+ All Categories
Home > Documents > Part Entrainment

Part Entrainment

Date post: 07-Apr-2018
Category:
Upload: neopba
View: 242 times
Download: 0 times
Share this document with a friend

of 21

Transcript
  • 8/3/2019 Part Entrainment

    1/21

    Partial entrainment in the finite

    Kuramoto-Sakaguchi model

    Filip De Smet, Dirk Aeyels

    SYSTeMS Research Group, Dept. of Electrical Energy, Systems and Automation,

    Ghent University, Technologiepark-Zwijnaarde 914, 9052 Zwijnaarde, Belgium

    Abstract

    Although modifications of the Kuramoto model have been the subject of extensiveresearch, the model itself is not yet fully understood. We offer several results andobservations, some analytic, others through simulations. We derive a sufficient con-dition for the existence of a solution exhibiting partial entrainment with respect toa given subset of oscillators; the result also implies persistence of the entrainmentbehavior under perturbations.

    The critical values of the coupling strength, defining the transitions between dif-ferent forms of partial entrainment, are predicted by an analytical approximation,based on the fact that oscillators with large differences in their natural frequencieshave little influence on each others entrainment behavior; the predictions agreewith the actual values, obtained by simulations.

    We indicate (by simulations) that entrainment can disappear with increasingcoupling strength, and that, in arrays of Josephson junctions, a similar phenomenoncan be observed, where it is also possible that a junction leaving one entrained subset joins another entrained subset.

    Pacs numbers: 05.45.Xt, 89.75.Fb, 74.81.Fa

    Key words: coupled oscillators; Kuramoto model; entrainment

    1 Introduction

    The Kuramoto model [6] is a prototype model for the study of systems of cou-pled oscillators. A discussion of illustrative examples, such as flashing fireflies,coupled laser arrays and pacemaker cells in the heart, can be found in [17].We will consider the extension introduced in [15] by Kuramoto and Sakaguchi,

    Preprint submitted to Elsevier Preprint 1 June 2007

  • 8/3/2019 Part Entrainment

    2/21

    which is described by the differential equations

    i(t) = i +K

    N

    Nj=1

    sin(j(t) i(t) ), (1)

    i {1, . . . , N }, t R, where N is the number of oscillators, K 0 is thecoupling strength, ||

    2and the i are drawn from a distribution g. (The

    Kuramoto model corresponds to the case = 0.) The parameters i representthe natural frequencies of the oscillators and determine the behavior of thesystem for K = 0. For a system with an infinite number of oscillators Ku-ramoto and Sakaguchi showed that there is a critical value Kc of the couplingstrength above which a solution exists exhibiting partial synchronization. ForK > Kc this solution is characterized by two different groups of oscillators;those in the first group are locked at a frequency while the other oscillators

    are moving with long term average frequencies different from . The stabilityproperties of this solution are not yet fully understood.

    The analysis by Kuramoto and Sakaguchi cannot simply be transposed to thecase with a finite number of oscillators. Analytical results are hard to obtainand treat special cases such as identical natural frequencies [18] or the caseof complete phase locking behavior and its stability properties [2, 5]. Mostresearch focuses on simulations [7, 12] and is concerned with modificationssuch as an alternative interaction structure [4, 11, 3, 13, 10, 9, 14], althoughthe unmodified Kuramoto model has also received some attention in recent

    years [7, 12, 8]. For an overview see e.g. [16, 1].

    In this paper we will investigate general partial entrainment in the Kuramoto-Sakaguchi model with non-identical natural frequencies, providing both an-alytical results and simulations. In the next section we describe the generalbehavior of the Kuramoto-Sakaguchi model as it is observed in simulationswith small N, and in support of these observations we formulate a sufficientcondition (for general finite N) for the entrainment of a given subset of thepopulation of oscillators. The proof implies persistence of the entrainmentbehavior under perturbations in the initial condition and the result remains

    non-trivial in the limit N .Section 4 deals with an estimation for the critical values of the couplingstrength defining the transitions between different forms of partial entrain-ment for the case = 0. In section 5 we illustrate the phenomenon for which,for both the Kuramoto-Sakaguchi model (with general ) and a system ofJosephson junction arrays, entrainment may disappear with increasing cou-pling strength.

    2

  • 8/3/2019 Part Entrainment

    3/21

    2 The general scenario

    Let be a solution of the system (1) and let Se {1, . . . , N } be non-empty.Definition 1. If

    C > 0 : |i(t) j(t)| < C, t 0, i, j Se,

    then the solution exhibits partial entrainment with respect to Se, and Se iscalled an entrained subset.

