω.e., gω + ω =gω ω. By makng a sutable choce of a rotatng frame, θ θ +Ωt, where Ω s the frst moment mean of gω, the dynamcs 1 get transformed to an equvalent system of phase oscllators whose natural frequences have a zero mean. Therefore we have that gω =g ω for all ω. To get a nce ntuton about the problem, the oscllators may also be thought of as ponts movng on a unt crcle. The problem s then to characterze the couplng gan so that the oscllators synchronze. The oscllators are sad to synchronze f θ θ j as t, j =1,..., or n other words the phase dfferences gven by θ θ j, j = 1,,..., become constant asymptotcally. Imagnng these oscllators on crcle as ponts, the ponts then move wth the same angular frequency and hence, angular dstance phase dfference between the ponts reman constant wth tme. Defne the order parameter r as re Ψ = 1 e θ j j=1 The order parameter rt wth rt 1 s a measure of phase coherence of the oscllator populaton. If the oscllators synchronze, then the parameter converges to a constant r 1, but f the oscllators add ncoherently then the order parameter r remans close to zero. Usng the order parameter, the model 1 can be rewrtten as [11] θ = ω + rsnθ Ψ, =1,,..., In the contnuum lmt case where, uramoto showed that there exsts a value of the couplng gan such that for all < c, the oscllators are ncoherent or reman unsynchronzed, but for > c the ncoherent state becomes unstable, the oscllators start synchronzng and eventually rt settles at some r < 1. uramoto calculated closed form solutons for the gan c the crtcal gan for the onset of synchronzaton, and r. Furthermore t has been shown va smulatons that for > c, the populaton of oscllators dvdes nto two groups. The oscllators whose natural frequences s close to the mean frequency, lock on to form a synchronzed cluster and start rotatng wth the mean frequency Ω, whle those whose natural frequences are far way from the mean of the group, drft relatve to the synchronzed cluster oscllators. However there are some mportant questons stll assocated wth the uramoto model and they form the motvaton for ths paper. We lst here one of the open problems whch we address n ths paper. It has been shown va numercal smulatons that for when > c, the parameter rt grows exponentally and saturates at some r < 1. Tll date there s no analyss whch shows that the oscllators n the orgnal uramoto model where the oscllators have dfferent natural frequences synchronze exponentally even locally and quotng Strogatz [11] obody has even touched the problems of global stablty and convergence. The paper s organzed as follows. In the next secton we calculate the crtcal gan c whch s necessary for the onset of synchronzaton n the whole populaton of oscllators. In Secton IV we develop a lower bound on the couplng gan = nv whch s suffcent for oscllator synchronzaton wthn an arbtrary compact set of π, π, and then n Secton V t s demonstrated that the oscllators locally exponentally synchronze. The results are valdated by smulatons n VI and summarzed n VII. III. OSET OF SYCHROIZATIO As we are nterested n the evoluton of the phase dfferences, the phase dfference dynamcs can be wrtten down usng 1 as θ θ j = ω ω j + { snθ θ j + snθk θ +snθ j θ k } If the oscllators are to synchronze.e. θ θ j as t, j = 1,...,, the R.H.S of must go to zero. In ths secton, we calculate a lower bound on the couplng gan so that there s a possblty of the R.H.S of to go to zero. In other words, we calculate a necessary condton for the onset of synchronzaton n 1. The oscllators can only synchronze f equaton 1 has at least one fxed pont, j =1,..., j. The fxed pont equaton can be wrtten down as ω j ω = { snθj θ + snθk θ +snθ j θ k }

