Low-dimensional dynamics of phase oscillators driven by Cauchy noise
LLow-dimensional dynamics of phase oscillators driven by Cauchynoise
Takuma Tanaka Graduate School of Data Science, Shiga University,1-1-1 Banba, Hikone, Shiga 522-8522, Japan ∗ (Dated: September 10, 2020) Abstract
Phase oscillator systems with global sine-coupling are known to exhibit low-dimensional dynam-ics. In this paper, such characteristics are extended to phase oscillator systems driven by Cauchynoise. The low-dimensional dynamics solution agreed well with the numerical simulations of noise-driven phase oscillators in the present study. The low-dimensional dynamics of identical oscillatorswith Cauchy noise coincided with those of heterogeneous oscillators with Cauchy-distributed nat-ural frequencies. This allows for the study of noise-driven identical oscillator systems throughheterogeneous oscillators without noise and vice versa.
PACS numbers: ∗ [email protected] a r X i v : . [ n li n . AO ] S e p . INTRODUCTION The synchronized rhythmic flashing of fireflies is a spectacular example of a collectivephenomenon [1]. Fireflies exhibit different and fluctuating flashing frequencies and can beregarded as heterogeneous and noisy oscillators. Both heterogeneity and noise are essentialproperties of systems that display collective phenomena. Coupled phase oscillators have beenused to examine how heterogeneity and noise affect the synchronization of physical, chemical,and biological systems [2, 3]. Phase oscillator systems with heterogeneous natural frequencieshave been studied since the invention of the phase oscillator model. Ott and Antonsen [4]showed that the behavior of globally sine-coupled oscillators, the natural frequencies ofwhich obey a family of rational distribution functions, can be described by low-dimensionaldynamics. Specifically, if the natural frequencies obey the Cauchy or Lorentzian distribution,the dynamics of an infinite number of oscillators are described by a Stuart–Landau equation,i.e., a two-dimensional dynamical system. If the coupling strength takes on several values orthe natural frequencies obey the mixture of Cauchy distributions, the dynamics are describedby coupled Stuart–Landau oscillators. This is an exact result for a specific initial conditionand not an approximation obtained by ignoring higher order terms. This type of low-dimensional description has accelerated the study of the heterogeneous oscillator systems[5, 6].However, investigating noise-driven oscillator systems appears to be more challengingthan studying heterogeneous oscillator systems. Previous studies have approximated thedynamics with circular cumulants to obtain low-dimensional dynamics similar to those pro-posed by Ott and Antonsen [7, 8]. Although this approach has been implemented withsome success, it is not always free from approximation error. Determining low-dimensionaldescriptions with fewer approximation errors will be useful in understanding the collectivephenomena in various fields, although it may not be as general as approximation with cir-cular cumulants. This may be possible using a noise that adheres to the assumption of theanalysis by Ott and Antonsen.This paper reports that systems driven by Cauchy noise can be described by closed-formlow-dimensional dynamical equations. Phase oscillator systems driven by Cauchy or moregenerally by non-Gaussian noise have not been studied in as much detail as those driven byGaussian noise. However, non-Gaussian noise is known to be prevalent in biological systems29]. For example, a circular auto-regressive model with wrapped Cauchy noise has beenproposed to model animals’ direction of travel [10]. Thus, the behavior of phase oscillatorsdriven by Cauchy noise is worthy of further examination. In addition, as harmonic oscillatorsdisplay nontrivial phase distribution under L´evy noise [11], the dynamics of phase oscillatorsdriven by Cauchy noise is of interest.This paper is organized as follows. First, the Watanabe–Strogatz theory is reviewed andused to derive the low-dimensional dynamics of the order parameter of identical sine-coupledoscillators driven by Cauchy noise. Second, the Ott–Antonsen ansatz is reviewed, and thedynamics of the order parameter of heterogeneous noise-driven oscillators are derived. Itis shown that the amplitude of Cauchy noise and the scale parameter of natural frequencyare equivalent in the low-dimensional description, and the implications of the model arediscussed.