    Notice that according to this definition (which slightly differs from the onein [2], where the system was said to exhibit partial entrainment if at leasttwo oscillators exist with bounded phase differences) there is always a trivialform of entrainment corresponding to the singletons {i} {1, . . . , N }. Partialentrainment with respect to the entire population is called full entrainment.

    Simulations indicate that, for most values of the natural frequencies and thecoupling strength, the entrainment behavior is independent of the initial con-dition allowing us to refer to the entrainment behavior of the system andfurthermore that for each oscillator the long term average frequency convergesto a constant (i.e. limi limt

    i(t)t

    exists), also independent of the initialcondition for most parameter values and as follows from definition 1 equal for all oscillators in the same entrained subset.

    2.1 Stepwise entrainment buildup for increasingK

    For most simulations of (1) with N small and || small ( 0.5) the entrainmentbehavior in terms of the coupling strength can be described as follows. If alli are different then for K = 0 the only entrained subsets are trivially thesingletons {i}, 1 i N.

    With increasing K, oscillators start to become entrained, enlarging sets al-ready entrained. In general there are N 1 bifurcation values Kc,k (k {1, . . . , N 1}), each value representing a coupling strength where two en-trained subsets (which subsets depends on the actual values of the i) merge.

    After N1 transitions full entrainment occurs, which is investigated for = 0in [2].

    This scenario is clearly illustrated in figure 1, where the different long termaverage frequencies of the oscillators are plotted (horizontal axis) for varyingcoupling strength (vertical axis), for a system with = 0, consisting of fouroscillators with natural frequencies given by 1 = 2.32, 2 = 0.89, 3 =0.68 and 4 = 1.23. Later on we indicate that this scenario might break down;

    3

  • 8/3/2019 Part Entrainment

    4/21

    see section 5 for details.

    2.5 2 1.5 1 0.5 0 0.5 1 1.50

    0.5

    1

    1.5

    2

    2.5

    3

    3.5

    4

    lim

    K

    Fig. 1. Long term average frequencies for varying coupling strength for a system offour oscillators. Different forms of partial entrainment can be distinguished: fromno entrainment for small K, to full entrainment for large K.

    2.2 A sufficient condition for partial entrainment

    In support of the scenario described in the previous section, which is basedon simulations, we provide an analytical result (for general finite N), provingthat the Kuramoto-Sakaguchi model is able to exhibit partial entrainment: wederive a sufficient condition for the existence of a solution exhibiting partialentrainment with respect to a given set of oscillators. This result is not to beconsidered as an attempt to estimate critical values for the coupling strength this will be dealt with in section 4 but as an analytical proof of theexistence of partial entrainment.

    We first formulate a preliminary result, pertaining to the model (1) with = 0,to illustrate the technique of proof. A stronger but analytically more technicalresult follows.Proposition 1. LetSe be a proper subset of {1, . . . , N } with M elements andsuch that M > N2 . Assume that

    |i j | < K

    N

    M

    4M 2N

    3N

    32

    , i, j Se.

    4

  • 8/3/2019 Part Entrainment

    5/21

    Then there exists a solution of (1) with = 0 which exhibits partial entrain-ment with respect to Se.

    Proof. For any a (0, 2 ) let Ra denote the region

    Ra = { RN : |i j | a, i, j Se}.We will determine a value for a for which Ra is a trapping region for (1): wewill show that the vector field points into Ra at the boundary ofRa, such thatany solution with an initial condition in Ra remains in Ra.

    Assume that for a particular t0 R the solution of (1) at time t0 is locatedat the boundary of Ra: (t0) Ra and i(t0) j(t0) = a for some i, j Se.From (1) it follows that

    i(t0)

    j(t0) = i j 2

    K

    N sini(t0)

    j(t0)

    2

    Nk=1

    cos

    k(t0) i(t0) + j(t0)

    2

    . (2)

    In the summation we can bound the terms for which k Se by

    cos

    k(t0) i(t0) + j(t0)

    2

    cos a

    since k(t0) i(t0)+j(t0)

    2 a. (In fact k(t0) i(t0)+j(t0)

    2 a2 , but this

    bound would result in more complicated calculations.)If k Se then cos

    k(t0) i(t0)+j(t0)2

    1, and thus

    i(t0) j(t0) i j 2KN

    sina

    2(Mcos a (N M)) .

