From the Kuramoto-Sakaguchi model to the Kuramoto-Sivashinsky equation

1.7 MB·

by  Yoji Kawamura

From the Kuramoto-Sakaguchi model to the Kuramoto-Sivashinsky equation
Yoji Kawamura∗
InstituteforResearchonEarthEvolution,JapanAgencyforMarine-EarthScienceandTechnology, Yokohama236-0001, Japan
(Dated: January 14, 2014)
We derive the Kuramoto-Sivashinsky-type phase equation from the Kuramoto-Sakaguchi-type
phasemodelviatheOtt-Antonsen-typecomplexamplitudeequationanddemonstrateheterogeneity-
inducedcollective-phase turbulencein nonlocally coupled individual-phaseoscillators.
PACSnumbers: 05.45.Xt
Introduction. Large populations of coupled limit- independently drawnfrom an identical distribution g(ω)
4
cycle oscillators exhibit various types of collective be- at each point. Equation (3) can be called a nonlocal
1
0 havior [1–3]. Among them, the following two types of Kuramoto-Sakaguchi model because of the similarity to
2 collective dynamics emerging from a system of coupled Eq. (1).
oscillators have received considerable attention: collec- Introducing a complex order parameter A(r,t) with
n
a tive synchronization in globally coupled systems [4–14] modulus R(r,t) and phase Θ(r,t) through
J andpatternformationincluding spatiotemporalchaosin
4 locally coupled systems [15–21]. The phase description A(r,t)=R(r,t)eiΘ(r,t) = dr′G(r−r′)eiφ(r′,t), (4)
1
method [1–3], which enables us to describe the dynam- Z

ics of an oscillator by a single variable called the phase, we rewrite Eq. (3) as
]
O is commonly used to analyze the system of coupled os-
1 r
A cillators. On one hand, a phase description approach ∂tφ(r,t)=ω(r)− A¯(r,t)eiφ( ,t)eiα−c.c. , (5)
2i
to collective synchronization resulted in the Kuramoto
. (cid:16) (cid:17)
n model [2, 4], which was generalizedby including a phase whereA¯(r,t)isthecomplexconjugateofA(r,t). Apply-
i
l shift to the Kuramoto-Sakaguchimodel [5]: ing mean-fieldtheory to Eq. (5), we obtain the following
n
continuity equation:
[
N
K
1 φ˙j(t)=ωj − N sin(φj −φk+α). (1) ∂ f(φ,ω,r,t)
v k=1 ∂t
X
9
∂ 1
8 This type of phase model has been experimentally re- =− ω− A¯eiφeiα−c.c. f(φ,ω,r,t) , (6)
0 alized using electrochemical oscillators [10, 11] or dis- ∂φ"( 2i ) #
(cid:16) (cid:17)
3 crete chemical oscillators [12–14]. On the other hand,
1. a phase description approach to pattern formation re- where the probability density function f(φ,ω,r,t) satis-
fies the following normalizationconditions for each loca-
0 sulted in the Kuramoto-Sivashinskyequation[2,16] (see
4 also Refs. [22–24]): tion r and each time t:
1 ∞ 2π
v: ∂tψ(r,t)=−∇2ψ−∇4ψ+(∇ψ)2, (2) dω dφf(φ,ω,r,t)=1, (7)
i Z−∞ Z0
X which exhibits spatiotemporal chaos called phase tur- 2π
dφf(φ,ω,r,t)=g(ω), (8)
r bulence [19–21]. In this Letter, we consider a nonlo-
a Z0
cal Kuramoto-Sakaguchi model and derive a Kuramoto-
and the complex order parameter A(r,t) is given by
Sivashinsky equation from it. Namely, we clarify a con-
nection between the above two phase equations. ∞ 2π
Derivation. We consider a system of nonlocally cou- A(r,t)= dr′G(r−r′) dω dφeiφf(φ,ω,r′,t).
