Local sensitivity analysis for the Kuramoto model with ...jin/PS/Kuramoto-2.pdf · 1 local sensitivity analysis for the kuramoto model 2 with random inputs in a large coupling regime

LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL1

WITH RANDOM INPUTS IN A LARGE COUPLING REGIME∗2

SEUNG-YEAL HA† , SHI JIN‡ , AND JINWOOK JUNG§3

Abstract. Synchronization phenomenon is ubiquitous in strongly correlated oscillatory systems,4and the Kuramoto model serves as a prototype synchronization model for phase-coupled oscillators.5In this paper, we provide local sensitivity analysis for the Kuramoto model with random inputs in6initial data, distributed natural frequencies and coupling strengths, which exhibits the interplay be-7tween random effects and synchronization nonlinearity. Our local sensitivity analysis provides some8understanding on the robustness of emergent dynamics of the random Kuramoto model in a large9coupling regime, including “propagation and vanishment of uncertainties” and “continuous depen-10dence” of phase and frequency variations in random parameter space with respect to the variations11on the initial data.12

Key words. Kuramoto model, synchronization, local sensitivity analysis, random communica-13tion, uncertainty quantification14

AMS subject classifications. 15B48, 92D2515

1. Introduction. Complex oscillatory systems often exhibit collective coherent16

behaviors, e.g., flashing of fireflies, chorusing of crickets, synchronous firing of cardiac17

pacemaker and metabolic synchrony in yeast cell suspension etc [1, 7, 34, 43]. The18

rigorous mathematical treatment of such problems began from the pioneering works19

[30, 43] by Kuramoto and Winfree about half century ago. They introduced simple20

phase models for weakly coupled limit-cycle oscillators, and showed how collective21

coherent behavior can emerge from the interplay between intrinsic randomness in22

natural frequency and nonlinear attractive phase couplings. This coherent motion is23

often called ”synchronziation” which means the adjustment of rhythms in an ensemble24

of weakly coupled oscillators. Recently, the synchronization of oscillators on networks25

became an emerging research area in different disciplines such as biology, control the-26

ory, statistical physics and sociology. After Kuramoto and Winfree’s seminal works,27

several phase models have been used in the phenomenological study of synchroniza-28

tion. Among them, our main interest in this paper lies on the Kuramoto model. We29

first briefly introduce the Kuramoto model (see Section 2 for its basic mathematical30

structures).31

Let θi = θi(t) be the phase of the i-th limit-cycle oscillator, and we assume32

that the Kuramoto oscillators are located on a symmetric network whose interac-33

tion(connection) topology is denoted by the coupling matrix K = (κij). In this34

setting, the evolution of phases is governed by the first-order system of ordinary dif-35

ferential equations [31, 30]:36

(1.1) θi = νi +1

N

N∑j=1

κij sin(θj − θi), t > 0,37

∗Submitted to the editors 2018.03.01.†Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National

University, Seoul 08826 and Korea Institute for Advanced Study, Hoegiro 87, Seoul 02455, Korea(Republic of) ([email protected]).‡School of Mathematical Sciences, MOE-LSC, and Institute of Natural Sciences, Shanghai Jiao

Tong University, Shanghai 200240, China ([email protected]).§Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea (Republic

of) ([email protected]).

1

This manuscript is for review purposes only.

mailto:[email protected]



2 S.-Y. HA, S. JIN AND J. JUNG

where νi is the random natural frequency and κij is the symmetric coupling strength38

between j and i-th oscillators. In previous literature in applied mathematics, control39

theory, say [1, 16, 24] and references therein, system (1.1) has been studied after40

the randomness in the natural frequencies is quenched, i.e., νi is treated as a time-41

independent parameter (see [2, 5, 4, 6, 8, 12, 14, 16, 15, 17, 22, 23, 24, 25, 37, 42, 41,42

40]) for deterministic data and coupling strengths. However, as one can easily imagine,43

initial data, natural frequencies and mutual coupling strengths can be uncertain due44

to incomplete measurement of data and ignorance of exact interaction mechanism45

between oscillators.46

In this paper, in order to address this uncertainty, we employ a UQ (uncertainty47

quantification) formalism in [26, 27, 32] (and references therein) in the context of48

synchronization. Recently, UQ analysis has been applied in the collective models in49

the context of flocking in [3, 9, 20, 21]. In previous studies, most analytical works50

for (1.1) were restricted to situation where the randomness in natural frequency νi is51

quenched and the coupling strengths are the same constant. Throughout the paper,52

we consider a more realistic case where the natural frequencies and mutual coupling53

strengths contain a kind of random component. For this, we introduce random pa-54

rameters z whose probability density function is given by g = g(z). In this setting,55

the random phase process θi(t, z) satisfies a random dynamical system:56

(1.2) ∂tθi(t, z) = νi(z) +1

N

N∑j=1

κij(z) sin(θj(t, z)− θi(t, z)), 1 ≤ i ≤ N.57

Note that if randomness in natural frequency and coupling strengths are quenched,58

then system (1.2) reduces to the deterministic Kuramoto model (1.1) on a symmetric59