    For Ra to be a trapping region we need the right hand side to be negative. Min-

    imizing this expression by choosing a appropriately leads to sin a2 =

    2MN6M

    ,resulting in

    i(t0)

    j(t0)

    i

    j

    KNM4M 2N

    3N

    3

    2

    < 0,

    and thus for this value of a Ra is a trapping region. Since Ra is non-emptywe can choose an initial condition in Ra and the resulting solution of (1) willexhibit partial entrainment with respect to Se.

    To extend proposition 1 we will invoke extra knowledge about oscillators notin Se to provide a better bound for the term cos

    k(t0) i(t0)+j(t0)2

    in (2)

    5

  • 8/3/2019 Part Entrainment

    6/21

    with k / Se. Although this term can attain its minimal value of 1, if kdiffers at least 2K from i

    +j2 , then it cannot remain 1 and it will also attain

    positive values. Using this property, we will provide a condition for which asolution of (1), starting within a region Ra, with a

    (0, a), cannot leave Ra.For the proof we refer to the appendix.

    Proposition 2. Let {S1, S2, S3} be a partition of {1, . . . , N }, with S2 andS3 possibly empty. Pick m, M S1 such that m = miniS1 i and M =maxiS1 i and assume that

    i m+M2 > 2K > 0, i S2. Define j, 1,

    2, 3 and T by

    j arccos

    4K

    4K+ j m+M2

    2K , j S2,

    i |Si |N

    , i {1, 3},

    2 1

    NjS2

    cos j,

    T exp

    2K

    N

    jS2

    sin j j cos jj m+M2 2K

    ,

    assume that || < 6 , 21 cos2 (2 + 3)2 > 0 and that the inequality

    M mK

    4

    3 T

    3

    2 3 1 |sin |

    (3T2(1 cos + 2 + 3) + 1 cos 2 3)(21 cos + 2 + 3)

    holds. Then there exists a solution of (1) exhibiting partial entrainment withrespect to S1.

    2.3 An asymptotic stability result

    We will show that, in case the entrained subset S1 contains oscillators withequal natural frequencies, the submanifolds where the oscillators in S1 haveequal phases are asymptotically stable.

    Assume all oscillators in S1 have the same natural frequency: i = , i S1,for some R. Let S3 contain the oscillators with natural frequency equal to which are not included in S1; then S2 contains all oscillators with naturalfrequency different from . (Notice that every solution of (1) will exhibitpartial entrainment with respect to S1: if the difference i(t) j(t), withi, j S1, would cross a multiple of 2 at some time t, then i(t) j(t) = 0,and the oscillators i and j would coincide (modulo 2) for all t R.) Assumethat K is sufficiently small for the condition of proposition 2 on the oscillators

    6

  • 8/3/2019 Part Entrainment

    7/21

    in S2 to be valid. As is shown in the appendix we can adapt the proof ofproposition 2 to obtain the following.Proposition 3. Assume that i1 = = iP = for some i1, . . . , iP {1, . . . , N }, with 2 P N, and that no other oscillators have an i-valueequal to . Then for any M Z, withM > P

    1+cos, there exists an > 0, such

    that K (0, ) the submanifolds defined by i1 +2mi1 = = iM + 2miM,(mi1, . . . , miM) ZM, are locally asymptotically stable under the flow of (1).

    3 Discussion

    In this section we discuss some consequences of proposition 2. (Most of theseobservations also apply to proposition 1.)

    Interpretation. If, in proposition 2, the set S1 contains a considerable frac-tion of the oscillators, is sufficiently close to zero, and the other oscillatorshave natural frequencies that differ largely from those in S1, such that is pos-itive, then for sufficiently small frequency differences of the oscillators in S1a solution exhibiting partial entrainment with respect to S1 is guaranteed toexist. If || 6 a similar sufficient condition can still be derived by choosings defined in the proof (see appendix) in a different way.

    Relation with proposition 1. To verify that proposition 2 is an extensionof proposition 1, set = 0, |S1| = M, S2 = and thus 2 = 0 and T = 1 andassume that M > N2 to obtain the condition (from proposition 2)

    M mK

    4

    3

    N(2M N) 322(M + N)2

    .

    Since

    43 N(2M N) 322(M + N)2

    = 23M(3N)3

    2

    23

    2 (M + N)2

    NM

    4M 2N

    3N

    32

    ,

    with

    2

    3

    M(3N)3

    2

    23

    2 (M + N)2 2

    3

    N2

    (3N)3

    2

    23

    2 (2N)2=

    9

    8,

    it follows that proposition 2 entails proposition 1.

