Order parameter analysis of synchronization transitions on star networks

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by  Hongbin Chen

Order Parameter Analysis of Synchronization transitions on star networks
Hongbin Chen,1,2 Yuting Sun,3 Jian Gao,3 Can Xu,3,∗ and Zhigang Zheng1,2,†
1Institute of Systems Science, Huaqiao University, Xiamen 361021, China
2College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
3Department of Physics and the Beijing-Hong Kong-Singapore Joint Centre for Nonlinear and Complex Systems (Beijing),
Beijing Normal University, Beijing 100875, China
(Dated: January 11, 2017)
Collectivebehaviorsofpopulationsofcoupledoscillatorshaveattractedmuchattentioninrecent
years. Inthispaper,anorderparameterapproachisproposedtostudythelow-dimensionaldynam-
icalmechanismofcollectivesynchronizationsbyadoptingthestar-topologyofcoupledoscillatorsas
a prototype system. The order parameter equation of star-linked phase oscillators can be obtained
in terms of the Watanabe-Strogatz transformation, Ott-Antonsen ansatz, and the ensemble order
parameterapproach. Differentsolutionsoftheorderparameterequationcorrespondtodiversecol-
7
lectivestates,anddifferentbifurcationsrevealvarioustransitionsamongthesecollectivestates. The
1
properties of various transitions are revealed in the star-network model by using tools of nonlinear
0
dynamics such as time reversibility analysis and linear stability analysis.
2

n PACSnumbers: 05.45.Vx,89.75.Hc,68.18.Jk
a
J

0 I. INTRODUCTION large numbers of oscillators. However, strictly speaking,
1 the OA manifold analysis cannot be applied to finite-
oscillator systems. Watanabe and Strogatz introduced
] Understanding the intrinsic microscopic mechanism
O the M¨obius transformation for finite-size systems with
embedded in collective macroscopic behaviors of popula-
specificsymmetriestoobtainanexactthree-dimensional
A tions of coupled units on heterogenous networks has be-
dynamics [21, 22], but this scheme cannot be extended
n. come a focus in a variety of fields, such as the biological to general finite systems. The mechanism of the valid-
neurons circadian rhythm, chemical reacting cells, and
i ity of the OA approach was recently studied, and the
l even society systems [1–8]. Numerous different emerging
n ensemble order parameter approach is proposed, which
macroscopic states/phases have been revealed, and vari-
[ extends the OA approach to more general cases such as
ous non-equilibrium transitions among these states have
a finite-number of oscillators and more general coupling
1 been observed and studied on heterogenous networks [9–
v forms [23].
19].
2 Abruptorexplosivetransitionfromincoherentstateto
9 The transitions among different collective states on
synchronization may occur on networks if the frequen-
5 heterogeneous networks exhibit the typical feature of
cies of oscillators on nodes are positively correlated to
2 multistability, i. e., these states may coexist for a group
the node’s degrees [13], which has been observed nu-
0 of given parameters and depend on the choice of ini-
. merically on scale-free networks and experimentally in
1 tial conditions. This interesting behavior is closely re-
electronic circuits [24, 25]. The first-order transition can
0 lated to the first-order phase transition, and multistabil-
bechangedandmorewaysoftransitionscanbeobserved
7 ityinthediscontinuoustransitionsindicatethecompeti-
by adjusting the phase shift among oscillators [26]. Nu-
1 tions of miscellaneous attractors and their correspond-
: merous efforts have been made to understand the mech-
v ing basins of attraction in phase space. For a net-
anism of explosive synchronization from different view-
i work of coupled oscillators, the microscopic description
X pointssuchasthetopologicalstructuresofnetworks,the
of the dynamics of oscillators should be made in a high-
r coupling functions among nodes, and so on [17, 25, 27–
dimensional phase space, which is very difficult to deal
a 32].
with. The key point in understanding macroscopic tran-
Itisvaluabletoanalyticallyunderstandthetransitions
sitions is the projection of the dynamics from this high-
among various synchrony states on heterogeneous net-
dimensionalspacetoamuchlower-dimensionalsubspace.
works. The star topology is the simplest while the key
This can be executed by introducing appropriate or-
topologyindescribingtheheterogeneitypropertyofcom-
der parameters and building their dynamical equations.