II. ANALYSIS AND RESULTS
This section first considers the system of identical oscillators and then that of hetero-geneous oscillators. Using the notation of Pikovsky and Rosenblum [12], we consider thesystem of N noise-driven phase oscillators with identical natural frequency ω , in which thedynamics of oscillator k are given by˙ φ k = ω + (cid:61) [ H ( t ) exp( − i φ k )] + σ ( t ) ξ k ( t )= ω + | H ( t ) | sin[arg H ( t ) − φ k ] + σ ( t ) ξ k ( t ) , (1)where H ( t ) is the common forcing, σ ( t ) > ξ k ( t ) isthe noise. This paper uses the Cauchy distribution instead of the Gaussian distribution,which has been used in earlier studies [7, 8]. It is assumed that ξ k ( t ) obeys the independentstandard Cauchy distribution without temporal correlation; the probability density functionof ξ k ( t ) is p [ ξ k ( t )] = 1 π ξ k ( t ) + 1 . (2)3he common forcing, H ( t ), can be an external forcing or mutual interaction between oscil-lators. For example, the dynamics with σ ( t ) = σ and H ( t ) = Kz ( t ), where z ( t ) = 1 N N (cid:88) k =1 exp(i φ k ) (3)is the complex-valued order parameter and K is the coupling strength, lead to the followingdynamics ˙ φ k = ω + KN N (cid:88) j =1 sin( φ j − φ k ) + σξ k ( t ) . (4)In this system, the oscillators are driven by the Cauchy noise and attracted to each other.The system of Eq. (1) can be numerically implemented by the Euler method as φ k ( t + ∆ t ) = φ k ( t ) + ∆ t { ω + | H ( t ) | sin[arg H ( t ) − φ k ( t )] + σ ( t ) ξ k ( t ) } , (5)where ξ k ( t ) follows the standard Cauchy distribution [Eq. (2)]. Let us note that the noiseterm is multiplied by ∆ t instead of √ ∆ t because the Cauchy distribution is the stabledistribution of index 1.Here, what has been clarified by the previous studies on the behavior of the systemwithout noise is reviewed. Inserting σ ( t ) = 0 into Eq. (1) yields˙ φ k = ω + (cid:61) [ H ( t ) exp( − i φ k )] . (6)Watanabe and Strogatz [13, 14] demonstrated that this system is described using threevariables and N − φ k ( t ) (1 ≤ k ≤ N )of oscillators driven by the common forcing H ( t ) are given by a three-parameter functionof the initial phases, φ k (0). Using the Watanabe–Strogatz theory, the function that maps φ k (0) to φ k ( t ) is defined by the real and imaginary components of the order parameter anda parameter corresponding to the rotation of the initial phases [13, 15]. This allows usto obtain a closed-form description of the dynamics of order parameter. In the followinganalysis, it is assumed that the constants of motion are uniformly distributed in the limit ofan infinite number of oscillators. This assumption has successfully described the behaviorof the finite number of phase oscillators whose initial phases are drawn from the uniformdistribution on [0 , π ]. The order parameter z ( t ) becomes Z ( ω, t ) = (cid:90) π p ( φ, t | ω ) exp(i φ ) d φ (7)4n the limit of N → ∞ , where p ( φ, t | ω ) is the density of the phases of oscillators with naturalfrequency ω at time t . For the system of Eq. 6, the dynamics of the order parameter havebeen shown to follow ∂Z ( ω, t ) ∂t = i ωZ ( ω, t ) + H ( t )2 − ¯ H ( t )2 Z ( ω, t ) (8)[15, 16]. Because, if the initial phases are uniformly distributed, the rotation of the initialphase does not affect the final distribution of the phases, the phase distribution of oscillatorsat t is determined solely by the order parameter [15]. Thus, it has been shown that thedensity of the oscillators’ phase obeys the Poisson kernel [4] p ( φ, t | ω ) = 12 π − | Z ( ω, t ) | − | Z ( ω, t ) | cos[ φ − arg Z ( ω, t )] + | Z ( ω, t ) | . (9)Having reviewed the previous results, we are prepared