    7

  • 8/3/2019 Part Entrainment

    8/21

    Persistence under perturbations. Proposition 2 also implies that en-trainment will persist under perturbations: for a perturbed initial condition(0), with (0) Ra and m(0) i(0) M(0), i S1, it follows from theproof that the entrainment with respect to S1 will be maintained.

    The limit N . Notice also that proposition 2 remains non-trivial forN : if S1, S2 and S3 contain oscillators with natural frequencies in pre-scribed intervals, then in general M m, 1, 2, 3, and T will approachnon-zero constant values for N .

    In view of this remark, the persistence of the entrainment under perturbationsmay contribute to the understanding of the stability properties of the partiallysynchronized solutions for the system (1) with N = . These stability prop-erties are not yet fully understood.

    Analytical identification of the entrainment behavior. The condition|i j | > 2K for some i, j S1 is obviously sufficient to exclude partialentrainment with respect to S1 since this implies that |i(t) j(t)| will growunbounded. For some configurations this condition together with proposition2 allows to determine a maximal entrained subset, i.e. a subset of {1, . . . , N }for which partial entrainment can occur and which is not included in anotherentrained subset. If the population can be partitioned in maximal entrained

    subsets, then the entire entrainment behavior can be determined on analyticalgrounds and all entrained subsets can be identified.

    4 Estimation of the transition values for = 0

    The value of the coupling strength calculated from proposition 2, guarantee-ing partial entrainment of a given subset, may be quite conservative. This isa consequence of the fact that only little information can be obtained about

    the interaction between oscillators from different entrained subsets. As a re-sult, the discrepancy between the value calculated from proposition 2 for theentrainment of a subset Se and the transition value for K obtained from sim-ulations decreases with increasing size of Se (for the same population), as canbe seen in figure 2 below.

    In this section a better estimation for the actual transition value of the cou-pling strength is given for the case = 0.

    8

  • 8/3/2019 Part Entrainment

    9/21

    4.1 Estimation procedure

    Proposition 2 (and its proof) suggest(s) that oscillators which differ largelyin natural frequency will have small mutual influence as to the entrainment

    behavior. Simulations confirm this and indicate that this is already true formuch smaller frequency differences than suggested by the analytical results.We estimate the critical values for the coupling strength, defining the transi-tions between different forms of partial entrainment behavior, by neglectingthe interactions between oscillators from different entrained subsets and usinganalytical results from [2]. Since the latter results are concerned with the case = 0, we will restrict to be zero throughout this section.

    We estimate the entrainment behavior with respect to a subset Se by dis-regarding all oscillators not belonging to Se, and we determine the coupling

    strength Kc for which full entrainment of Se would occur. This is done bynumerically solving the following equations for K and r, obtained from [2].

    r =1

    N

    jSe

    1

    jKr

    2,

    jSe

    1 2

    j

    Kr

    2

    1

    j

    Kr

    2 = 0. (3)

    In these equations j j 1|Se | iSe i. The variable r is up to a factor

    |Se |

    N equal to the order parameter corresponding to the subsystem (1) withi Se and with the summation index j also restricted to Se. For this subsystemthe order parameter r is defined as

    r(t)

    1

    |Se|jSe

    eij(t)

    , t R,

    and for K sufficiently large, r is time-invariant and r r(t) |Se |N

    satisfies thefirst equation from (3). The second equation is satisfied at the transition to

    full entrainment and determines Kc. (See [2] for more details.)

    This procedure is also supported by observations from the model with aninfinite number of oscillators. For = 0 and with the distribution g of thenatural frequencies even, the partially synchronized subset in the solutionmentioned in the introduction is independent of the shape of the distributiong outside the region corresponding to the frequencies of the oscillators in thepartially synchronized subset. (See e.g. [16] for mathematical details.)

    9

  • 8/3/2019 Part Entrainment

    10/21

    4.2 Comparison and discussion

    We compare the critical values for K resulting from the above analytical esti-mation based on equalities (3) with the values for K derived from propositions

    1 and 2, and from simulations. We considered (1) for = 0 and with N = 100,and we randomly picked 100 natural frequencies from the distribution g de-fined by

    g() =

    C

    10.0001||0.0001+| |

    , || 10000,0, || > 10000,

    where C > 0 is such that+ g()d = 1. (The expression for g() is a

    slight modification of 1/, to guarantee that g can be normalized to 1.) Thefrequencies were ordered by their absolute values.

