1

Low-dimensional dynamics of phase oscillators driven by Cauchy

2

noise

3

Takuma Tanaka1
1

4

Graduate School of Data Science, Shiga University,5

1-1-1 Banba, Hikone, Shiga 522-8522, Japan∗

6(Dated: October 11, 2020)

7

Abstract

8

Phase oscillator systems with global sine-coupling are known to exhibit low-dimensional dynam-9

ics. In this paper, such characteristics are extended to phase oscillator systems driven by Cauchy

10

noise. The low-dimensional dynamics solution agreed well with the numerical simulations of noise-

11

driven phase oscillators in the present study. The low-dimensional dynamics of identical oscillators

12with Cauchy noise coincided with those of heterogeneous oscillators with Cauchy-distributed nat-13

ural frequencies. This allows for the study of noise-driven identical oscillator systems through14

heterogeneous oscillators without noise and vice versa.

15

PACS numbers: 05.45.Xt, 05.40.Ca∗

tanaka.takuma@gmail.com

1

16

I.

INTRODUCTION17

The synchronized rhythmic flashing of fireflies is a spectacular example of a collective

18

phenomenon [1]. Fireflies exhibit diﬀerent and fluctuating flashing frequencies and can be

19regarded as heterogeneous and noisy oscillators. Both heterogeneity and noise are essential

20

properties of systems that display collective phenomena. Coupled phase oscillators have been

21

used to examine how heterogeneity and noise aﬀect the synchronization of physical, chemical,

22

and biological systems [2, 3]. Phase oscillator systems with heterogeneous natural frequencies

23

have been studied since the invention of the phase oscillator model. Ott and Antonsen [4]24showed that the behavior of globally sine-coupled oscillators, the natural frequencies of25

which obey a family of rational distribution functions, can be described by low-dimensional

26

dynamics. Specifically, if the natural frequencies obey the Cauchy or Lorentzian distribution,27

the dynamics of an infinite number of oscillators are described by a Stuart–Landau equation,

28

i.e., a two-dimensional dynamical system. If the coupling strength takes on several values or29

the natural frequencies obey the mixture of Cauchy distributions, the dynamics are described30

by coupled Stuart–Landau oscillators. This is an exact result for a specific initial condition31

and not an approximation obtained by ignoring higher order terms. This type of low-32

dimensional description has accelerated the study of the heterogeneous oscillator systems

33[5, 6].

34

However, investigating noise-driven oscillator systems appears to be more challenging

35

than studying heterogeneous oscillator systems. Previous studies have approximated the36

dynamics with circular cumulants to obtain low-dimensional dynamics similar to those pro-37

posed by Ott and Antonsen [7, 8]. Although this approach has been implemented with38

some success, it is not always free from approximation error. Determining low-dimensional39

descriptions with fewer approximation errors will be useful in understanding the collective

40

phenomena in various fields, although it may not be as general as approximation with cir-41

cular cumulants. This may be possible using a noise that adheres to the assumption of the42

analysis by Ott and Antonsen.

43This paper reports that systems driven by Cauchy noise can be described by closed-

44

form low-dimensional dynamical equations. Non-Gaussian noise is known to be prevalent in45

biological systems [9]. For example, a circular auto-regressive model with wrapped Cauchy46

noise has been proposed to model animals’ direction of travel [10]. Thus, the behavior of2

47phase oscillators driven by Cauchy noise is worthy of further examination. Kallionatis and48

Roberts reported the behavior of the phase oscillator systems driven by Lévy noise [11, 12].

49

In addition, as harmonic oscillators display nontrivial phase distribution under Lévy noise50[13], the dynamics of phase oscillators driven by Cauchy noise is of interest.

51

This paper is organized as follows. First, the Watanabe–Strogatz theory is reviewed and52

used to derive the low-dimensional dynamics of the order parameter of identical sine-coupled53

oscillators driven by Cauchy noise. Second, the Ott–Antonsen ansatz is reviewed, and the54

dynamics of the order parameter of heterogeneous noise-driven oscillators are derived. It

55

is shown that the amplitude of Cauchy noise and the scale parameter of natural frequency

56

are equivalent in the low-dimensional description, and the implications of the model are

57

discussed.

58

II.