to examine the dynamics of noise-driven oscillators. Because the phase distribution in the system without noise, which has alow-dimensional description, is determined by the order parameter, the system with noisecan have a low-dimensional description if the phase distribution is determined by a fewparameters. To obtain a low-dimensional description, it is useful to note that the Poissonkernel is identical to the wrapped Cauchy distribution [17] p ( φ ) = ∞ (cid:88) n = −∞ λπ [ λ + ( φ − µ + 2 πn ) ]= 12 π sinh λ cosh λ − cos( φ − µ ) (10)if the following is set: µ = arg Z ( ω, t ) , (11) λ = sinh − (cid:18) | Z ( ω, t ) | − − | Z ( ω, t ) | (cid:19) = − log | Z ( ω, t ) | . (12)Before considering noise-driven sine-coupled oscillators, uncoupled oscillator systems drivenby Cauchy noise are examined, that is, σ ( t ) > H ( t ) = 0. In this system, assumingthat the oscillators are initially distributed according to the wrapped Cauchy distribution5Eq. (10)], the Cauchy noise ensures that the oscillators obey the wrapped Cauchy distribu-tion. This is clarified by the Euler method φ k ( t + ∆ t ) = φ k ( t ) + ∆ tσ ( t ) ξ k ( t ) . (13)If φ k ( t ) obeys the Cauchy distribution with the scale parameter λ and the location param-eter µ , φ k ( t + n ∆ t ), where n >
0, obeys the Cauchy distribution with the scale parameter λ + ∆ t (cid:80) n − j =0 σ ( t + j ∆ t ) and the location parameter µ owing to the reproductive property.Therefore, the Cauchy noise ξ k ( t ) increases the scale parameter λ as˙ λ = σ ( t ) , (14)while keeping the location parameter constaint as˙ µ = 0 . (15)Inserting Eqs. (11) and (12) gives ∂Z ( ω, t ) ∂t = − σ ( t ) Z ( ω, t ) . (16)This equation means that the Cauchy noise causes the exponential decay of the order pa-rameter.Because Eqs. (8) and (16) are exact closed-form descriptions of Z ( ω, t ) in the limit of N → ∞ , we can combine these two equations to obtain the dynamics of Cauchy noise-drivencoupled oscillators. This is justified by the fact that, the oscillators obeying a wrappedCauchy distribution remain obeying a wrapped Cauchy distribution if driven by either thesine-coupling or Cauchy noise. Combining Eq. (8) with Eq. (16) yields the dynamics of thesystem with H ( t ) (cid:54) = 0 and σ ( t ) > ∂Z ( ω, t ) ∂t = [i ω − σ ( t )] Z ( ω, t ) + H ( t )2 − ¯ H ( t )2 Z ( ω, t ) . (17)This is equivalent to the system of oscillators driven alternately by Eq. (8) and Eq. (16). Forglobally sine-coupled phase oscillator systems [Eq. (4)], the common forcing of the oscillatorsis proportional to the order parameter, that is, H ( t ) = KZ ( ω, t ). Hence, Eq. (17) can beused to describe the dynamics of the order parameter of the system of Eq. (4) with ∂Z ( ω, t ) ∂t = (cid:18) i ω − σ + K (cid:19) Z ( ω, t ) − K Z ( ω, t ) Z ( ω, t ) . (18)6 .0 0.1 0.2 0.3 0.4 0.5 0.60.00.20.40.60.81.0 | z | FIG. 1. Numerical and theoretical results of (cid:104)| z |(cid:105) for the system of Eq. (4) with K = 1. Thenumerical and theoretical results are indicated by the circles and the solid line, respectively. It has a closed-form stable solution Z ( ω, t ) = (cid:113) − σK exp[i ω ( t − t )] ( σ ≤ K )0 (cid:0) K < σ (cid:1) , (19)where t is a constant. This means that a weak noise allows for the synchronization whereasnoise stronger than a threshold value abolishes the synchronization.To numerically confirm the above theoretical prediction, the simulation of Eq. (4) wasperformed by the Euler method φ k ( t + ∆ t ) = φ k ( t ) + ∆ t (cid:32) ω + KN N (cid:88) j =1 sin[ φ j ( t ) − φ k ( t )] + σξ k ( t ) (cid:33) (20)with the parameter values N = 10 000, ω = 0, and K = 1 and the simulation time step∆ t = 0 . ξ k ( t ) was drawn from the independent standard Cauchy distribution[Eq. (2)]. The initial phase was uniformly distributed on [0 , π ]. The average (cid:104)| z |(cid:105) of theabsolute value of the order parameter [Eq. (3)] was obtained during 100 ≤ t ≤ (cid:104)| z |(cid:105) (circles) and the theoretical value of | Z ( ω, t ) | [solid line,Eq. (19)]. The numerical and theoretical values agreed relatively well. The continuoustransition from the synchronized state to the desynchronized state is observed.In the literature on phase oscillators, oscillator heterogeneity is often represented byheterogeneous natural frequencies. The dynamics of oscillator k are˙ φ k = ω k + (cid:61) [ H ( t ) exp( − i φ k )] + σ ( t ) ξ k ( t ) , (21)7here the natural frequency ω k is drawn from the probability density function g ( ω ). In thissystem, the order parameter of the whole system is defined by Y ( t ) = (cid:90) ∞−∞ g ( ω ) Z ( ω, t ) d ω, (22)which is the center of mass of all oscillators in the system. In other words, the orderparameter of the whole system, Y ( t ), is the average of the order parameters, Z ( ω, t ), of theoscillators with the natural frequency ω . It has been shown that the Ott–Antonsen ansatzcan reduce the dynamics of the order parameter of phase oscillators whose natural frequenciesobey a family of rational distribution functions into low-dimensional dynamical equations.In the most commonly studied version of this system, g ( ω ) is the Cauchy distribution, g ( ω ) = 1 πγ γ ( ω − ω ) + γ , (23)where γ is the scale parameter and ω is the location parameter. In this case, the Ott–Antonsen low-dimensional dynamics are shown to be given by inserting Y ( t ) = Z ( ω + i γ, t ) (24)into Eq. (8) [4, 12, 16]. Again, it is assumed that the density of the phases of oscillatorswith frequency ω follows the wrapped Cauchy distributions and that Eq. (17) holds for theoscillators with the natural frequency ω . This assumption approximately holds if the initialphases of a finite number of oscillators are uniformly distributed on [0 , π ]. This results in ∂Y ( t ) ∂t = [i ω − γ − σ ( t )] Y ( t ) + H ( t )2 − ¯ H ( t )2 Y ( t ) . (25)Specifically, in globally-coupled phase oscillator systems˙ φ k = ω k + KN N (cid:88) j =1 sin( φ j − φ k ) + σξ k ( t ) (26)with Cauchy-distributed natural frequencies, the mutual interaction is H ( t ) = KY ( t ).Therefore, the dynamics of the order parameter are given by ∂Y ( t ) ∂t = (cid:18) i ω − γ − σ + K (cid:19) Y ( t ) − K Y ( t ) Y ( t ) . (27)Its stable solution is Y ( t ) = (cid:113) − σ + γK exp[i ω ( t − t )] ( σ + γ ≤ K )0 (cid:0) K < σ + γ (cid:1) . (28)8 .0 0.1 0.2 0.3 0.4 0.5 0.60.00.10.20.30.40.50.6 0.00.20.40.60.81.0 | z | FIG. 2. Numerical results of (cid:104)| z |(cid:105) for the system of Eq. (26) with K = 1. The theoreticallyderived boundary between the synchronized state and the desynchronized state is indicated by thedashed line. Replacing Y ( t ), ω , and γ + σ with Z ( ω, t ), ω , and σ in Eq. (28) yields Eq. (18). Themacroscopic behavior of the system can be perfectly represented as a function of σ + γ ;that is to say, the noise amplitude and the scale parameter of the natural frequency areequivalent in the dynamics of the order parameter. This means that weak noise and narrowlydistributed natural frequencies allow for the synchronization whereas strong noise and widelydistributed natural frequencies abolish the synchronization.To test this analytical result, the simulation of Eq. (26) was performed by the Eulermethod φ k ( t + ∆ t ) = φ k ( t ) + ∆ t (cid:32) ω k + KN N (cid:88) j =1 sin[ φ j ( t ) − φ k ( t )] + σξ k ( t ) (cid:33) (29)with the same parameter values as in Fig. 1. In Fig. 2, the black and white colors correspondto (cid:104)| z |(cid:105) = 1 and 0, respectively. The dashed line represents the boundary between thesynchronized state and the desynchronized state (i.e., σ + γ = K/ σ + γ . Thissupports the equivalence of the noise amplitude and the scale parameter of the naturalfrequency in the present model. 9 II. DISCUSSION
