Low-dimensional description for ensembles of identical phase oscillators subject to Cauchy noise
LLow-dimensional description for ensembles of identical phase oscillators subject toCauchy noise
Ralf T¨onjes and Arkady Pikovsky
1, 2 Institute of Physics and Astronomy, Potsdam University, 14476 Potsdam-Golm, Germany Department of Control Theory, Nizhny Novgorod State University,Gagarin Avenue 23, 603950 Nizhny Novgorod, Russia
We study an ensembles of globally coupled or forced identical phase oscillators subject to inde-pendent white Cauchy noise. We demonstrate, that if the oscillators are forced in several harmonics,stationary synchronous regimes can be exactly described with a finite number of complex order pa-rameters. The corresponding distribution of phases is a product of wrapped Cauchy distributions.For sinusoidal forcing, the Ott-Antonsen low-dimensional reduction is recovered.
One of the central challenges in synchronization the-ory is finding a possibility to describe the dynamics ofpopulations of globally coupled oscillators in terms of afew order parameters. Indeed, generally, following thepioneering approach of Kuramoto [1], one derives self-consistency conditions for the order parameters, whichare formulated as integral equations and constitute aninfinite hierarchy of coupled nonlinear relations. Suchself-consistency equations have also been derived for os-cillators with coupling heterogeneity [2], or for identicaloscillators subject to Gaussian white noise and couplingin the first or higher harmonics [3, 4]. Self consistencymeans that given a stationary prior phase distribution,one calculates the force acting on oscillators of frequency ω , the phase distribution of these oscillators and, by aver-aging over a frequency distribution g ( ω ), the phase distri-bution of the whole oscillator ensemble. The latter mustbe equal to the prior distribution. The self-consistencycondition is often formulated in terms of integrals whichhave to be evaluated numerically.Only in the case of coupling in the first harmonics,Lorentzian frequency distribution and zero noise strengththe integrals can be evaluated explicitly. A Lorentzianfrequency distribution is also central to the low di-mensional dynamics of oscillator ensembles on the Ott-Antonsen manifold [5]. Indeed, this is the main reasona Lorentzian frequency distribution is used by default inmany studies on nonidentical phase oscillators.Although Gaussian white noise acts qualitatively sim-ilar to frequency heterogeneity, for ensembles driven byGaussian noise there is no exact low-dimensional reduc-tion, except for several approximate approaches basedon moment closures of the infinite hierarchy [6–8]. Aswe demonstrate in this letter, the situation is differentwhen the noise is not Gaussian but L´evy stable with ex-ponent α = 1, i.e. Cauchy noise. Using Cauchy whitenoise instead of Gaussian one simplifies the analysis ina similar way as using a Lorentzian instead of a Gaus-sian frequency distribution, while keeping the bifurca-tion scenario qualitatively similar. In the simplest caseof purely harmonic coupling, Cauchy noise allows forthe Ott-Antonsen reduction (cf. [9]). Furthermore, we demonstrate that a low-dimensional reduction is also pos-sible for multi-mode coupling , albeit being restricted tostationary distributions which we show to be fully char-acterized by a finite number of modes.We start with formulating general equations for an en-semble of identical phase oscillators driven by indepen-dent noise forces ˙ ϑ n = F ( ϑ n ) + η n ( t ) . (1)For ensembles of coupled oscillators the driving force F ( ϑ ) depends on the phase distribution P ( ϑ, t ), whichmakes the problem of finding a stationary phase dis-tribution nonlinear. This is for instance the case withKuramoto-Daido mean field coupling F ( ϑ ) = (cid:90) π H ( ϑ − ϑ (cid:48) ) P ( ϑ (cid:48) ) dϑ (cid:48) . (2)The noise terms η n ( t ) are assumed to be Poisson pro-cesses of delta pulses with rate ν and amplitude distribu-tion W (∆ ϑ ). Then the evolution equation for the phasedensity is given by the integro-differential equation ∂ t P = − ∂ ϑ ( F ( ϑ ) P ) + ν (cid:90) π ( W ( φ ) − δ ( φ )) P ( ϑ − φ ) dφ. (3)As special cases we will consider a wrapped Gaussian dis-tribution of pulse amplitudes W G ∼ N (0 , Dν − ), anda wrapped Cauchy distribution W C ∼ C (0 , σν − ). Inthe limit ν → ∞ the shot noise η ( t ) becomes, respec-tively, Gaussian (cid:82) τ η G ( t ) dt ∼ √ Dτ N (0 ,