pled phase oscillators described by the following equa- Z Z−∞ Z0
(9)
tion [6]:
Now, we utilize the Ott-Antonsen ansatz [25, 26]:
∂ φ(r,t)=ω(r)− dr′G(r−r′)sin φ(r,t)−φ(r′,t)+α , f(φ,ω,r,t)
t
Z (cid:0) (3(cid:1)) = g(ω) 1+ ∞ a(ω,r,t) neinφ+c.c. . (10)
where φ(r,t) ∈ S1 is the phase at location r and time 2π
" ( )#
t. The nonlocal coupling function G(r) is isotropic and nX=1 (cid:16) (cid:17)
normalized as drG(r) = 1. The type of the phase Substituting Eq. (10) into Eq. (6), we obtain the follow-
couplingfunctionisassumedtobein-phasecoupling,i.e., ing equation:
R
|α| < π/2. We note that this system is heterogeneous 1
owingtothespatiallydependentfrequencyω(r),whichis ∂ta(ω,r,t)=−iωa− Aa2e−iα−A¯eiα , (11)
2
(cid:16) (cid:17)
2
where the complex order parameter A(r,t) is given by Owingtothelong-wavedynamicsofthecomplexvariable
z(r,t), the complex order parameter A(r,t) is approxi-
∞
A(r,t)= dr′G(r−r′) dωg(ω)a¯(ω,r′,t). (12) mated as follows:
Z Z−∞ A(r,t)=R(r,t)eiΘ(r,t) ≃z(r,t). (22)
In the case of the Lorentzian frequency distribution
Therefore, the phase of z(r,t) can also be considered as
γ 1
g(ω)= , (13) the collective phase Θ(r,t).
π (ω−ω )2+γ2
0 The uniformly oscillating solution X (Θ) of Eq. (21)
0
the complex order parameter A(r,t) is given by or Eq. (17) is described by
ε
A(r,t)= dr′G(r−r′)z(r′,t), (14) z(r,t)=X0(Θ)= eiΘ, Θ˙(r,t)=Ω, (23)
Reβ
Z r
where the complex variable z(r,t) is defined as where the collective frequency Ω is obtained as
z(r,t)=a¯(ω =ω −iγ,r,t). (15) Imβ
0 Ω=Ω −ε =ω −sinα+γtanα. (24)
0 0
Reβ
FromEq.(11),wethusobtainthefollowingcomplexam-
plitude equation for z(r,t) in a closed form: The left and right Floquet eigenvectors associated with
the zero eigenvalue Λ =0 are respectively given by
0
1
∂ z(r,t)=(−γ+iω )z− A¯z2eiα−Ae−iα . (16)
t 0 2 U∗(Θ)=i Reβ β eiΘ, U (Θ)=i ε eiΘ,
(cid:16) (cid:17) 0 ε Reβ 0 Reβ
We note that this complex amplitude field is homoge- r r
(25)
neous. Equation (16) can be called a nonlocal Ott-
where U (Θ) = dX (Θ)/dΘ. The left and right Floquet
Antonsenequation,whichwasfirstderivedbyLaing[27]. 0 0
eigenvectors associated with another eigenvalue Λ =
This type of equation has been derived and investigated 1
−2ε are respectively given by
by several authors [27–32], but we clarify yet another
point mentioned below.
Reβ ε β
Equation(16)canalsobewritteninthefollowingform: U1∗(Θ)= ε eiΘ, U1(Θ)= ReβReβeiΘ.
r r
(26)
∂ z(r,t)=(ε+iΩ )z−β|z|2z+ β¯(A−z)−βz2(A¯−z¯) ,
t 0 These eigenvectors satisfy the following orthonormaliza-
h (17i) tion condition:
where the parameters are given by
Re U¯∗(Θ)U (Θ) =δ , (27)
cosα sinα 1 p q pq
ε= −γ, Ω0 =ω0− , β = eiα. (18) h i
2 2 2 wherep,q=0,1. AlthoughtheFloqueteigenvectorsand
We note that the first and second terms on the right- their inner product are expressed by complex numbers
handside ofEq.(17)representthe localdynamics called for the sake of convenience in the analytical calculations
a Stuart-Landau oscillator [2], and the remaining term performed below, they exactly coincide with the known
represents the coupling. From the condition of ε > 0, results for the Stuart-Landau oscillator [2].