plexnetworkssuchasthescale-freenetworks [33–36]. In
Ott and Antonsen [20] proposed an ansatz to project
this paper, we study the collective states and the abun-
the infinite-dimensional dynamics to a low-dimensional
danttransitionsamongthesestatesonastarnetworkby
manifold called the Ott-Antonsen (OA) manifold, which
considering the effect of the phase shift among coupled
has been successfully applied to systems composed of
oscillators [23, 26, 37–39]. The dynamics of star net-
worksofoscillatorsisanalyticallystudiedbybuildingthe
equations of motion of the order parameter for networks
∗Electronicaddress: [email protected] with a finite size, which accomplishes a great reduc-
†Electronicaddress: [email protected] tion from microscopic high-dimensional phase dynamics
2
of coupled oscillators to a macroscopic low-dimensional thecomplexnumberz,orequivalently,bythepolarcoor-
dynamics. Based on the order parameter dynamics, we dinatesrandΦ. AnintriguingpointdiscoveredbyOttet
further reveal numerous transitions among different col- al. [20]istheinvarianceofthePoissonsubmanifold,i. e.,
lective states in this model by using tools of nonlinear if the initial phase density is a Poisson kernel, it remains
dynamics such as time reversibility analysis [15] and lin- a Poisson kernel for all the time. This can be verfied by
ear stability analysis. We found three typical processes substituting the velocity field (3) and the ansatz (4) into
of the transitions to the synchronous state, i. e., the the continuity equation (2). It can be found that the
transitions from the neutral state, the in-phase state or amplitude equations for each harmonic einφ are simul-
the splay state to the synchronous state, and a continu- taneously satisfied if and only if z(t) evolves according
ous process of desynchronization and a group of hybrid to
phasetransitionsthatarediscontinuouswithnohystere-
sis. z˙ =i(fz2+gz+f¯). (7)
This equation can be recast in a more physically mean-
II. THE OTT-ANTONSEN ANSATZ AND THE ingful form in terms of the complex order parameter de-
WATANABE-STROGATZ APPROACH fined as the centroid of the phases φ regarded as points
eiφ on the unit circle:
WefirstillustratetheOtt-Antonsenansatz[20]briefly
(cid:90) 2π
by analyzing the following class of identical oscillators <eiφ >= eiφρ(φ,t)dφ. (8)
governed by the equations of motion 0
ϕ˙ =feiϕj +g+f¯e−iϕj, j =1,··· ,N, (1) By substituting Eq. (4) into Eq. (8) one may find that
j
where f is a smooth, complex-valued 2π-periodic func- z =<eiφ >=reiΦ (9)
tion of the phases ϕ ,··· ,ϕ and the overbar denotes
1 N
complexconjugate, g isarealvaluedfunctionsinceϕ˙ is forallstatesonthePoissonsubmanifold. Thenthemean-
j
real. In the limit N −→∞, by introducing the distribu- ing of z is clear that it represents the order parameter of
tion of phases of oscillators, the evolution of the system the system, r is the modulus of it and Φ is the mean
(1) is given by the continuity equation phase of it. However, whether the governing equation
Eq. (7) can be used for system with finite size can not
∂ρ ∂(ρν) be implied from Ott-Antonsen ansatz.
+ =0, (2)
∂t ∂φ For a finite number of oscillators N, the original mi-
croscopic dynamical state can be reduced to a macro-
where ρ(φ,t) is the phase distribution function, and scopic collective state by the Watanabe-Strogatz ap-
ρ(φ,t)dφ gives the fraction of phases that lie between proach[21,22],andthegoverningequationsofthesystem
φandφ+dφattimet. ThevelocityfieldistheEulerian canalsobegeneratedbytheM¨obiusgroupaction[40,41].
version of Eq. (1), The class of identical oscillators still governed by the
equations of motion Eq. (1), then the oscillators’ phases
ν(φ,t)=feiϕ+g+f¯e−iϕ. (3) ϕ (t)evolveaccordingtotheactionoftheM¨obiusgroup
j
on the complex unit cycle
Suppose ρ is of the form
eiϕj(t) =M (eiθj) (10)
∞ t
1 (cid:88)
ρ(φ,t)= {1+ (z¯(t)neinφ+z(t)ne−inφ)} (4)
2π for j = 1,...,N, where M is a one-parameter family
n=1 t
of M¨obius transformations and θ is a constant angle.