network. Since the R.H.S. of (1.2) is 2π-periodic, the system (1.2) can be regarded as60

a dynamical system on N -tori TN . However, if necessary, by lifting the system (1.2)61

in its covering space RN , we will regard (1.2) as a dynamical system on RN . For the62

proposed random dynamical system (1.2), we are mainly interested in the following63

questions:64

• (Q1): How do the randomness in natural frequencies and cou-65

pling strength affect synchronization process?66

• (Q2): Are phase-locked states for (1.1) robust in the presence of67

randomness?68

While the mathematical and computational study for self-organization has received69

tremendous interests in the last decade (see for examples review articles [33, 38]), as70

far as the authors know, the study of uncertainty quantification for such problems71

has not been fully addressed in literature until several recent works [3, 9, 20, 21] for72

the Cucker-Smale model of flocking. The type of analysis conducted here is similar73

to the ones done in [20], in which the flocking conditions, as well as local sensitivity74

analysis were studied from the viewpoint of random initial data and communication75

weights between particles. Such a study not only helps to understand the impact of76

uncertainty in the dynamic behavior of system under investigation, but it also helps77

to understand the behavior of numerical approximations for such random systems,78

since indeed the sensitivity results imply the regularity of the solution in the Sobolev79

norms which is important to understand the convergence of stochastic algorithms [44].80

For notational simplicity, we set81

z ∈ Ω ⊂ R, ωi := ∂tθi, Θ := (θ1, · · · , θN ), V := (ω1, · · · , ωN ), V := (ν1, · · · , νN ).82


LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 3

To see the random effect in (1.2), we expand the phase and frequency processes83

θi(t, z + dz) and via Taylor’s expnasion:84

θi(t, z + dz) = θi(t, z) + ∂zθi(t, z)dz +1

2∂2zθi(t, z)(dz)

2 + · · · ,

ωi(t, z + dz) = ωi(t, z) + ∂zωi(t, z)dz +1

2∂2zωi(t, z)(dz)

2 + · · ·(1.3)85

86

Thus, the local sensitivity estimates [35, 36] deal with the dynamic behaviors of the87

sensitivity vectors ∂rzΘ and ∂rzV consisting of coefficients in the R.H.S. of (1.3).88

89

The main results of this paper are two-fold: First, we provide a sufficient frame-90

work (F) leading to the uniform bound estimate for the diameter and `1-stability91

property of ∂rzΘ. Under the framework (F) formulated in terms of initial data, nat-92

ural frequencies and coupling strengths which are computable from given data and93

parameters, our results provide the following local sensitivity estimates for ∂rzΘ:94

95

• (Uniform bound for the diameter of ∂rzΘ): Our first estimate provides the96

estimate like97

D(∂lzΘ(t, z)) ≤ D(∂lzΘ0(z))e−κm(z) cosD(Θ0(z))t

+ Cl(z)(1− e−κm(z) cosD(Θ0(z))t),(1.4)98

where the random function Cl(z) depends only on given random data and99

parameters D(∂rzV(z)), ∂rzκij(z) and D(∂rzΘ0(z)) for r = 0, 1, · · · , l (see100

Theorem 3.2 and 4.5). In particular, for the ensemble of identical Kuramoto101

oscillators, we will show a more refined estimate than (1.4):102

D(∂zΘ(t, z)) ≤ Dl(z)e−κm(z) cosD(Θ0(z))t

2 , for every t ≥ 0,103

where the random function Dl(z) depends only on given random data and104

parameters ∂rzκij(z) and D(∂rzΘ0(z)) for r = 0, 1, · · · , l (see Corollary 3.6105

for details).106

• (`1-stability): For two solutions Θ and Θ to (1.2) with initial data Θ0 and107

Θ0 in random space, respectively: for every l ∈ N, there exists a nonnegative108

random variable El = El(z) independent of t and a non-negative functional109

Λl := Λl(t, z) such that110

∂

∂t‖∂lz(Θ− Θ)(t, z)‖1 + Λl(t, z) ≤ El(z)

l−1∑p=0

‖∂pz (Θ− Θ)(t, z)‖1,111

for every t ≥ 0. In addition, if we further assume that θc(0, z) = θc(0, z)(for112

the definition of θc, see (2.1)), we can find the exponential decay of `1-113

difference between two solutions as follows: for every l ∈ N, there exists114

a nonnegative random variable El := El(z) such that115

‖∂lz(Θ− Θ)(t, z)‖1 ≤ El(z)e−κm(z)γ(z)(cosD(Θ0(z)))tl∑

p=0

‖∂pz (Θ0 − Θ0)(z)‖1.116



In general, the aforementioned local sensitivity estimates for ∂rzΘ do not hold in a117

low coupling regime (see the discussion right after the framework (F) in Section 3).118

Second, we provide a synchronizing property of the frequency variations ∂rzV (see119

Theorem 5.2) under the same framework (F):120

D(∂lzV (t, z)) ≤ Fl(z)e−κm(z) cosD(Θ0(z))t

2 , for every t ≥ 0,121

where the random function Fl(z) depends only on given random data and parameters122