1) or Cauchywhite noise (cid:82) τ η C ( t ) dt ∼ στ C (0 , W U = (2 π ) − . Equation (3) can berewritten in terms of the Fourier components of F with F ( ϑ ) = (cid:80) ∞ k = −∞ f k exp[ − ikϑ ] and order parameters (cir-cular moments) z k = (cid:10) e ikϑ (cid:11) P and w k = (cid:10) e ikφ (cid:11) W as aninfinite system of coupled ordinary differential equations˙ z k = ik ∞ (cid:88) l = −∞ f − l z k + l + ν ( w k − z k . (4) a r X i v : . [ n li n . AO ] S e p The circular moments w Gk of the wrapped Gaussian dis-tribution, w Ck of the wrapped Cauchy distribution and w Uk of the uniform distribution are w Gk = exp (cid:18) − Dν k (cid:19) , w Ck = exp (cid:16) − σν | k | (cid:17) , w Uk = δ k . (5)In the limit ν → ∞ we recover the Fourier representa-tion of the second derivative ν (cid:0) w Gk − (cid:1) → − Dk forGaussian white noise, and of the fractional derivative ν (cid:0) w Ck − (cid:1) → − σ | k | for Cauchy white noise [10, 11].As a first result we show how the type of noise affectsstability of the asynchronous system state with a uni-form phase distribution for Kuramoto-Daido coupling (2)where the Fourier modes of the force F depend on themoments of the phase distribution as f k = h k z k . In thiscase the incoherent state P ( ϑ ) = (2 π ) − or z k = δ k isalways a stationary solution of (4). Linearization yieldsdecoupled equations for different order parameters˙ z k = [ ik ( h k + f ) + ν ( w k − z k . (6)The condition for stability of a mode k > ikh k + ν ( w k − < . (7)This gives three different stability criteria: Im (cid:2) ¯ h k /k (cid:3) 1, only the λ m with | λ m | < L roots λ m with | λ m | < 1, the coefficients c m are theunique solutions of the following set of linear equationsfor k = 1 . . . L − L (cid:88) m =1 c m = 1 , L (cid:88) m =1 c m λ km − ¯ c m ¯ λ − km = 0 . (15)The first inhomogeneous equation expresses z = 1,and the L − z stk = ¯ z st − k expressed in terms of (13) for k = 1 . . . L − 1. This means all z k are fully deter-mined by the set of eigenvalues λ m . This is the desiredlow-dimensional reduction: possible stationary distribu-tions in a population of oscillators forced in L harmonicsare fully determined by L complex parameters λ m with | λ m | < 1. Furthermore, (14) constitutes a linear set ofequations for the Fourier modes f l , the noise intensity σ and the frequency Ω which are thus known explicitly asfunctions of the λ m .Next, we demonstrate that this low-dimensional solu-tion for a stationary phase density is in fact a product ofWCDs, which can be dubbed poly-WCD (cf. [14]) P st ( ϑ ) = 1 M π L (cid:89) m =1 − | λ m | | e iϑ − λ m | . (16)Given the values of λ m we explicitly calculate the coeffi- cients c m z stk = 1 M (cid:90) π e ikϑ π L (cid:89) m =1 − | λ m | | e iϑ − λ m | dϑ (17)= 12 πi (cid:73) | z | =1 z k M L (cid:89) m =1 z (cid:16) − | λ m | (cid:17) ( z − λ m )(1 − ¯ λ m z ) dzz = L (cid:88) m =1 λ km M L (cid:89) n (cid:54) = m λ m (cid:0) − | λ n | (cid:1) ( λ m − λ n ) (cid:0) − ¯ λ n λ m (cid:1) ( k ≥ − L ) . The integrand has exactly the L poles λ m on the unitdisc if k ≥ − L . This begets (13) with coefficients c m = 1 M L (cid:89) n (cid:54) = m λ m (cid:0) − | λ n | (cid:1) ( λ m − λ n ) (cid:0) − ¯ λ n λ m (cid:1) (18)and with normalizing weight M = L (cid:88) l =1 L (cid:89) p (cid:54) = l λ l (cid:0) − | λ p | (cid:1) ( λ l − λ p ) (cid:0) − ¯ λ p λ l (cid:1) . (19)Equations (18)-(19) together with z stk = L (cid:88) m =1 c m λ km , k ≥ − L (20)and Eq. (14) (where only roots with modulus smallerthan 1 are taken) form the self consistency conditionsfor the stationary phase density of identical phase oscil-lators subject to Cauchy white noise and under a forcing F ( ϑ ) with L harmonics. Equation (20) can be regardedas a generalization of the Ott-Antonsen ansatz, althoughit is restricted to stationary solutions. While algebraicself-consistency equations still require numeric root find-ing, the evaluation of these equations is much faster andnumerical errors are much smaller than for integral self-consistency equations. Before proceeding to an exam-ple we mention that a circle distribution having moment z k = cλ k has been considered by Kato and Jones [15].Expression (20) means that our stationary distributionsare weighted sums of Kato-Jones distributions.Let us discuss the simplest nontrivial example: an en-semble of phase oscillators with a phase difference cou-pling (2) H (∆ ϑ ) = ε sin( α − ∆ ϑ ) + ε sin( α − ϑ ) (21)in the first and the second harmonics, i.e. h k = i ε k e iα k ,subject to Cauchy white noise. According to (20), sta-tionary rotating wave solutions have the form z stk = c λ k + c λ k (22)where, as it follows from (18) c = 1 − c = (cid:20) − λ (1 − | λ | )(1 − ¯ λ λ ) λ (1 − | λ | )(1 − ¯ λ λ ) (cid:21) − (23) | λ | a r g ( λ ) (a) σ R , R IIIIII IIIIII (b) σ R , R (c) t R , R (d)FIG. 1. (a) Zero lines for the imaginary (dark blue) andthe real part (light red) of ∆ as a function of λ (amplitudeand complex argument) given λ = 0 . ε = 1 . ε = 0 . α = − . α = − . 0. The points λ (cid:54) = λ where theselines cross (marker symbols) correspond to stationary solu-tions. In the shaded region σ < 0. (b) Collection of stationarysolutions upon variation of λ from zero to one. The dark blueand the light yellow lines mark the order parameters R , R ,respectively. Bold lines are stable solutions and dotted linesunstable solutions. The markers correspond to the same sta-tionary solutions depicted in (a), they are located on the threedifferent branches (triangle markers are on the same branch).(d) Zoom of the transition region from branch I to branch II .Branch I becomes unstable in a supercritical Hopf bifurcationwhen σ ≈ . σ is slowly decreased. Between 0 . < σ < . 163 no station-ary solution exists whereas between 0 . < σ < . 154 theperiodic solution co-exists with a stable equilibrium. Below σ < . 148 the periodic solution has disappeared in a SNICbifurcation. (d) Integration of the Langevin equations (1) for N = 5000 phase oscillators with phase difference coupling inthe first an second harmonics and Cauchy noise of strength σ = 0 . 151 in a small bistable regime. Stochastic switching be-tween stationary and oscillating order parameters is observed. and from (14), λ and λ are simultaneous solutions ofthe algebraic equationsΩ − iσ = ¯ h ¯ z st λ , + ¯ h ¯ z st λ , + h z st λ − , + h z st λ − , . (24)One parameter in the problem can be eliminated by arescaling of time. In this example we choose ε = 1. Fur-thermore, because of rotational invariance ϑ → ϑ + const ,we can choose λ ∈ [0 , 1] to be real. Given this free pa-rameter we can calculate the right hand sides of (24)asfunctions of λ . At points λ (cid:54) = λ where the difference∆ between the r.h.s. of (24) vanishes, a stationary solu- θ ) θ E rr o r σ (a) (b)FIG. 2. (a) Comparison of the empirical probability densityobtained in simulations of an ensemble of N = 10 oscillators(red circles) with theoretical prediction of poly-WCD (solidcurve under the circles), for branch I σ = 0 . 