j
for some unknown function z that is independent of φ. By parameterizing the one-parameter family of M¨obius
Note that Eq. (4) is just an algebraic rearrangement of transformations as
the usual form for the Poisson kernel
eiψw+η
1 1−r2 Mt(w)= 1+η¯eiψw, (11)
ρ(φ)= , (5)
2π1−2rcos(φ−Φ)+r2
where |η(t)|<1 and ψ(t)∈R, and let
where the complex number z can be expressed in the
complex plane as w =eiθj. (12)
j
z =reiΦ. (6) One then obtains
The ansatz (4) defines a submanifold in the infinite- η˙ =i(fη2+gη+f¯), (13a)
dimensionalspaceofthedensityfunctionρ. ThisPoisson
ψ˙ =fη+g+f¯η¯. (13b)
submanifold is two-dimensional and is parameterized by
3
With these new variables, one could rewrite the order synchronycandecayorincoherencecanregainitsstabil-
parameter as ity with increasing coupling and multistability between
partially synchronized and/or the incoherent state can
1 (cid:88)N eiψeiθj +η(t) appear in the globally coupled network.
z(t)= , (14)
The coupled phase oscillator system with α=0 of the
N 1+η¯(t)eiψeiθj
j=1 starnetworkwasoriginallytostudythecharacteristicsof
theexplosivesynchronization[13],howevertheprocessof
Eqs. (13) and (14) can describe the system with ar-
thesynchronizationmaybeinfluencedwiththeintroduc-
bitrary initial conditions as η(0),ψ(0) and N constants
tion of phase shift [14]. By introducing the phase differ-
θ ,1 ≤ j ≤ N. The order parameter (14) could be sim-
j encesbetweenthehubandleavesϕ =θ −θ ,thephase
plified further by choosing the constants j h j
dynamics on star networks can be transformed to the
following phase difference dynamics on an all-connected
j−1
θj =2π N ,1≤j ≤N, (15) network,
K
with which, the order parameter (14) reads (cid:88)
ϕ˙ =∆ω−λ sin(ϕ +α)−λ sin(ϕ −α), (19)
i j i
z(t)=η(t)(1+I), (16) j=1
where1≤i≤K. Wefurtherdefinetheorderparameter
where I = (1−|η(t)|−2)/(1±(eiψη¯(t))−N), ”−” for the of the all-connected network to describe the degree of
case with even N and ”+” for the case with odd N. One synchronization as
can verify that for large N, I (cid:28) 1, the order parameter
could be approximated as 1 (cid:88)K
z(t)≡r(t)eiΦ(t) = ei(ϕj). (20)
K
z(t)≈η(t),N (cid:29)1. (17) j=1
Itisworthnotingthatthestarnetworkbecomesglobally
Basedontheanalysisabove, thedynamicsofthesystem
synchronousifthemodulusr(t)=1andthemeanphase
withfinitesizecanbedescribedbythesameequationas
Φ(t) = const. If the modulus r(t) = 1 while the mean
thegoverningequation(7)whichobtainedfromtheOtt-
phaseΦ(t)isperiodicwhichcorrespondstothestatewith
Antonsen ansatz for the system with infinite size. Then
ϕ (t) = ϕ(t), all the leaf nodes are synchronous to each
Eq. (7) can still be used to explore the low-dimensional j
other while they are asynchronous to the hub oscillator.
collective behaviors of the system with finite size.
It is instructive to rewrite Eq. (19) as
ϕ˙ =feiϕj +g+f¯e−iϕj, j =1,··· ,K, (21)
j
III. THE SAKAGUCHI-KURAMOTO MODEL
ON STAR NETWORKS: THE ORDER λ
where i denotes the imaginary unit and f = i e−iα,
PARAMETER EQUATION 2
g =∆ω−λKrsin(Φ+α).
We start with a star network of coupled phase oscilla- For finite K, due to the high topological symmetry of
tors with nonzero phase shift as our working model. In the star network, the collective behaviors of the system
the star network with one hub and K leaves, the degree canbeanalyzedbyderivingthelow-dimensionaldynam-
of the leaves is k = 1 (i = 1,...,K) and the degree of ical equations in terms of both the ensemble order pa-
i
the hub is k =K. Suppose that the natural frequencies rameter approach [23] and the the Watanabe-Strogatz
h
of the oscillators are proportional to their degrees, the transformation[21,22]. Iftheinitialphasesofoscillators
equations of motion for the hub and leaf nodes read are chosen as (15), we have the dynamical equation for
the order parameter z(t) as
K
θ˙h =ωh+λ (cid:88)sin(θj −θh−α), z˙ =−λe−iαz2+i(∆ω−λKrsin(Φ+α))z+λeiα, (22)
(18) 2 2
j=1
θ˙ =ω+λ sin(θ −θ −α),1≤j ≤K, which is just the OA result (18) in terms of the order
j h j
parameter. Different solutions of Eq. (22) build corre-
whereθ ,θ andω ,ω areinstantaneousphasesandnat- spondences with diverse collective states of the coupled
h j h
uralfrequenciesofthehubandleafnodesrespectively. λ oscillator system.