collective oscillations exist in the following region: Applyingthe second-orderphasereductionmethod[2]
to Eq. (21), we derive the following Kuramoto-
cosα>2γ. (19) Sivashinsky equation for Θ(r,t):
Consideringthelong-wavedynamicsofthecomplexvari- ∂ Θ(r,t)=Ω+ν∇2Θ+µ(∇Θ)2−λ∇4Θ. (28)
t
able z(r,t), we expand the nonlocal coupling term as
Defining the following operator,
∞
A(r,t)−z(r,t)= G2n∇2nz(r,t), (20) Dˆ U(Θ)=G β¯U(Θ)−β X (Θ) 2U¯(Θ) , (29)
2n 2n 0
n=1
X h (cid:0) (cid:1) i
where G is the 2n-th moment of G(r). Substituting we can write the coefficients as follows. First, the coeffi-
2n
Eq. (20) into Eq. (17), we obtain the following equation: cient ν is given by
γ
∂tz(r,t)=(ε+iΩ0)z−β|z|2z+G2 β¯∇2z−βz2∇2z¯ ν =Re U¯0∗(Θ)Dˆ2U0(Θ) =G2 cosα− cos2α . (30)
h i (cid:16) (cid:17)
h i
+G β¯∇4z−βz2∇4z¯ Second, the coefficient µ is given by
4
+O h∇6z . (21i) µ=Re U¯∗(Θ)Dˆ U′(Θ) =G sinα. (31)
0 2 0 2
(cid:0) (cid:1) h i
3
Finally, the coefficient λ is given by which gives G = 1. The truncation of the Kuramoto-
2n
Sivashinsky equation (28) holds if and only if the collec-
λ=Λ−1Re U¯∗(Θ)Dˆ U (Θ) Re U¯∗(Θ)Dˆ U (Θ) tive phase diffusion coefficient ν is small and negative.
1 0 2 1 1 2 0
The parameter values are thus chosen to be α = 1.08
h i h i
−Re U¯0∗(Θ)Dˆ4U0(Θ) and γ = cos(α)/4, which give ν ≃ −0.06, µ ≃ 0.88, and
h i 2 λ≃0.99. Thecentralvalueofthe frequencydistribution
1 γtanα
=G22 cosα−2γ cosα isfixedatω0 =sin(α)−γtan(α),whichgivesΩ=0. The
(cid:18) (cid:19)(cid:18) (cid:19) system size is 512, and a periodic boundary condition is
γ
−G cosα− . (32) imposed.
4 cos2α Numerical simulations of the nonlocal Kuramoto-
(cid:16) (cid:17)
Sakaguchi model (3) [36], the nonlocal Ott-Antonsen
By introducing the following collective phase gradient,
V(r,t)=2∇Θ(r,t), Eq. (28) is also written as equation (16), and the Kuramoto-Sivashinsky equa-
tion(28)areshowninFigs.2,3,and4,respectively. The
∂ V(r,t)=ν∇2V +µV ·∇V −λ∇4V, (33) spatiotemporal evolutions of the collective phase gradi-
t
ent V(x,t) shown in Figs. 2(d), 3(d), and 4 are remark-
which is also called the Kuramoto-Sivashinsky equation. ably similar to each other. Equivalently, the spatiotem-
Here, we note the derivation of the coefficients in poral evolutions of the collective phase Θ(x,t) shown in
another manner [24]. The linear stability analysis of Figs. 2(b) and 3(b) are also similar to each other. As
Eq.(17)aroundtheuniformlyoscillatingsolutionX (Θ) seen in Figs.2(a) and 2(b), the spatialpattern of the in-
0
given in Eq. (23) provides two linear dispersion curves dividualphase φ(x,t) isnon-smooth,butthatofthe col-
Λ (q): one is the phase branch, which satisfies Λ (0)= lectivephaseΘ(x,t)issmooth. AsseeninFigs.3(a)and
± +
Λ = 0; the other is the amplitude branch, which sat- 3(b), the phase ofz(x,t) canbe consideredas the collec-
0
isfies Λ (0) = Λ = −2ε. The phase branch is ex- tive phase Θ(x,t), namely, Eq. (22) is actually valid. As
− 1
panded with respect to the wave number q as follows: seeninFigs.2(c)and3(c), the orderparametermodulus
Λ (q) = −νq2−λq4+O(q6). The linear coefficients, ν R(x,t) is almost constant, so that the phase reduction
+
andλ, arealso obtainedin this way. Inaddition, the co- approximation is also valid. We thus conclude that this
efficientµisfoundfromthe dependence ofthe frequency spatiotemporalchaosistheso-calledphaseturbulencein
Ω on the wave number k for plane wave solutions to the complex order parameter field. Here, we note that
k
Eq. (17) as follows: Ω =Ω+µk2+O(k4). amplitude turbulence also occurs in the large negative ν
k
We also note that the collective phase diffusion coef- region.