3 Assumng wthout loss of generalty that ω j >ω, to calculate a lower bound on the couplng gan satsfyng 3, we need to maxmze the expresson E =snθ j θ + snθ k θ +snθ j θ k 4 Usng elementary calculus, the frst order necessary condtons for maxmzng 4 are gven by = cosθ j θ cosθ k θ = 5 θ =cosθ j θ + cosθ j θ k = 6 θ j = cosθ k θ cosθ j θ k = 7 θ k Usng 7, we get that ether θ k = θ + θ j or θ = θ j It s easly seen that θ = θ j =mples that E =,aswe are lookng for a maxmum we nvestgate the other soluton. Substtutng θ k as θ +θ j n 5 we get the condton θj θ cosθ j θ + cos = θj θ cosθ j θ + cos = 4cos θ j θ θj θ + cos = It s to be noted that the same equaton wll be obtaned by substtutng θ k as θ +θ j n 6. Solvng the above quadratc equaton we get θj θ cos = ± +3 8 As cosx 1 x R, a well defned soluton for all s gven by θj θ cos = Denote the optmal value of θ j θ maxmzng 4 by θ j θ opt. It can also be verfed that the second order necessary condton for optmalty maxmum n ths case gven by m, n =1,..., θ m θ n s also satsfed by θ j θ opt, and hence the optmal maxmum value of E s gven as θj θ opt E max =snθ j θ opt + sn Thus the crtcal gan couplng gan requred for onset of synchronzaton n s gven by c = ω j ω E max 3918 If the natural frequences belong to a compact set, then the crtcal gan couplng gan requred for onset of synchronzaton n 1 s gven as c = ω max ω mn 9 E max where ω max >,ω mn s the maxmum and mnmum frequences n the set of natural frequences. The phrase crtcal gan couplng gan requred for onset of synchronzaton does not mply that at c the oscllators synchronze. The crtcal gan c s the gan below whch the oscllators cannot synchronze. It s nterestng to compare the condton 9 wth that for the crtcal gan obtaned n [3]. The value for the crtcal couplng n [3] s gven as L = ω max ω mn 1 1 Therefore, comparng 9 wth 1 we fnd that n 1 E max = 1 mplctly. We contend that E max = 1 s a value not achevable by the functon E. Ths s so because n [3] the authors assumed that at E max, θ m θ n = π m, n =1,...,. Ths clearly s not possble as the phase dfferences θ m θ n m, n =1,..., are not ndependent. Thus the onset of synchronzaton s not possble for all couplng gans satsfyng L < c. Only for c, the dynamcal system gven by 1 may synchronze. IV. SYCHROIZATIO OF URAMOTO OSCILLATORS In the prevous secton we developed the lower bound on the crtcal gan denoted by c whch s necessary for the onset of synchronzaton n the tradtonal uramoto model. In ths secton we develop a lower bound on the couplng gan whch s suffcent for synchronzaton of the oscllators wthn an arbtrary compact set of π, π. The assumpton n the analyss that follows s that the ntal phase of all oscllators le wthn the set descrbed by D = {θ,θ j R θ θ j π ɛ} where ɛ< π 4 s an arbtrary postve number. We wll develop a lower bound on the couplng gan denoted by nv whch makes ths set postvely nvarant for all oscllators,.e. θ θ j Dat t= θ θ j D t>. Then havng phase-locked the oscllators n D, we wll show that the oscllators synchronze. The phase dfference dynamcs as descrbed by can be rewrtten as θ θ j = { ω ω j snθ θ j + 1 snθ θ j +snθ k θ +snθ j θ k } 11 Consder the term 1 snθ θ j +snθ k θ +snθ j θ k

4 Ths can be rewrtten usng some trgonometrc rearrangements as 1 snθ θ j 1 cosθ k θ +θ j cos θ θ j = 1 snθ θ j C k where C k = 1 cosθ k θ +θ j. It s easy to see that cos θ θ j θ θ j D, C k < 1. Usng ths, 11 can be rewrtten as θ θ j = { ω ω j snθ θ j + 1 C k snθ θ j } = { ω ω j snθ θ j 1 1 } C k 1 We are now n a poston to state the frst result of ths secton. Theorem 4.1: Consder the system dynamcs as descrbed by 1. Let all ntal phase dfferences at t= be contaned n the compact set D = {θ,θ j θ θ j π ɛ, j = 1,...,}. Then there exsts a couplng gan nv > such that θ θ j D t>. Proof: Let the postve defnte Lyapunov functon for the dynamc system governed by 1 be gven as V = 1 θ θ j The dervatve of the Lyapunov functon along trajectores of the system 1 s gven as V = 1 θ θ j θ θ j ω ω j =θ θ j snθ θ j 1 1 θ θ j ω ω j C k θ θ j snθ θ j 1 C k θ θ j ω ω j θ θ j snθ θ j 1 where t has been used n the last equaton that C k < 1 and that C k =for k =, j. Thus the dervatve can be wrtten as V θ θ j ω ω j θ θ j snθ θ j It s to be noted that the functon snθ θ j θ θ j s always nonnegatve n the consdered doman. Therefore, f > ω