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 lence. We hope that the connection between these two [26] E. Ott and T. M. Antonsen,Chaos 19, 023117 (2009). landmark phase equations will facilitate the theoretical [27] C. R.Laing, Physica D 238, 1569 (2009). analysis of coupled oscillators and that heterogeneity- [28] C. R.Laing, Physica D 240, 1960 (2011). [29] W. S. Lee, J. G. Restrepo, E. Ott, and T. M. Antonsen, induced turbulence will be experimentally confirmed in Chaos 21, 023122 (2011). the near future. [30] G. Bordyugov, A. Pikovsky, and M. Rosenblum, Phys. Rev. E82, 035205(R) (2010). TheauthorisgratefultoYoshikiKuramotoandHiroya [31] M. Wolfrum, O. E. Omel’chenko, S. Yanchuk, and Nakaoforvaluablediscussions. Thisworkwassupported Y. L. Maistrenko, Chaos 21, 013112 (2011). by JSPS KAKENHI Grant Number 25800222. [32] O. E. Omel’chenko, Nonlinearity 26, 2469 (2013). [33] Y. Kawamura, H. Nakao, and Y. Kuramoto, Phys. Rev. E 75, 036209 (2007). [arXiv:nlin/0702042] [34] Y. Kawamura, H. Nakao, K. Arai, H. Kori, and Y. Ku- ramoto, Chaos 20, 043109 (2010). [arXiv:1007.4382] ∗ Electronic address: [email protected] [35] Y. Kawamura, H. Nakao, K. Arai, H. Kori, and Y. Ku- [1] A. T. Winfree, The Geometry of Biological Time ramoto, Chaos 20, 043110 (2010). [arXiv:1007.5161] (Springer,Second Edition, New York,2001). [36] Thissimulationrequiresacarefulsetuptosuppressfinite- [2] Y.Kuramoto,ChemicalOscillations,Waves,andTurbu- sizeeffects.Thefrequencydistributionwithinthecharac- lence (Springer, NewYork, 1984). teristiccouplingwidthateachpointshouldbeLorentzian [3] A.Pikovsky,M.Rosenblum,andJ.Kurths,Synchroniza- as well as possible, and each distribution at each point tion: A Universal Concept in Nonlinear Sciences (Cam- should be thesame as each other. 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[22] G. I. Sivashinsky,Acta Astronautica 4, 1177 (1977). [23] Y.Kuramoto, Prog. Theor. Phys. 63, 1885 (1980). [24] Y.Kuramoto, Prog. Theor. Phys. 71, 1182 (1984). [25] E. Ott and T. M. Antonsen,Chaos 18, 037113 (2008). 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).