ANALYSIS AND RESULTS

59

This section first considers the system of identical oscillators and then that of hetero-60

geneous oscillators. Using the notation of Pikovsky and Rosenblum [14], we consider the61system of N noise-driven phase oscillators with identical natural frequency ω, in which the62

dynamics of oscillator k are given byϕ̇k = ω + ℑ[H(t) exp(−iϕk )] + σ(t)ξk (t)= ω + |H(t)| sin[arg H(t) − ϕk ] + σ(t)ξk (t),

(1)

63

where H(t) is the common forcing, σ(t) > 0 is the amplitude of the noise, and ξk (t) is the64

noise. This paper uses the Cauchy distribution instead of the Gaussian distribution, which65

has been used in earlier studies [7, 8]. It is assumed that ξk (t) is the Cauchy white noise,

66

i.e.,

∫ t+∆tξk (τ ) dτãk (t; ∆t) =

(2)

t
67

follows the Cauchy distribution with the scale parameter ∆t and the location parameter 0,p[ãk (t; ∆t)] =1∆t.
π [ãk (t; ∆t)]2 + ∆t2

(3)

68

Defining ak (t) = ãk (t; 1), we can see that ak (t) follows the standard Cauchy distribution. The

69

common forcing, H(t), can be an external forcing or mutual interaction between oscillators.
3

70

For example, the dynamics with σ(t) = σ and H(t) = Kz(t), where
1 ∑z(t) =
exp(iϕk )
N k=1
N

(4)

71

is the complex-valued order parameter and K is the coupling strength, lead to the following

72

dynamics
K∑
ϕ̇k = ω +sin(ϕj − ϕk ) + σξk (t).
N j=1
N

(5)

73

In this system, the oscillators are driven by the Cauchy noise and attracted to each other.

74

The system of Eq. (1) can be numerically implemented by the Euler method as
ϕk (t + ∆t) = ϕk (t) + ∆t {ω + |H(t)| sin[arg H(t) − ϕk (t)] + σ(t)ak (t)} ,

(6)

76

where ak (t) follows the standard Cauchy distribution. Let us note that the noise term is√multiplied by ∆t instead of ∆t because the Cauchy distribution is the stable distribution

77

of index 1.

75

78

79

Here, what has been clarified by the previous studies on the behavior of the system
without noise is reviewed. Inserting σ(t) = 0 into Eq. (1) yields
ϕ̇k = ω + ℑ[H(t) exp(−iϕk )].

(7)

80

Watanabe and Strogatz [15, 16] demonstrated that this system is described using three81

variables and N − 3 constants of motion. More specifically, the phases ϕk (t) (1 ≤ k ≤ N )82

of oscillators driven by the common forcing H(t) are given by a three-parameter function

83of the initial phases, ϕk (0). Using the Watanabe–Strogatz theory, the function that maps84

ϕk (0) to ϕk (t) is defined by the real and imaginary components of the order parameter and85

a parameter corresponding to the rotation of the initial phases [15, 17]. This allows us

86

to obtain a closed-form description of the dynamics of order parameter. In the following

87

analysis, it is assumed that the constants of motion are uniformly distributed in the limit of88

an infinite number of oscillators. This assumption has successfully described the behavior

89of the finite number of phase oscillators whose initial phases are drawn from the uniform

90

distribution on [0, 2π]. The order parameter z(t) becomes∫ 2πZ(ω, t) =p(ϕ, t|ω) exp(iϕ) dϕ
0

4

(8)

91

in the limit of N → ∞, where p(ϕ, t|ω) is the density of the phases of oscillators with natural

92

frequency ω at time t. For the system of Eq. 7, the dynamics of the order parameter have93

been shown to follow∂Z(ω, t)H(t) H̄(t)= iωZ(ω, t) +−Z(ω, t)2
∂t
2
2

(9)

94

[17, 18]. Because, if the initial phases are uniformly distributed, the rotation of the initial

95

phase does not aﬀect the final distribution of the phases, the phase distribution of oscillators96

at t is determined solely by the order parameter [17]. Thus, it has been shown that the97

density of the oscillators’ phase obeys the Poisson kernel [4]p(ϕ, t|ω) =1
1 − |Z(ω, t)|2.
2π 1 − 2|Z(ω, t)| cos[ϕ − arg Z(ω, t)] + |Z(ω, t)|2