ωj cosɛ, the dervatve of the Lyapunov functon s negatve at θ θ j = π ɛ and thus the phase dfference θ θ j cannot leave the set D. Fnally, f = nv > ω max ω mn cosɛ, all phase dfferences θ θ j =1,,..., are postvely nvarant wth respect to the compact set D. Havng trapped the phase dfferences wthn the desred compact set D by approprately choosng the couplng gan, we demonstrate that the oscllators synchronze. Theorem 4.: Consder the system dynamcs as descrbed by 1. Let all ntal phase dfferences at t= be contaned n the compact set D. If the couplng gan s chosen such that = nv, then all the oscllators synchronze.e. θ θ j as t, j =1,..., Proof: Consder the postve functon, S = 1 θ T θ where θ =[ θ 1... θ ] T Dfferentatng along trajectores of the system 1 we get Ṡ = θ 1 θ1 + θ θ θ n θn cosθ 1 θ θ θ cosθ n θ 1 θ n θ 1 = θ 1 β + θ β. + θ n β cosθ 1 θ θ 1 θ cosθ n θ θ n θ cosθ θ n θ θ n cosθ 1 θ n θ 1 θ n where β =. On rearrangng terms and smplfyng we have that, Ṡ = cosθ θ j θ θ j 13 j=1 =1 Due to Theorem 4.1 we have that θ θ j D,, j. Ths gves us that cosθ θ j >, j and hence Ṡ. Hence all angular frequences.e. θ are bounded. Consder the set E = {θ θ j, θ R, j Ṡ =}. The set E s characterzed by all trajectores such that θ = θ j,, j. Let M be the largest nvarant set contaned n E. Usng Lasalle s Invarance Prncple, all trajectores startng n D converge to M as t. Hence the oscllators synchronze asymptotcally. The above theorem tells us that all the oscllators start movng wth the same angular frequency, but what s the consensus value of the group? Or n other words what s the common angular frequency to whch all the oscllators converge? We provde an answer to ths queston n the next result. Corollary 4.3: Consder the system represented by 1. If = nv, then the oscllatory asymptotcally converge to the mean natural frequency of all oscllators.e., θ = θ j = =1 ω =Ω, j =1,..., as t. Proof: It s easy to see from 1 that θ = ω 14 =1 =1 As θ θ j, j =1,..., as t, we have that =1 θ ω =1,,..,, and hence asymptotcally all oscllators start movng wth the mean natural frequency of the group. 3919

5 V. EXPOETIAL SYCHROIZATIO In the prevous secton we demonstrated that wth sutable choce of the couplng gan, the oscllators synchronze. In ths secton we demonstrate that the oscllators converge exponentally to the mean natural frequency of the group. In the uramoto model, all oscllators are connected va all to all topology,.e. every oscllator or node s connected to every other oscllator node. These nodes form a graph and our results n ths secton use some algebrac propertes of the underlyng graph. We provde a bref ntroducton of the graph theory tools used n ths secton. A graph theoretc approach to the uramoto Oscllator problem was used recently n [3] and we adapt ther concse ntroducton n ths secton. The graph can be descrbed by two matrces whch encode the topology of the nterconnecton. The ncdence matrx of an orented graph G α wth vertces and e edges s the e matrx such that: B j =1 f the edge s ncomng to vertex, B j = 1 s the edge j s outcomng from the vertex, and otherwse. The symmetrc matrx defned as: L = BB T s called the Laplacan of G and s ndependent of the choce of orentaton α. The Laplacan has several mportant propertes: L s always postve semdefnte wth a zero egenvalue; the algebrac multplcty s equal to the connected components n the graph; the dmensonal vector assocated wth the zero egenvalue s the vector of ones 1. The spectrum of the Laplacan matrx of the graph captures many topologcal propertes of the graph. It was shown by Fedler that the frst non-zero egenvalue λ L also referred to as the algebrac connectvty and the Felder egenvalue gves a measure of connectedness of the graph. If we assocate a postve number W to each edge and we form the dagonal matrx W e e := dagw, then the matrx L W G =BWB T s a weghted Laplacan whch fulflls the aforementoned propertes. In the uramoto model, all nodes are connected to all other nodes and hence the dynamcs whch were