is the coupling strength. K is the number of leaf nodes
connected with this hub and α is the phase shift. The
effect of phase shift among coupled oscillators has been IV. COLLECTIVE DYNAMICS OF
extensively investigated in recent years, while this has STATIONARY STATES
been seldom discussed in star networks [14, 42]. Abun-
dant collective dynamics appear in the global coupled In the following we start discussing the collective dy-
model in the presence of a finite phase shift [14], where namics of the star network in terms of the Eq. (22). By
4
setting z = x+iy, we can describe the order parameter
dynamics in the x−y plane as
1 λ
x˙ =λ( +K)cosαy2− cosαx2
2 2
λ
+λ(K−1)sinαxy−∆ωy+ cosα,
2
(23)
1 λ
y˙ =λ( −K)sinαx2− sinαy2
2 2
λ
−λ(K+1)cosαxy+∆ωx+ sinα.
2
Thesteady-statesolutionsaredeterminedbysettingx˙ =
0 and y˙ = 0, which results in four fixed points noted by
(x ,y ) with
i i
−sinα∆ω±Asinα
x = ,
1,2 λ(2Kcos2α+1) FIG. 1: (a) The time evolution of sinϕ (t) with α =
j
−0.4π,λ = 2, j = 1,2. (b) The Lyapunov exponents of the
−cosα∆ω±Acosα
y =− , network with α = 0.1π. (c) The time evolution of sinθ (t)
1,2 λ(2Kcos2α+1) i
with α=0.1π,λ=0.5,i=1,··· ,K. (d) The order parame-
(24)
sinα sin2αB±K(sin2αB−sin2α2) ter against the coupling strength with different initial states
x3,4 = λ + λ2sinα(K2+22cos(2α)K+1) , for α=0. The size of the star network is N =11.
−∆ω(−cosα±sinαB−Kcosα)
y = ,
3,4 λ(K2+2cos(2α)K+1) of the 2×2 Jacobian matrix J of the fixed points with
elements
where ”+” represents the fixed points (x ,y ), ”−”
1,3 1,3
J =−λcosαx +λ(K−1)sinαy ,
represents the fixed points (x ,y ) and 11 i i
2,4 2,4
J =λ(1+2K)cosαy +λ(K−1)sinαx −∆ω,
12 i i
(cid:112)
A= −2Kλ2cos2α−λ2+∆ω2, J =λ(1−2K)sinαx −λ(K+1)cosαy +∆ω,
21 i i
(25)
(cid:112) J =−λsinαy −λ(K+1)cosαx .
B = λ2+K2λ2+2Kλ2cos2α−∆ω2. 22 i i
(29)
The eigenvalues of the Jacobian matrix are
The existence condition for the fixed points are deter-
minedbyEq. (25),where−2Kλ2cos2α−λ2+∆ω2 ≥0, (cid:112)
J +J ± (J +J )2−4(J J −J J )
and λ2+K2λ2+2Kλ2cos2α−∆ω2 ≥ 0. For the fixed β = 11 22 11 22 11 22 12 21 .
points (x ,y ), the existence condition can be given 1,2 2
1,2 1,2 (30)
as
Thestability conditions ofthefourfixedpoints aresum-
∆ω marizedinTableI,whereTheparametersinthetableare
λ≤λ1 = √2Kcos2α+1, (26) λˆfc = ∆ω/√2Kcos2α+1, λ+sc = −∆ω/(Kcos2α+1),
λ− = ∆ω/(Kcos2α+1), α− = −arccos(−1/K)/2,
sc 0
and for the fixed points (x ,y ), the existence condi- α+ =arccos(−1/K)/2.
3,4 3,4 0
tion is
Fixed point Stability condition
∆ω
λ≥λ = √ . (27) (x ,y ) λ<λˆf,α∈(α−,0)
2 K2+2Kcos2α+1 1 1 c 0
λ>0,α∈(−π/2,α−)
0
For the fixed points (x3,4,y3,4), there is an additional (x2,y2) λ>λ+sc,α∈(α0+,π/2)
natural restriction relation (x3,y3) λ>λ−sc,α∈(α0−,0)
λ<λ+,α∈(α+,π/2)
sc 0
x2+y2 =1, (28) (x ,y ) always unstable
4 4
while for the fixed points (x ,y ), x2 + y2 may be TABLE I: The stability conditions of the four fixed points.