ficient ν can be negative despite the in-phase coupling, Discussion. In this Letter, we studied heterogeneity-
|α| < π/2, and that negative diffusion results in spa- inducedturbulenceinnonlocallycoupledoscillators(i.e.,
tiotemporal chaos. In fact, from the condition of ν < 0, the nonlocal Kuramoto-Sakaguchi model), where the
spatiotemporal chaos can occur in the following region: Kuramoto-Sivashinskyequationwasderivedviathenon-
local Ott-Antonsen equation. In Ref. [33], we have
cos3α<γ. (34) studied noise-induced turbulence in nonlocally coupled
oscillators (i.e., a nonlocal noisy Kuramoto-Sakaguchi
At the onset of collective oscillation, i.e., cosα = 2γ, model), where the Kuramoto-Sivashinsky equation has
spatiotemporal chaos can occur in the following region: been derived via the complex Ginzburg-Landau equa-
tion [2, 15, 18] or the nonlinear Fokker-Planck equa-
1
cos2α< , (35) tion [2]. There exists a remarkable connection be-
2
tween the nonlocal Kuramoto-Sakaguchi model and the
which gives |α| > π/4. Figure 1 shows the phase dia- Kuramoto-Sivashinsky equation.
gram, which is composed of Eqs. (19), (34), and (35) in As mentioned inRef. [34], noise-inducedturbulence in
the parameter plane, α and γ. When π/4 < |α| < π/2, a system of nonlocally coupled oscillators [33] is closely
thesignofthe coefficientν changesfrompositivetoneg- related to noise-induced anti-phase synchronization be-
ative as the dispersion parameter γ increases; this tran- tween two interacting groups of globally coupled oscilla-
sition phenomenon can be called heterogeneity-induced tors [34]. Similarly, heterogeneity-induced turbulence in
turbulence. a system of nonlocally coupled oscillators is also closely
Simulation. Forthe sakeofsimplicity,wecarriedout relatedto heterogeneity-inducedanti-phasesynchroniza-
numerical simulations in one spatial dimension. As the tion between two interacting groups of globally coupled
nonlocalcouplingfunctionG(x), weusedtheHelmholtz- oscillators[35]; namely, Eqs. (30) and (31) in this Letter
type Green’s function: correspond to Eq. (31) in Ref. [35].
In summary, we derived the Kuramoto-Sivashinsky
1 ∞ eiqx 1 equation from the nonlocal Kuramoto-Sakaguchi model
G(x)= dq = e−|x|, (36)
2π 1+q2 2 and demonstrated heterogeneity-induced phase turbu-
Z−∞
4
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5
π / 2
no collective oscillation
α
ν < 0
ν > 0
π / 4
collective oscillation
0
0.0 0.1 0.2 0.3 0.4 0.5
γ
FIG. 1: (color online). Phase diagram in the parameter
plane, α and γ. The red solid curve indicates the Hopf
bifurcation line, γ = cos(α)/2. The blue broken curve
indicates the transition line to turbulence, γ = cos3(α).
The dotted line (α = π/4) is a guide for the eye. The
plussign(+)indicatesα=1.08andγ =cos(α)/4,which
were used in all thenumerical simulations.
FIG. 2: (color online). Numerical simulation of the non-
local Kuramoto-Sakaguchi (nKS) model. The number of
grid points is 219. (a) Local phase φ(x,t). (b) Order
parameter phase Θ(x,t). (c) Order parameter modulus
R(x,t). (d) Orderparameter phase gradient V(x,t).
6
FIG. 4: (color online). Numerical simulation of the
Kuramoto-Sivashinsky (KS) equation. The number of
grid pointsis 211.
FIG. 3: (color online). Numerical simulation of the non-
local Ott-Antonsen(nOA)equation. Thenumberofgrid
pointsis211. (a) Local variableargumentargz(x,t). (b)
Order parameter phase Θ(x,t) = argA(x,t). (c) Order
parametermodulusR(x,t)=|A(x,t)|. (d)Orderparam-
eter phase gradient V(x,t)=2∂xΘ(x,t).