prevously descrbed by 1 can be equvalently wrtten down as θ = ω BsnBT θ 15 where B s the ncdence matrx of the unweghted graph, θ and ω are 1 vectors. It s also helpful to defne the e 1 vector of the phase dfferences φ := B T θ. Let us revst Theorem 4., where t was shown that the oscllators synchronze. The postve functon S s gven as S = 1 θ T θ 16 The dervatve of ths functon along trajectores of 15 can be wrtten as Ṡ = θ T Bdag cosφ B T θ The matrx L G =Bdag cosφ B T s the weghted Laplacan and s descrbed as follows L W G = cosθ k θ =1,...,,k L W G j = cosθ θ j, j =1,..., j Clearly, f all phase dfferences φ D, then the weghted Laplacan matrx L G s postve-semdefnte, and hence the result of Theorem 4. follows. In the next theorem we extend ths result by developng an exponental bound on the synchronzaton rate of the oscllators. Theorem 5.1: Consder the dynamcs of the system as descrbed by 15. If the phase dfferences gven by φ Dat t =and the couplng gan s selected such that = nv, then the oscllators synchronze exponentally at a rate no worse that snɛ. Proof: It follows from 14 that =1 Ω= θ =1 = ω whch mples that Ω s an nvarant quantty. Followng [7], the vector θ can be wrtten down as θ =Ω1+δ 18 where 1 s the dmensonal vector of ones assocated wth the zero egenvalue of the weghted Laplacan L W G, δ R n satsfes =1 δ =as θ =1 = Ω. The vector δ s orthogonal to 1 and was referred to as the group dsagreement vector n [7]. Substtutng 18 n 16, we have that dδ T δ = dt δt L W Gδ 19 where we have used the fact that Ω s an nvarant quantty and that 1 T L W G =as 1 s an egenvector assocated wth the zero egenvalue of L W G. It s easy to see from the above equaton and the postve defnteness of the matrx L W G n the projected space orthogonal to 1 that the dsagreement vector δ exponentally converges to the orgn. The exponental convergence of δ and 18 tells us that the oscllators start movng wth the mean frequency of the group. As λ L G s the Fedler egenvalue smallest non-zero egenvalue of the weghted Laplacan λ L G, we have from 19 that dδ T δ dt δt λ L W Gδ δt λ Bdag cosφ B T δ δt snɛλ BB T δ snɛδ T δ as mn{cosφ} : φ D= cos π ɛ = snɛ and for an all-to-all connected topology λ BB T =. Thus the exponental convergence rate for synchronzaton s no worse that snɛ. = θ T L G θ 17 39

6 Phase Dfferences Tme Fg. 1. The oscllators do not synchronze when L << c. VI. SIMULATIOS In ths secton we smulate the uramoto oscllator model wth =3oscllators. The oscllators are chosen such that ther natural frequency are as follows, ω 1 =1, ω =3, and ω 3 =7unts are n rad/s. The mean frequency of the group s then gven by Ω= The couplng gan = c whch s necessary for the onset of synchronzaton s gven by 9, and as ω max =7, ω mn =1,wehave that c = Usng 1, the necessary lower bound provded n [3] equals L =45. To defne the set n whch we want to confne our phase dfferences, choose ɛ =.5, and thus the desred compact set D s gven as D = {θ,θ j R θ θ j π ɛ} = {θ,θ j R θ θ j.578} The desred couplng gan s gven by the formula = nv > ω max ω mn cosɛ and thus substtutng the relevant values, we select nv = 167. The smulatons were performed for three values of the couplng gan L <=5< c, c <=53< nv and = nv. The ntal phase dfferences at t= were selected so that they were n D. In the frst smulaton, when the couplng gan satsfes L <=5< c, the phase dfferences dverge and the oscllators are unsynchronzed as seen n Fgure 1. In the next smulaton scenaro, the couplng gan s chosen to be c <=53< nv =53and as seen n Fgure, the oscllators synchronze. On the other hand we see that the set D s not able to attract the phase dfferences. Fnally settng the couplng gan = nv, we fnd fnd that the oscllators synchronze, and the phase dfferences are ndeed nvarant wth respect to the compact set D as seen n Fgure 3. Also the angular frequences of all oscllators θ = 1,, 3 exponentally converge to the mean frequency Ω as seen n Fgure 4. The next smulaton we perform s wth the ntal phase dfferences outsde the set