1,2 1,2
greater or lower than 1. The definition of z implies that
only those fixed points satisfying x2+y2 ≤1 are reason- The fixed points of order parameter equation are re-
able. lated to the collective states of coupled phase oscillators.
Linear stability analysis can be applied to the fixed Fixed points (x ,y ) with |z| = 1 correspond to the
3,4 3,4
points (x ,y ),i=1,2,3,4 by computing the eigenvalues synchronous state (SS) of the system where all the
i i
5
phase differences between hub and leaf nodes are the
same and keep constant as
ϕ (t)=Const,1≤j ≤K, (31)
j
implyingtheglobalsynchronizationofleavesandthehub
in a star network. The above stability analysis indicates
thatfixedpoint(x ,y )correspondstotheunstablesyn-
4 4
chronous state and the fixed point (x ,y ) corresponds
3 3
to the stable synchronous state, and their stability can FIG. 2: (a) The Lyapunov exponents of the network with
be easily studied. α = 0,N = 11. (b) Phase plane of Eq. (23) with ∆ω = 9,
The fixed points (x ,y ) with the modulus |z| = K =10,α=0, λ=0.1. Red lines are x˙ =0, and green lines
1,2 1,2
(cid:112) are y˙ =0. The intersection of x˙ =0 and y˙ =0 is fixed point
x2+y2 >1isunphysicalbecausetheorderparameter
D.Trajectorieswithdifferentinitialvaluesaremarkedby’∗’.
z of coupled oscillators is bounded by |z| ≤ 1. If |z| <
1, the related collective state is called the splay state
(SPS) [11, 43], the phase differences between the hub
forlimitcycleasitisstablefor0<α<π/2andunstable
and leaf nodes satisfy a function relation as
for −π/2<α<0.
2. The neutral state
jT
ϕ (t)=ϕ(t+ ),1≤j ≤K (32) Thedynamicsatα=0,±π/2whenλ<λ arespacial
j K ec
cases and correspond to the critical dynamical states of
with T the period of ϕ(t), as shown in Fig. 1(a). This the system, where the related fixed point is found to be
kind of state physically represents the collective state neutrally stable where Re(β1,2) = 0 and Im(β1,2) (cid:54)= 0
where all the leaf oscillators in the star network move in Eq. (30). In this case there are a large class of
synchronously with a constant time shift. states in the critical cases with the order parameter r
are determined by initial values of (x,y), where long-
term behaviors of z depend crucially on initial phases,
V. COLLECTIVE DYNAMICS OF as shown in Fig. 1(d). We call this state the neu-
TIME-DEPENDENT STATES tral state(NS) [15], and the corresponding fixed point
is neutrally stable, where all Lyapunov exponents of the
Long-term solutions of the order parameter equation fixedpointarezerowhenλ<λec, asshowninFig. 2(a).
contains not only the states given by fixed points but Because the Kuramoto system is dissipative [44], the ex-
also time-dependent states corresponding to periodic so- istence of a large class of neutral states are counterin-
lutions. There are two periodic regimes, i.e., the regime tuitive, which implies that the phase space of this state
0<α<π/2 and λ<λ =λ , and the critical line with contains an integrable Hamiltonian system family of pe-
ec 2
α=0,±π/2: riodic orbits, shown in Fig. 2(b).
1. The in-phase state To understand the mechanism of these neutral states,
Whenλ<λ and0<α<π/2,asshowninFig. 1(b), weresorttotheanalysisoftheorderparameterequations
ec
the largest Lyapunov exponent is zero and the other ex- (23). Forthecaseofα=0,Eq. (23)canbesimplifiedto
ponents are negative, implying a stable limit-cycle solu-
1 λ λ
tion. This solution can be conveniently found by trans- x˙ =λ(K+ )y2− x2−∆ωy+ ,
forming the Eq. (22) to polar coordinates as z =reiΦ: 2 2 2 (35)
y˙ =−λ(K+1)xy+∆ωx.
λ
r˙ =− (r2−1)cos(Φ+α), The fixed points are determined by setting x˙ = 0 and
2
y˙ = 0. When λ < λ only the fixed point (x ,y ) ex-
λ 1 ec 1 1
Φ˙ =− (r+ )sin(Φ−α)+∆ω−λKrsin(Φ+α). ists inside the unit cycle in the plane as shown in Fig.