(10)

98

Having reviewed the previous results, we are prepared to examine the dynamics of noise-99

driven oscillators. Because the phase distribution in the system without noise, which has a100

low-dimensional description, is determined by the order parameter, the system with noise101

can have a low-dimensional description if the phase distribution is determined by a few

102

parameters. To obtain a low-dimensional description, it is useful to note that the Poisson

103kernel is identical to the wrapped Cauchy distribution [19]p(ϕ) =
=
104

∞
∑

λ

n=−∞

π[λ2 + (ϕ − µ + 2πn)2 ]1sinh λ2π cosh λ − cos(ϕ − µ)(11)

if the following is set:

µ = arg Z(ω, t),)(|Z(ω, t)|−1 − |Z(ω, t)|−1
λ = sinh2= − log |Z(ω, t)|.

(12)

(13)

105

Before considering noise-driven sine-coupled oscillators, uncoupled oscillator systems driven

106

by Cauchy noise are examined, that is, σ(t) > 0 and H(t) = 0. In this system, assuming

107

that the oscillators are initially distributed according to the wrapped Cauchy distribution
5108[Eq. (11)], the Cauchy noise ensures that the oscillators obey the wrapped Cauchy distribu-

109

tion. This is illustrated by the Euler method
ϕk (t + ∆t) = ϕk (t) + ∆tσ(t)ak (t).

110

111

112

113

If ϕk (t0 ) obeys the Cauchy distribution with the scale parameter λ and the location parameter µ, ϕk (t0 + n∆t), where n > 0, obeys the Cauchy distribution with the scale parameter∑λ(t0 + n∆t) = λ + ∆t n−1
j=0 σ(t0 + j∆t) and the location parameter µ owing to the reproductive property. Therefore, the Cauchy noise ξk (t) increases the scale parameter λ as
λ̇ = σ(t),

114

115

(14)

(15)

while keeping the location parameter constant asµ̇ = 0.

(16)∂Z(ω, t)= −σ(t)Z(ω, t).
∂t

(17)

Inserting Eqs. (12) and (13) gives

116This equation means that the Cauchy noise causes the exponential decay of the order pa-

117

rameter.

118

Because Eqs. (9) and (17) are exact closed-form descriptions of Z(ω, t) in the limit of119

N → ∞, we can combine these two equations to obtain the dynamics of Cauchy noise-driven

120

coupled oscillators. This is justified by the fact that, the oscillators obeying a wrapped

121

Cauchy distribution remain obeying a wrapped Cauchy distribution if driven by either the122

sine-coupling or Cauchy noise. Combining Eq. (9) with Eq. (17) yields the dynamics of the123

system with H(t) ̸= 0 and σ(t) > 0,
H(t) H̄(t)∂Z(ω, t)= [iω − σ(t)]Z(ω, t) +−Z(ω, t)2 .
∂t
2
2

(18)

124

This is equivalent to the system of oscillators driven alternately by Eq. (9) and Eq. (17). A

125much more mathematically rigorous and general derivation is presented in a recent paper

126

by Tönjes and Pikovsky [20]. For globally sine-coupled phase oscillator systems [Eq. (5)],127the common forcing of the oscillators is proportional to the order parameter, that is, H(t) =128

KZ(ω, t). Hence, it is suggested that inserting it into Eq. (18) yields the dynamics of the129

order parameter of the system of Eq. (5) with()
K
K
∂Z(ω, t)= iω − σ +Z(ω, t) − Z̄(ω, t)Z(ω, t)2 .
∂t
2
2
6

(19)

1.0
0.8

|z|

0.6
0.4
0.2
0.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

FIG. 1. Numerical and theoretical results of ⟨|z|⟩ for the system of Eq. (5) with K = 1. The
numerical and theoretical results are indicated by the circles and the solid line, respectively.

130

It has a closed-form stable solution√ 1 − 2σ exp[iω(t − t0 )] (σ ≤ K )K
2
Z(ω, t) =
(K
),

0
<σ
2

(20)

131

where t0 is a constant. This means that a weak noise allows for the synchronization whereas

132

noise stronger than a threshold value abolishes the synchronization as is general with the133noise driven systems.