    For the calculation of the K-values according to propositions 1 and 2 and equa-

    tion (3), we then considered the entrainment of the subsets Se = {1, . . . , M },with M > 1. For each value ofM, propositions 1 and 2 provide a minimal valueofK for which entrainment of Se is guaranteed. Together with the estimationresulting from the equations (3) and the value given by the simulations, theseare shown in figure 2 for varying M. In the simulations, the entrained subsetsmay differ from the sets {1, . . . , M }, but in most cases they are equal. (In thecase represented by figure 2 this holds for 95% of the M-values.)

    0 20 40 60 80 10010

    4

    102

    100

    102

    104

    106

    M

    K

    Proposition 1Proposition 2Equation (3)Simulations

    Fig. 2. Comparison of values for the coupling strength related to entrainment of thesubset {1, . . . ,M } for different procedures.Since proposition 1 requires that M > N /2, the corresponding curve only

    10

  • 8/3/2019 Part Entrainment

    11/21

    starts at M = 51. Also proposition 2 does not generate results for any valueof M, but the condition is less stringent and the corresponding curve startsfor a value ofM < 51. (Although for several other distributions of the naturalfrequencies this curve also starts at M = 51, the present form ofg shows thatthis does not always hold.) For the curve associated with proposition 2 the

    K-values are closer to those obtained from the simulations than for the curveassociated with proposition 1, but, considering the logarithmic scale, both areunsuitable as estimations for the transition values between different forms ofpartial entrainment.

    For this choice of the natural frequencies, the estimation based on equations(3) corresponds quite well with the simulation results, justifying the assump-tion that oscillators outside an entrained subset have little influence on theentrainment of this subset.

    5 Entrainment break up with increasing K

    The general scenario does not always hold; one of the points we want to drawattention to is that entrained oscillators may break up with increasing K, aphenomenon reported before in [8, p. 46] for the case = 0. We consider aparticular system with four oscillators, and also = 0. Observe in figure 3that there is a critical value for the coupling strength (0.313) above whichthe entrainment of a subset breaks up. Further increase of K reestablishes theentrainment.

    We offer an intuitive explanation. Denote the oscillators by 1, 2, 3 and 4,according to the order of their natural frequencies i (i.e. 1 < 2 < 3 < 4).

    As we have already mentioned in previous sections, the interaction betweentwo oscillators appears to decrease with increasing difference in their natu-ral frequencies. This implies that oscillator 3 will be subject to a strongerattraction towards oscillators 1 and 2 than oscillator 4. With increasing cou-pling strength K this attraction, and also the difference in attraction fromoscillators 1 and 2 on oscillator 3 and on oscillator 4, will increase. For somevalue ofK oscillators 3 and 4 become entrained, but when K is increased fur-

    ther, the increase in attraction difference becomes more important than theincreased attraction between oscillators 3 and 4, making it possible for partialentrainment to disappear.

    For = 0 simulation results indicate that entrainment of two oscillators ina system of only three oscillators cannot disappear with increasing couplingstrength. The probability of entrainment disappearing with increasing cou-pling strength seems to increase with ||, and for || sufficiently large it can

    11

  • 8/3/2019 Part Entrainment

    12/21

    0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.750

    0.05

    0.1

    0.15

    0.2

    0.25

    0.3

    0.35

    lim

    K

    Fig. 3. Long term average frequencies for varying coupling strength for a systemof four oscillators. When K is increased from 0.3 to 0.35 there is a transition inwhich entrainment of two oscillators disappears. It reappears at the transition tofull entrainment.

    also be observed in a system of three oscillators.

    5.1 Persistence of full entrainment with increasingK

    The explanation offered in the previous paragraph implies that a given en-trained subset can break up with increasing coupling strength only if otheroscillators are present, and suggests that entrainment of the entire populationcannot disappear with increasing coupling strength. For = 0 this is con-

    firmed as follows by analytical results that can be derived from [2]. In thispaper, it was proven that the existence of a locally stable phase-locked solu-tion (which corresponds to full entrainment) is equivalent with the existenceof a solution r (0, 1] of (see also (3) in section 4)

    1(r, K) r 1N

    Ni=1

    1

    iKr

    2= 0 (4)