2 r
2(b). Thefixedpointisneutrallystable,andalltheLya-
(33)
punov exponents of the the neutral state with α=0 are
There is a limit cycle solution with r = 1 and periodic
zero. It is worthy to note that if we define a time re-
phase Φ(t), which is called the in-phase state(IPS),
versal transformation as R : (t,x,y) (cid:55)→ (−t,−x,y), the
where all the phase differences between leaves and the
dynamical equations (35) remain invariant. Hence they
hub are the same and time dependent, i. e.,
are called the time-reversible dynamical system or the
ϕ (t)=ϕ(t),1≤j ≤K. (34) quasi-Hamiltonian system [15]. This symmetry endows
j
the system many interesting properties.
This state corresponds to the phases of all leaf nodes Note that, the time reversal transformation R can be
evolve synchronously, while they are not synchronous to resolved intoR=TW with T :t(cid:55)→−t and W :(x,y)(cid:55)→
the hub, as shown in Fig. 1(c). In this case, the model (−x,y). Hence the invariant set for W is the y axis with
Eq. (18) is reduced to the case with K = 1 and the x = 0,y > 0. For any trajectory crossing this invariant
stability of the state can be obtained by Floquet theory set,accordingtothetimereversalsymmetry,theforward
6
trajectoryandthebackwardtrajectoryaresymmetric. If
the forward trajectory evolves to an attractor, the back-
ward trajectory will evolve to the symmetric repeller of
the system. Then the attractor and the repeller of the
system emerge in pairs. When the trajectory crosses the
invariant set more than once, the forward and backward
trajectorywillcoincidewitheachother,formingtheperi-
odicsolutionforthesystem,whichiscalledthereversible
trajectory [15]. For any reversible trajectory, the Lya-
punov exponents have the sign-symmetry form and the
volume of phase space in the vicinity of it are conserved
in average as we discovered in numerical simulations.
FIG. 3: Phase diagram of the Sakaguchi-Kuramoto model.
Regimes SS, SPS and IPS are stable region for the syn-
chronousstate,thesplaystateandthein-phasestaterespec-
For our order parameter plane of the system, it is
tively. The stable region for the neutral state is too narrow
bounded by the unit circle with the invariant set as to plot with only α = 0,±π. The coexistence regime of the
x = 0,y > 0, the attractor and the repeller emerge in splay state and the synchro2nous state is plotted by shadow.
the same time, implying that if the plane only exist one
fixed point, it is neither the attractor nor the repeller,
i. e., the only fixed point is neutrally stable. Suppose VI. SCENARIOS OF SYNCHRONIZATION
that there only exist one neutrally stable fixed point in TRANSITIONS
theplane,thetrajectoriesarevagrantandmustcrossthe
invariant set more than once, then those trajectories are
The phase diagram shown in Fig. 3 presents a great
closed and periodic. This is what happens when α = 0
variety of transitions among the different collective dy-
withtheregionλ<λ asshowninFig. 2(b). Forα=0
ec namicalstates. Onecanalsofindthatsomeofthestates
and λ > λ , there is a coexisting region for the syn-
ec coexist with each other at the same parameter. These
chronous state and the neutral state as the critical cases
coexisting states may lead to abrupt transitions among
for the coexistence region for the neutral state and the
them and hysteresis behaviors, while the others lead to
synchronous state.
continuous transitions.
1. Synchronization transition from the neutral
state
We first investigate the synchronization process from
Alltheabovepossiblecollectivestatesaresummarized the neutral state to the synchronous state for α = 0.
in the parameter space (α,λ) as a phase diagram Fig. The synchronization process when α = 0 is discontin-
3 with the boundaries we get analytically above both uous known as the explosive synchronization which has
from existence and stability conditions. In Fig. 3, four attractedmuchattentionrecently[13]. Ithasbeenshown
regions of the phase shift α can be identified. For the that with changing the coupling strength λ, this kind of
first region −π/2<α<α0−, the splay state exists and is transition is abrupt, and there is a hysteretic behavior
stable for any λ. With the increase of coupling strength at the onset of synchronization, and λb and λf are the
c c
λ, unstable synchronous state exists above the threshold backwardandforwardcriticalcouplingstrengthsrespec-
λ > λec. For the second region α0− < α < 0, the splay tively, where λbc = λ2 and λfc depends on initial states
state exists and is stable within 0 < λ < λˆf, and the as shown in Fig. 4(a). The upper limit of λf is denoted
c c
synchronousstateexistwithλ>λec butunstableunless by λˆf. As λ > λˆf, the synchronization state is globally
c c
λ > λ−sc. Obviously there exist a co-existing region for attractive. It is difficult to understand this process on
the splay state and the synchronous state in the region the basis of the self consistent method, especially for the
−α− <α<0,withinthecouplingintervalλ− <λ<λˆf. hysteresis behavior and coexisting region.