134

135

To numerically confirm the above theoretical prediction, the simulation of Eq. (5) was
performed by the Euler method
(

)
N
K∑
ϕk (t + ∆t) = ϕk (t) + ∆t ω +sin[ϕj (t) − ϕk (t)] + σak (t)
N j=1

(21)

136

with the parameter values N = 10 000, ω = 0, and K = 1 and the simulation time step137

∆t = 0.005. The noise ak (t) was drawn from the independent standard Cauchy distribution.

138

The initial phase was uniformly distributed on [0, 2π]. The average ⟨|z|⟩ of the absolute

139

value of the order parameter [Eq. (4)] was obtained during 100 ≤ t ≤ 200. Figure 1 shows the140

numerical results of ⟨|z|⟩ (circles) and the theoretical value of |Z(ω, t)| [solid line, Eq. (20)].

141

The numerical values for N = 10 000 and the theoretical values for an infinite number of142

oscillators agreed quite well. The continuous transition from the synchronized state to the143

desynchronized state is observed.

144

In the literature on phase oscillators, oscillator heterogeneity is often represented by
7

145

heterogeneous natural frequencies. The dynamics of oscillator k areϕ̇k = ωk + ℑ[H(t) exp(−iϕk )] + σ(t)ξk (t),

146

147

(22)

where the natural frequency ωk is drawn from the probability density function g(ω). In thissystem, the order parameter of the whole system is defined by∫ ∞Y (t) =
g(ω)Z(ω, t) dω,

(23)

−∞
148

which is the center of mass of all oscillators in the system. In other words, the order149

parameter of the whole system, Y (t), is the average of the order parameters, Z(ω, t), of the150

oscillators with the natural frequency ω. It has been shown that the Ott–Antonsen ansatz

151

can reduce the dynamics of the order parameter of phase oscillators whose natural frequencies

152

obey a family of rational distribution functions into low-dimensional dynamical equations.

153

In the most commonly studied version of this system, g(ω) is the Cauchy distribution,g(ω) =1γ2,
πγ (ω − ω0 )2 + γ 2

(24)

154

where γ is the scale parameter and ω0 is the location parameter. In this case, the Ott–

155

Antonsen low-dimensional dynamics are shown to be given by inserting
Y (t) = Z(ω0 + iγ, t)

(25)

156

into Eq. (9) [4, 14, 18]. Again, it is assumed that the density of the phases of oscillators

157

with frequency ω follows the wrapped Cauchy distributions and that Eq. (18) holds for the158

oscillators with the natural frequency ω. This assumption could not necessarily be needed

159

because the order parameter of the system of oscillators with Cauchy-distributed natural160

frequencies is described by the low-dimensional dynamics in the limit of t → ∞ [21]. This

161

results in

162

H(t) H̄(t)∂Y (t)= [iω0 − γ − σ(t)]Y (t) +−Y (t)2 .
∂t2
2Specifically, in globally-coupled phase oscillator systems
K∑sin(ϕj − ϕk ) + σξk (t)
N j=1

(26)

N

ϕ̇k = ωk +
163

164

(27)

with Cauchy-distributed natural frequencies, the mutual interaction is H(t) = KY (t).
Therefore, the dynamics of the order parameter are given by()
KK∂Y (t)= iω0 − γ − σ +Y (t) − Ȳ (t)Y (t)2 .
∂t
2
2
8

(28)

165

Its stable solution is√ 1 − 2 σ+γ exp[iω0 (t − t0 )] (σ + γ ≤ K )K
2
Y (t) =
(
).

K
0
<
σ
+
γ
2

(29)

166

Replacing Y (t), ω0 , and γ + σ with Z(ω, t), ω, and σ in Eq. (29) yields Eq. (19). The

167

macroscopic behavior of the system can be perfectly represented as a function of σ + γ;

168that is to say, the noise amplitude and the scale parameter of the natural frequency are

169

equivalent in the dynamics of the order parameter. This means that weak noise and narrowly

170

distributed natural frequencies allow for the synchronization whereas strong noise and widely

171

distributed natural frequencies abolish the synchronization.