    12

  • 8/3/2019 Part Entrainment

    13/21

    (with i i 1NNj=1 j ; Kr maxi i), satisfying

    2(r, K) r 1N

    Ni=1

    i

    Kr

    2

    1 i

    Kr 2

    > 0. (5)

    Assume that r and K satisfy both (4) and (5) and take K > K. It followsthat 1(r, K

    ) 0, while clearly 1(1, K) 0, implying the existence of anr [r, 1] with 1(r, K) = 0. From r r and Kr > Kr and 1(r, K) = 0it then follows that

    2(r, K) 2(r, K) > 0,

    implying the existence of a solution of (1) for a coupling strength K whichexhibits entrainment of the entire population. This result implies that thesolution of (1) corresponding to entrainment of the entire population willpersist with increasing coupling strength (for the case = 0).

    5.2 Josephson junctions

    The relation between the Kuramoto-Sakaguchi model and arrays of Joseph-son junctions [19] suggests that the phenomenon of destruction of entrain-ment with increasing K may also be observed in Josephson junction arrays.Simulations confirm this. For a Josephson junction characterized by a phasedifference the voltage and current across the junction are given by 2e andIC sin respectively, where is Plancks constant divided by 2, e denotes the

    elementary electrical charge and the constant IC is the critical current of thejunction. We consider the same circuit as in [19]: a (parallel) connection of abias current IB, N different junctions in series, and a load with inductance L,resistance R and capacitance C, with the charge on the capacitor denoted byQ. For junction i the phase difference, resistance and critical current are de-noted by i, ri and Ii respectively; we assume its capacitance can be neglected.The system equations can then be written as

    2erii + Ii sin i + Q = IB, i {1, . . . , N },

    LQ + RQ +Q

    C=

    2e

    N

    i=1

    i.

    In [19] it was shown that in the limit of weak coupling and weak disorderthis system can be cast into the model (1). Setting N = 4, IB = 1.5mA,R = 50, L = 25pH, C = 0.04pF, ri = 0.5 , i {1, . . . , N } and Ii =

    0.5 +IiI

    mA, with I = (0.057, 0.021, 0.0075, 0.0165, 0.021), (most values

    are taken from [19]), we calculate the long term average frequencies (i.e. the

    values limi limti(t)i(0)

    t) for varying I, representing different levels

    13

  • 8/3/2019 Part Entrainment

    14/21

    of disorder. The result is shown in figure 4. In spite of the irregular shapeof the graph (which persists when simulating with different time steps orover different time intervals, excluding simulation errors as the cause of theirregularity), different phenomena can be observed. Enumerating the junctionsfrom 1 to 5, such that lim1 lim2 lim3 lim4 lim5 (with strict inequalitiese.g. for I = 9.6, notice that for I = 9.2 junctions 2 an 3 have already becomeentrained), the following transitions (with increasing I) are clearly visible:

    I 9.36 junction 2 leaves junction 3, and joins junction 1 at I 9.71; I (10.03, 10.16): entrainment of junctions 1 and 2 temporarily disap-

    pears; I (9.31, 10): junctions 4 and 5 are entrained but for several subintervals

    the entrainment disappears; I 10.17: junction 4 (having left junction 5 at I 10) becomes entrained

    with junction 3.

    6 Conclusion

    We have investigated the partial entrainment behavior of the Kuramoto-Sakaguchi model. We derived a sufficient condition for partial entrainment ofa given subset of oscillators; the result implies persistence of the entrainmentbehavior under perturbations and remains non-trivial in the limit N .

    For the investigated distribution of natural frequencies, the critical value of the

    coupling strength defining the onset of partial entrainment of a given subsetcan be estimated analytically by neglecting oscillators which do not belong tothis subset.

    Simulations indicate that entrainment can disappear with increasing couplingstrength in the Kuramoto-Sakaguchi model, and that a similar phenomenoncan be observed in arrays of Josephson junctions, where it is also possible thata junction leaving one entrained subset joins another entrained subset.

    Acknowledgments

    This paper presents research results of the Belgian Programme on Interuniver-sity Attraction Poles, initiated by the Belgian Federal Science Policy Office.The scientific responsibility rests with its authors.

    Filip De Smet is a Research Assistant of the Research Foundation - Flanders(FWO - Vlaanderen).

    14

  • 8/3/2019 Part Entrainment

    15/21

    2.148 2.1485 2.149 2.1495 2.15 2.1505

    x 1012

    9.2

    9.4

    9.6

    9.8

    10

    10.2

    lim

    i(rad/s)

    I

    Fig. 4. Long term average frequencies for varying levels of disorder for an array of5 Josephson junctions. When I is increased there are different transitions in whichentrainment of two junctions disappears.