0 sc c
In the third region where 0 < α < α+, the splay state Thecriticalcouplingcorrespondstotheupperlimitof
0
is always unstable, the stable synchronous state emerges λf, which can be determined as
c
as the coupling strength λ > λ . For the forth region
ec
α0+ < α < π/2, the splay state always exist but only λˆf = √ ∆ω . (36)
stable when λ > λ+sc, and the synchronous state only c 2K+1
existandstableintheregionλ <λ<λ+. Theneutral
ec sc
state exists as a particular case for the phase shift, α = Theanalyticalcurveandthesimulationresultsaregiven
0,±π/2, and the in-phase state is always stable in the in Fig. 4(b), it is clear that the results conform with the
region 0 < α < π/2, within the coupling range 0 < λ < curve.
λ . The variety of states in the phase diagram leads to In the bistable regime, as shown in Fig. 4(c), the null-
ec
various transitions among them. clines x˙ = 0 (the red lines) and y˙ = 0 (the green lines)
7
FIG.5: (a)Theforwardandbackwardcontinuationdiagrams
FIG.4: (a)Theforwardandbackwardcontinuationdiagrams with α = −0.2π,N = 11. (b) The upper limit of forward
with α = 0,N = 11. (b) The upper limit of forward critical criticalcouplingstrengthwithα=−0.2π inEq. (37). Phase
coupling strength with α = 0 in Eq. (36). Phase plane of plane for ∆ω = 9, K = 10, α = −0.1π, (c) λ = 1.8, (d)
Eq. (23) with ∆ω = 9, K = 10,α = 0, (c) λ = 1.5, (d) λ=2.17. Red lines are x˙ =0 and green lines are y˙ =0. The
λ=1.9. Red lines are x˙ =0, and green lines are y˙ =0. The intersectionsofx˙ =0andy˙ =0arethefixedpointsA,B,C,D.
intersections of x˙ =0 and y˙ =0 are fixed points A, B, C, D. Trajectories with different initial values are marked as ’∗’.
Trajectories with different initial values are marked by ’∗’.
attraction of the splay state and the synchronous state
have four intersections labeled by A-D with A an attrac-
areseparatedbythesaddlepointB.Whencouplingλin-
tor, C a repeller and B, D neurally stable. Any orbits
creases,asshowninFig. 5(d),thesaddlepointBandthe
crossing the nullcline A-B-C will eventually fall to A,
attractor D collide and disappear via an inverse saddle-
andotherswillholdthepropertyasperiodicorbits. Itis
nodebifurcation,andthisdiscontinuoustransitionmakes
clearthatthestablefixedpointAcorrespondstothesyn-
thefixedpointAcorrespondstothesynchronousstatea
chronous state. And the basin for the neutral state can
global attractor.
be calculated approximated by the circle which has its
3. Synchronizationtransitionfromthein-phase
centerinpointD andradiusasthelengthoflineB−D.
state
As λ increases, points D and B close to each other and
The route of synchronization from the in-phase state
eventually collide at a critical coupling, as shown in Fig.
to the synchronous state for α>0 is shown in Fig. 6(a).
4(d),andthesynchronousstatebecomesgloballyattrac-
The critical coupling strength of this continuous transi-
tive.
tionλ isdeterminedbyEq. (27). Itcanbefoundfrom
2. Synchronization transition from the splay ec
Fig. 6(b) that the simulation results agree well with the
state
analytical curve.
Thesynchronizationprocessfromthesplaystatetothe
The dynamical manifestations of the transition from
synchronous state for α− <α<0 is found to be discon-
0 thein-phasestatetosynchronousstateareshowninFig.
tinuous. Numericalcomputationsrevealthatthiskindof
6(c,d). AsshowninFig. 6(c),thein-phasestateisalimit
transition is abrupt with hysteresis at the onset of syn-
cycle in the order parameter plane. As λ increases, the
chronization as shown in Fig. 5(a). The abrupt transi-
stable fixed point A corresponding to the synchronous
tionimpliesthattherearetwocriticalcouplingstrengths
state emerges on the limit cycle. The transition from
λb and λf, where λb = λ− and λf depend on the basin
c c c sc c the in-phase state to the synchronous state takes place
of attraction. The upper limit of λf can be determined
c continuouslythroughasaddle-nodebifurcation,asshown
by analyzing the inverse saddle-node bifurcation as
in Fig. 6(d).