D and wth the couplng gan = nv. It turns out that the phase dfferences stll converge to the desred set D Fgure 5. Ths behavor seems nterestng and shall be a topc of our future research. VII. COCLUSIOS In ths paper we studed the phenomenon of synchronzaton n the uramoto model wth an arbtrary but fnte number of oscllators. A necessary condton n the form of a lower bound on the couplng gan = c was establshed for the onset of synchronzaton n the uramoto model. A lower bound on the couplng gan = nv was developed whch s suffcent for oscllator synchronzaton wthn an arbtrary compact set of π, π, provded the oscllators phases are contaned n that compact set at t=. Fnally t was shown that the oscllators synchronze exponentally. In [3], exponental convergence was only demonstrated for the case when the natural frequences ω are same for all oscllators. In ths paper we have extended ths for the case when the natural frequences may be dfferent for all oscllators. Smulatons were also presented to justfy the proposed results. Future work nvolves extendng ths work to arbtrary swtchng topologes and for networks wth tme delays. VIII. ACOWLEDGEMETS The authors would lke to thank the anonymous revewers for ther helpful suggestons. REFERECES [1] J. Buck, Synchronous Rhythmc Flashng of Frefles. II, Quarterly Revew of Bology, 63:65, [] A. Jadbabae, J.Ln, and A.S. Morse, Coordnaton of groups of moble autonomous agents usng nearest neghbor rules, IEEE Transactons on Automatc Control, Vol. 48, pp , June 3. [3] A. Jadbabae,. Motee, and M. Barahona, On the stablty of the uramoto model of coupled nonlnear oscllators, Proceedngs of the Amercan Control Conference, pp , 4. [4] E.W. Justh and P.S. rshnaprasad, A smple control law for UAV formaton flyng, Techncal Report -38, Insttute for Systems Research,. [5] Y. uramoto, In Internatonal Symposum on Mathematcal Problems n Theoretcal Physcs, Lecture otes n Physcs,, Vol. 39, Sprnger, ew York, [6] Y. uramoto, Chemcal Oscllatons, Waves, and Turbulence, Sprnger, Berln, Phase Dfferences Fg.. nv Tme The oscllators synchronze when the couplng gan c << 391

7 Phase Dfferences.1.1 Phase Dfferences Tme Fg. 3. The oscllators synchronze and are nvarant wth respect to D Tme Fg. 5. The set D s able to attract the phase dfference even when they start outsde D. [7] R. Olfat-Saber and R.M. Murray, Consensus problems n networks of dynamc agents wth swtchng topology and tme-delays, IEEE Transactons on Automatc Control, Vol.49, pp , Sept. 4. [8] J.A. Rogge and D. Aeyels, Exstence of Partal Entranment and Stablty of Phase Lockng Behavor of Coupled Oscllators, Progress of Theoretcal Physcs, Vol. 11, pp , 4. [9] J.A. Rogge and D. Aeyels, Stablty of phase lockng n a rng of undrectonally coupled oscllators, Journal of Physcs A, Vol. 37, pp , 4. [1] R. Sepulchre, D. Paley, and. Leonard, Collecton moton and oscllator synchronzaton, Cooperatve Control, Lecture otes n Control and Informaton Scences, Vol. 39, Eds: V. umar,. Leonard, and A. S. Morse, Sprnger Verlag, 4. [11] S.H. Strogatz, From uramoto to Crawford: explorng the onset of synchronzaton n populatons of coupled oscllators, Physca D, 143:1-,. [1] S.H. Strogatz, SYC: The Emergng Scence of Spontaneous Order, Hyperon Press, ew York, 3. [13] T.J. Walker, Acoustc synchrony: two mechansms n the snowy tree crcket, Scence 166: , [14] W. Wang and J.J.E. Slotne, On partal contracton analyss for coupled nonlnear oscllators, Techncal Report, onlnear Systems Laboratory, MIT, 3. [15].Wener, onlnear Problems n Random Theory, MIT Press, Cambrdge, Ma, [16]. Wesenfeld, P. Colet, and S. Strogatz, Frequency lockng n Josephson arrays: Connecton wth the uramoto model, Phys. Rev. E, 57, pp , [17] A.T Wnfree, The Geometry of Bologcal Tme, Sprnger, ew York, 198. [18] A.T. Wnfree, Bologcal rhythms and the behavor of populatons of coupled oscllators, J. Theoretcal Bology, , Angular Frequency Tme Fg. 4. The angular frequences θ = 1,, 3 of the oscllators exponentally converge to the mean frequency Ω. 39