    A Proof of proposition 2

    The main idea of the proof is similar to the idea behind the proof of proposition1. Instead of one trapping region, we will consider two regions and show thata solution starting in the smaller region cannot leave the larger region (whichencompasses the smaller region). We first formulate and prove a lemma that

    allows us to provide a better bound for the term cos

    k(t0) i(t0)+j(t0)2

    in

    (2) with k / Se.Lemma 1. Assume the continuously differentiable function 0 : R R : t

    15

  • 8/3/2019 Part Entrainment

    16/21

    0(t) satisfies 0 < 1 |0(t)| 2, t R+. Thent0

    cos 0(t)dt t cos0 + 2

    1(sin 0 0 cos0) ,

    for all 0 (0,

    2 ) satisfying

    2

    1

    1 1

    2

    (sin0 0 cos0) 2

    2cos0 0.

    Proof. The result is invariant under the substitution 0 0, and thereforewe will only consider the case for which (t) > 0, t R+. Since 0 is strictlyincreasing we can perform the substitution t = 10 (

    0) in the following inte-

    gral.

    t0 (cos 0(t

    ) cos0) dt

    =0(t)0(0)

    (cos 0

    cos0) d

    0

    0(10 (0)) .

    Setting I+(t) {0 [0, 0(t)] : cos(0) cos0} and I(t) {0 [0, 0(t)] : cos(

    0) cos0}, we obtain the inequality

    t0

    cos 0(t)dt t cos0

    I+(t)

    (cos 0 cos0) d01

    +I(t)

    (cos 0 cos0) d02

    .

    Denoting by n(t) the number of connected components of I+(t) if I+(t) = and setting n(t) 1 for I+(t) = we obtaint0

    cos 0(t)dt t cos0 n(t)

    1

    00

    (cos 0 cos0) d0

    +n(t) 1

    2

    200

    (cos 0 cos0) d0

    =2n(t)

    1(sin 0 0 cos0)

    2(n(t) 1)2

    (sin 0 + ( 0)cos0)

    =2

    1(sin0 0 cos0) + (n(t) 1)

    2

    11

    12

    (sin0 0 cos0) 22 cos0

    ,

    proving the lemma.

    We will now prove proposition 2.

    16

  • 8/3/2019 Part Entrainment

    17/21

    For an a (0, 2 ||), yet to be determined, consider the region

    Ra { RN : |i j | a, i, j S1}.

    We will determine conditions such that a well-chosen initial condition leads to

    a solution of (1) which remains in Ra. The initial value (0) belongs to Ra

    ,with a (0, a) also yet to be determined, and is such that m(0) i(0) M(0), i S1. This implies that m(t) i(t) M(t), t R+, since iffor some t 0, i(t) = m(t) (resp. i(t) = M(t)), for some i S1, theni(t

    ) m(t) = im 0 (resp. i(t) M(t) 0). Consequently we onlyneed to derive conditions guaranteeing that M(t) m(t) will remain smallerthan or equal to a. With Mm2 and

    Mm2 it follows that

    (t) = KN

    sin (t)N

    j=1

    cos

    j(t) m(t)+M(t)2

    .

    We will investigate the behavior of for t-values for which (t) [a2

    , a2

    ]. Inthis situation the terms in the summation for which j S1 are bounded by

    cos

    j(t) m(t)+M(t)2

    cos(a2

    + ||),

    since m(t) j(t) M(t) and thusj(t) m(t)+M(t)2

    a2 . If j S3, we

    apply that cos

    j(t) m(t)+M(t)2

    1, and thus

    (t)

    K

    N

    sin (t)|S1

    |cos(a2 +

    |

    |)

    |S3

    |+jS2

    cos

    j(t) m(t)+M(t)2 .

    With (t) [a2 , a2 ], and i = |Si |N (i {1, 3}), we obtain

    (t)

    Ksin (t)

    Ksin a

    2

    1 cos(a2 + ||) + 3

    1

    NjS2

    cos j(t) m(t)+M(t)

    2

    ,

    or

    1

    Kln

    tan (t)2

    tan (t0)2

    Ksin a

    2

    1 cos(a2 + ||) + 3

    (t t0)

    1N

    jS2

    tt0

    cos

    j(t) m(t)+M(t)2

    dt,

    17

  • 8/3/2019 Part Entrainment

    18/21

    for all t0, t 0, such that (t) [a2 , a2 ], t [t0, t]. We want to establishconditions for which the right hand side is upper bounded by 1

    Kln

    tan a4

    tan a

    4

    ,

    guaranteeing that for increasing t, (t) can only leave the interval [ a

    2 ,a2

    ] by

    becoming smaller than a

    2.