λˆf = ∆ω . (37) 4. Scenario of desynchronization
c (cid:112)2Kcos(2α)+1 In the region α0+ < α < π2 of the phase diagram 3,
one may find the synchronous state is unstable when
As shown in Fig. 5(b), the simulation results are consis- λ > λ+ and the stable splay state emerges, which is
sc
tent with the analytical curve. contrary to our conventional belief that the system will
The dynamical manifestations of the discontinuous always be synchronous if the coupling strength is large
transition from the splay state to the synchronous state enough. The transition is called the desynchronization,
are shown in Fig. 5(c,d). Fig. 5(c) exhibits the coexis- and it is a continuous transition as shown in Fig. 7(a).
tence of the splay state and the synchronous state as the The order parameter r decrease rapidly at the thresh-
stable fixed points D and A respectively. The basins of old and effective frequencies of hub and leaf nodes are
8
Kuramotomodelbyresortingtothedynamicalorderpa-
rameterequationthatcanbeobtainedintermsofdiffer-
ent approaches, e.g.,the ensemble order parameter ap-
proach and the Watanabe-Strogatz approach. The order
parameterequationobtainedforstarnetworkcanalsobe
approximately described from the Ott-Antonsen ansatz,
whichisoriginatedfromthehighsymmetryofthetopol-
ogy. Byreducingfromahigh-dimensionalphasespaceto
FIG. 6: (a) The forward continuation diagrams with α =
0.3π,N =11. (b)Theforwardcriticalcouplingstrengthwith
α = 0.3π in Eq. (27). Phase plane for ∆ω = 9, K = 10,
α = 0.3π, (c) λ = 0.5, (d) λ = 1.5. Red lines are x˙ = 0 and
greenlinesarey˙ =0. Theintersectionsofx˙ =0andy˙ =0are
the fixed points A,B,C,D. Trajectories with different initial
values are marked as ’∗’.
divided at the same coupling λ. It is easy to know
FIG.7: (a)Theorderparameteragainstthecouplingstrength
the route of the de-synchronization is from the syn-
withα=0.3π,N =11. (b)Thecriticalcouplingstrengthλ+
chronous state to the splay state from the view of the sc
with α = 0.3π. Phase plane for ∆ω = 9, K = 10, α = 0.3π,
phase diagram. The threshold of the de-synchronization
(c) λ=3, (d) λ=5. Red lines are x˙ =0 and green lines are
is λ+sc =−∆ω/(Kcos2α+1) as shown in Fig. 7(b), the y˙ = 0. The intersections of x˙ = 0 and y˙ = 0 are the fixed
simulation results are consistent with it obviously. points A,B,C,D. Trajectories with different initial values are
The dynamical manifestations of the continuous tran- marked as ’∗’.
sition from the synchronous state to the splay state are
shown in Fig. 7(c,d). Fig. 7(c) exhibits the stable syn-
chronousstateofthesystemwhenλ<λ+,alltheorbits amuchlower-dimensionalorderparameterspacewithout
sc
in the phase of the order parameter will evolve to the additional approximation, one is able to grasp analyti-
fixed point A eventually. As λ increases and larger than cally the essential dynamical mechanism of different sce-
the critical coupling λ+, the two nullclines will intersect nariosofsynchronization. Differentsolutionsoftheorder
sc
in four fixed points as shown in Fig. 7(d), the point A parameter equation corresponds to the various collective
loses its stability and a new stable fixed point B which states of coupled oscillators, and different bifurcations
correspondstothesplaystateappears. Theprocessfrom reveal various transitions among those collective states.
thesynchronousstatetothesplaystateisfinishedbythis Theprocessofthosetransitionsarerevealedintheplane
bifurcation continuously. of order parameter and the critical coupling strengths of
them are obtained analytical which are verified by the
simulation results.
VII. CONCLUSION This work is partially supported by the National Nat-
ural Science Foundation of China (Grant No. 11075016
To summarize, in this paper we study the dynamics of and 11475022) and the Scientific Research Funds of
coupledoscillatorsonastarnetworkwiththeSakaguchi- Huaqiao University.
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