2. Dashedgreen lines show corresponding wrapped Cauchy densities (ar-bitrarily scaled). (b) Bifurcation diagram (red: span between R max and R min ) obtained from the solutions of system (11)at slowly decreasing parameter σ . Dashed blue line (see scaleat the right) shows the deviation from the poly-WCD. tion exists in a rotating reference frame with frequencyΩ and noise σ , which are thus defined parametrically asthe real part and the negative imaginary part of the r.h.s.of (24), respectively. Tuning λ from zero to one we canquickly pinpoint all λ where the difference ∆ is zero (seeFig. 1(a)), and continue these different solution branches.We illustrate the found bifurcation diagram in Fig. 1.Zeros of ∆ found according to Fig. 1(a) result in threebranches of solutions. Branch I , starting at the point ofinstability of the first mode σ c = cos(0 . / ≈ . 49, is astate with a bi-modal distribution, where both real orderparameters R , = | z , | are non-zero. Another branch III , which starts at the instability point of the secondmode σ c = 0 . / ≈ . 19, is a pure symmetricbi-modal solution, where only even modes are presentand R = 0. This branch in terms of the eigenvalues λ , correspond to the case λ = − λ , c = c = .There is also a third nontrivial mode II that bifurcatesat σ c ≈ . III and foldsback at larger values of σ to become the unique station-ary solution in the limit of small noise strength. Stabilityof stationary states (and periodic solutions) was checkednumerically from (11) with truncation at a large numberof Fourier modes. Remarkably, there is a stability changefrom mode I to mode II (mode III is always unstable).An interesting feature of the bifurcation diagram Fig. 1is that there is a range of noise intensities σ where allsteady states are unstable, and stable periodic oscilla-tions of the order parameters are observed. Furthermore,there is an even smaller range of bi-stability where stableperiodic oscillations coexists with the stable stationarystate of branch II . Only for stationary states we doexpect validity of the poly-WCD distribution. To char-acterize this, it is instructive to consider a Fourier trans-form of the logarithm of P st ( ϑ ) (16). In our examplethis is ln P st ( θ ) = const + (cid:80) k> ( s k e − ikϑ + ¯ s k e ikϑ ) with s k = k − ( λ k + λ k ). Remarkably, this representation doesnot contain the constants c m . Because all Fourier modes s k depend on two complex numbers only, Fourier modeswith k > s , . In particular, s = s s − s / 6. Thus, quantity Err = | s − s s + s / | serves as a measure of the deviation from the poly-WCD.We show this deviation together with the empirical bi-furcation diagram in Fig. 2(b). One can see that indeed,while stationary states are given by (16), a clear devia-tion occurs for periodic solutions.In conclusion, we have demonstrated that an ensem-ble of identical phase oscillators subject to independentCauchy white noise admits a finite-dimensional descrip-tion. In the simplest case of sinusoidal forcing, the re-sulting reduction is just the Ott-Antonsen ansatz witha wrapped Cauchy distribution of the phases. If forc-ing contains up to L Fourier modes, stationary statesare given by a poly-wrapped Cauchy distribution with L complex roots λ m on the unit disc as parameters.This finite-dimensional reduction is valid not only forKuramoto-Daido type coupling, but also for more gen-eral situations, such as Winfree-type models and ensem-bles with nonlinear coupling.We stress here that the special role of the Cauchydistribution is well-known in statistical physics, start-ing from the seminal work by Lloyd on the exact so-lution [16] for disordered Hamiltonians with Cauchy dis-tributed heterogeneities. In the context of populations ofdynamical elements, this property has been explored instudies of homographic maps [17]. Our study shows thatthe Cauchy noise significantly simplifies the descriptionof the dynamics, compared to the Gaussian noise, alsofor populations of phase oscillators.We thank D. Goldobin for fruitful discussions. A. 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