172

173

To test this analytical result, the simulation of Eq. (27) was performed by the Euler
method(K∑ϕk (t + ∆t) = ϕk (t) + ∆t ωk +sin[ϕj (t) − ϕk (t)] + σak (t)
N j=1
N

)
(30)

174

with the same parameter values as in Fig. 1. In Fig. 2, the black and white colors correspond175

to ⟨|z|⟩ = 1 and 0, respectively. The dashed line represents the boundary between the176

synchronized state and the desynchronized state (i.e., σ + γ = K/2). The figure clearly177

indicates that the steady-state value of the order parameter is a function of σ + γ. This178

supports the equivalence of the noise amplitude and the scale parameter of the natural

179

frequency in the present model.

180

III.

DISCUSSION

181

This paper examined phase oscillator systems driven by Cauchy noise and obtained the182

low-dimensional description of the dynamics of the order parameter using the Watanabe–

183

Strogatz theory and Ott–Antonsen ansatz. The low-dimensional dynamics agreed relatively

184

well with the numerical results of a system of a finite number of oscillators. In the derived

185

low-dimensional dynamics, the scale parameter of the natural frequency, γ, and the noise

186

amplitude, σ, were equivalent. The macroscopic dynamics of the system with heterogeneous187

natural frequencies were indistinguishable from those of the system driven by Cauchy noise.

188

The time evolution of the phases of sine-coupled oscillators is described by linear fractional

189

transformations [18]. Linear fractional transformations map the Cauchy distributions to

190

the Cauchy distributions and the wrapped Cauchy distributions on the unit circle to the
9

0.6

1.0

0.5

0.8

0.4

0.6

|z|

0.3

0.4

0.2

0.2

0.1
0.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.0

FIG. 2. Numerical results of ⟨|z|⟩ for the system of Eq. (27) with K = 1. The theoretically derived
boundary between the synchronized state and the desynchronized state is indicated by the dashed
line.

191

wrapped Cauchy distributions on the unit circle [22, 23]. The combination of the linear

192

fractional transformations and the Cauchy distribution has also been examined in the context

193

of coupled map [24]. As the Cauchy distribution is a stable distribution, it is continued to be194obeyed by the oscillators driven by Cauchy noise. Although the trajectory of the vector of

195

oscillators’ phases is not microscopically contained in a low-dimensional manifold (because

196

it is driven by independent noise), it can macroscopically be considered as being confined to197

a low-dimensional manifold. Within the framework of the circular cumulant approach [25],198

only the first circular cumulant is nonzero in the present model.

199

The present results shed light on the dynamics of phase oscillators driven by Cauchy

200

noise. For example, Martens et al. investigated the dynamics of phase oscillators whose201

natural frequencies followed a mixture of two Cauchy distributions [5]. The results of the202

present study combined with those of Martens et al. predict the low-dimensional dynamics of203

Cauchy-noise-driven phase oscillators whose natural frequencies take on one of two values.

204

The analysis of the conformist and contrarian oscillators [6] can also be applied to the205

analysis of noise-driven oscillators. The present results allow for the reinterpretation of

206

previous analyses on oscillator systems with Cauchy natural frequencies as the analyses on

207

oscillators driven by Cauchy noise. Whether or not the Gaussian noise facilitates the same

208

type of reinterpretation in certain problem settings is not within the scope of the present

209

research.
10

210

The present model assumes that the noise is temporally uncorrelated. The dynamics of211phase oscillators driven by correlated Gaussian noise were previously investigated [26]. The

212

eﬀect of the correlated Cauchy noise could be investigated by extending the present results.

213

Because the 1/f fluctuation is found in heartbeats [27] and in the activity of the central214

nervous system [28, 29], the analyses of systems driven by temporally correlated noise are215

likely to find applications in physiology research.

216

The present study showed that white Cauchy noise and Cauchy-distributed natural fre-217

quencies have the same eﬀect on the macroscopic behavior of a specific model. In the218

context of statistical physics, the critical behavior of a d-dimensional random field model is219

related to the critical behavior of a d − 2-dimensional model without disorder [30–32]. The

220

model presented in this study oﬀers a further example of the equivalence of annealed and221

quenched disorders. The interplay between annealed and quenched disorders or noise and222

heterogeneity could be explored further by extending the present model.

223

ACKNOWLEDGMENTS

224

This work was supported by JSPS KAKENHI Grant Number JP19K12184.

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