This paper examined phase oscillator systems driven by Cauchy noise and obtained thelow-dimensional description of the dynamics of the order parameter using the Watanabe–Strogatz theory and Ott–Antonsen ansatz. The low-dimensional dynamics agreed relativelywell with the numerical results of a system of a finite number of oscillators. In the derivedlow-dimensional dynamics, the scale parameter of the natural frequency, γ , and the noiseamplitude, σ , were equivalent. The macroscopic dynamics of the system with heterogeneousnatural frequencies were indistinguishable from those of the system driven by Cauchy noise.The time evolution of the phases of sine-coupled oscillators is described by linear fractionaltransformations [16]. Linear fractional transformations map the Cauchy distributions to theCauchy distributions and the wrapped Cauchy distributions on the unit circle to the wrappedCauchy distributions on the unit circle [18, 19]. As the Cauchy distribution is a stabledistribution, it is continued to be obeyed by the oscillators driven by Cauchy noise. Althoughoscillators are not microscopically contained in a low-dimensional manifold (because theyare driven by independent noise), the trajectories of oscillators driven by sine coupling andCauchy noise can macroscopically be considered as being confined to a low-dimensionalmanifold. Within the framework of the circular cumulant approach [20], only the firstcircular cumulant is nonzero in the present model.The present results shed light on the dynamics of phase oscillators driven by Cauchynoise. For example, Martens et al . investigated the dynamics of phase oscillators whosenatural frequencies followed a mixture of two Cauchy distributions [5]. The results of thepresent study combined with those of Martens et al . predict the low-dimensional dynamics ofCauchy-noise-driven phase oscillators whose natural frequencies take on one of two values.The analysis of the conformist and contrarian oscillators [6] can also be applied to theanalysis of noise-driven oscillators. The present results allow for the reinterpretation ofprevious analyses on oscillator systems with Cauchy natural frequencies as the analyses onoscillators driven by Cauchy noise. Whether or not the Gaussian noise facilitates the sametype of reinterpretation in certain problem settings is not within the scope of the presentresearch.The present model assumes that the noise is temporally uncorrelated. The dynamics ofphase oscillators driven by correlated Gaussian noise were previously investigated [21]. The10ffect of the correlated Cauchy noise could be investigated by extending the present results.Because the 1 /f fluctuation is found in heartbeats [22] and in the activity of the centralnervous system [23, 24], the analyses of systems driven by temporally correlated noise arelikely to find applications in physiology research.The present study showed that white Cauchy noise and Cauchy-distributed natural fre-quencies have the same effect on the macroscopic behavior of a specific model. In thecontext of statistical physics, the critical behavior of a d -dimensional random field model isrelated to the critical behavior of a d − ACKNOWLEDGMENTS
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