    Consider a j S2. Then

    0 3, (A.2a) holds with strict inequality ifa andK are chosen sufficiently small. From (A.1) we then obtain that limt (t) =0, leading to proposition 3.

    19

  • 8/3/2019 Part Entrainment

    20/21

    References

    [1] J. A. Acebron, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler.The Kuramoto model: a simple paradigm for synchronization phenomena.Reviews of Modern Physics, 77(1):137185, January 2005.

    [2] D. Aeyels and J. A. Rogge. Existence of partial entrainment and stabilityof phase locking behavior of coupled oscillators. Progress of TheoreticalPhysics, 112(6):921942, December 2004.

    [3] H. Daido. Lower critical dimension for populations of oscillators with ran-domly distributed frequencies: a renormalization-group analysis. PhysicalReview Letters, 61(2):231234, July 1988.

    [4] H. Hong, M. Y. Choi, and B. J. Kim. Synchronization on small-worldnetworks. Physical Review E, 65(2):026139, January 2002.

    [5] A. Jadbabaie, N. Motee, and M. Barahona. On the stability of the Ku-ramoto model of coupled nonlinear oscillators. In Proceedings of theAmerican Control Conference (ACC 2004), volume 5, pages 42964301,

    July 2004.[6] Y. Kuramoto. Cooperative dynamics of oscillator community. Supplement

    of the Progress of Theoretical Physics, 79:223240, 1984.[7] Y. Maistrenko, O. Popovych, O. Burylko, and P. A. Tass. Mechanism

    of desynchronization in the finite-dimensional Kuramoto model. PhysicalReview Letters, 93(8):084102, August 2004.

    [8] S. C. Manrubia, A. S. Mikhailov, and D. H. Zanette. Emergence of Dy-namical Order: Synchronization Phenomena in Complex Systems, vol-ume 2 of World Scientific Lecture Notes in Complex Systems. WorldScientific, 2004.

    [9] M. Marodi, F. dOvidio, and T. Vicsek. Synchronization of oscillatorswith long range interaction: phase transition and anomalous finite sizeeffects. Physical Review E, 66(1):011109, 2002.

    [10] L. G. Morelli, H. A. Cerdeira, and D. H. Zanette. Frequency clusteringof coupled phase oscillators on small-world networks. European PhysicalJournal B, 43(2):243250, January 2005.

    [11] Y. Moreno and A. F. Pacheco. Synchronization of Kuramoto oscillators inscale-free networks. Europhysics Letters, 68(4):603609, November 2004.

    [12] O. V. Popovych, Y. L. Maistrenko, and P. A. Tass. Phase chaos in coupledoscillators. Physical Review E, 71(6):065201(R), June 2005.

    [13] J. L. Rogers and L. T. Wille. Phase transitions in nonlinear oscillator

    chains. Physical Review E, 54(3):R2193R2196, September 1996.[14] J. A. Rogge and D. Aeyels. Stability of phase locking in a ring of unidi-

    rectionally coupled oscillators. Journal of Physics A: Mathematical andGeneral, 37(46):1113511148, November 2004.

    [15] H. Sakaguchi and Y. Kuramoto. A soluble active rotator model show-ing phase transitions via mutual entrainment. Progress of TheoreticalPhysics, 76(3):576581, September 1986.

    [16] S. H. Strogatz. From Kuramoto to Crawford: exploring the onset of

    20

  • 8/3/2019 Part Entrainment

    21/21

    synchronization in populations of coupled oscillators. Physica D, 143:120, 2000.

    [17] S. H. Strogatz. SYNC: The Emerging Science of Spontaneous Order.Hyperion Press, 2003.

    [18] S. Watanabe and S. H. Strogatz. Constants of motion for superconducting

    Josephson arrays. Physica D, 74(3-4):197253, July 1994.[19] K. Wiesenfeld, P. Colet, and S. H. Strogatz. Frequency locking in Joseph-

    son arrays: Connection with the Kuramoto model. Physical Review E,57(2):15631569, February 1998.


Recommended