∂rzκij(z) and D(∂rzΘ0(z)) for r = 0, 1, · · · , l.123

124

The rest of this paper is organized as follows. In Section 2, we provide con-125

servation laws, relative equilibria, gradient flow formulation and pathwise emergent126

dynamics for the random Kuramoto model (1.2). In Section 3, a uniform bound for127

the diameter for the phase variation is present, and a uniform `1-stability of ∂rzΘ is128

given in Section 4. In Section 5, we present a synchronizing property of ∂rzV . In129

Section 6, we provide some numerical simulation results to confirm our theoretical130

results. Finally, Section 7 is devoted to a brief summary of our main results and131

future directions.132

133

Notation: Throughout the paper, we use the following simplified notation: for Z :=134

(z1, · · · , zN ) and coupling matrix K = (κij), we set135

D(Z) := max1≤i,j≤N

|zi − zj |, ‖Z‖p :=( N∑i=1

|zi|p) 1p

, p ∈ [1,∞),

‖Z‖∞ := max1≤i≤N

|zi|, κm(z) := mini,j

κij(z), ‖∂rzκ(z)‖∞ := maxi,j|∂rzκi,j(z)|.136

Let π : Ω → R+ ∪ 0 be a nonnegative p.d.f. function, and let y = y(z) be a137

scalar-valued random function defined on Ω. Then, we define the expected value as138

E[ϕ] :=

∫Ω

ϕ(z)π(z)dz.139

2. Preliminaries. In this section, we study conservation laws and pathwise140

asymptotic dynamics for the random Kuramoto model (1.2). These estimates are141

crucial in the local sensitivity analysis in the following three sections.142

2.1. Conservation laws. First, we consider conservation laws associated with143

random dynamical system (1.2). In general, for a given dynamical system, it is im-144

portant to look for conserved quantities which govern overall dynamics of a system.145

For example, if a Hamiltonian system has enough conserved quantities, then it can146

be integrable. So far, it is known that the Kuramoto model (1.1) admits two conser-147

vation laws, namely the number of oscillators and total sum of phases. Thus, once148

the complete synchronization happens, where all oscillators rotate with the common149

frequency, then that constant is given by the average natural frequencies. For given150

Θ and V, consider a time-dependent random function C(Θ,V, t):151

C(Θ,V, t) :=

N∑i=1

θi − tN∑i=1

νi.152

Next, we show that the quantity C(Θ,V, t) is conserved along (1.2).153



Lemma 2.1. Let Θ = Θ(t, z) be a random phase vector whose dynamics is gov-154

erned by the random Kuramoto model (1.2). Then, the quantity C(Θ,V, t) is constant155

along the path of (1.2): for z ∈ Ω, t > 0,156

∂tC(Θ(t, z),V(z), t) = 0.157

Proof. We use the symmetry of κij = κji and (1.2) to obtain158

∂tC(Θ(t, z),V(z), t) = ∂t

( N∑i=1

θi(t, z)− tN∑i=1

νi(z))

=

N∑i=1

∂tθi(t, z)−N∑i=1

νi(z) = 0,159

which yields the desired estimate.160

Remark 2.2. Note that Lemma 2.1 implies161

N∑i=1

θi(t, z) = t

N∑i=1

νi +

N∑i=1

θ0i (z), t ≥ 0, z ∈ Ω.162

Hence, unless∑Ni=1 νi is zero, the total phase

∑Ni=1 θi itself is not a conserved quantity.163

Due to the translation invariant property of (1.2), the dynamics for averages and164

fluctuations around them are completely decoupled in the sense that if one sets165

θc :=1

N

N∑i=1

θi, νc :=1

N

N∑i=1

νi,

θi := θi − θc, νi := νi − νc, i = 1, · · · , N,

(2.1)166

167

then θc and θi satisfy168

∂tθc(t, z) = νc(z),

∂tθi(t, z) = νi(z) +1

N

N∑j=1

κij(z) sin(θj(t, z)− θi(t, z)).(2.2)169

170

Note that the fluctuations satisfy the same equation (1.2). Thus, without loss of171

generality, we will assume zero sum conditions:172

N∑i=1

νi = 0,

N∑i=1

θi = 0173

instead of the system (2.2)2.174

2.2. Relative equilibria. Note that the equilibrium solution Θ = (θ1, · · · , θN )175

to (1.2) is a solution to the following random system: for each z ∈ Ω,176

(2.3) νi(z) +1

N

N∑j=1

κij(z) sin(θj(t, z)− θi(t, z)) = 0, 1 ≤ i ≤ N.177

Due to Remark 2.2, if∑Ni=1 νi 6= 0, then system (2.3) does not have a solution. This178

forces us to consider relaxed equilibria.179



Definition 2.3. [1, 16, 23] Let Θ(t,z)=(θ1(t,z), · · · , θN (t,z)) be a time-dependent180

random phase vector.181

1. Θ is in a random phase-locked state if all relative phase differences are con-182

stant over time along the sample path: for z ∈ Ω,183

θi(t, z)− θj(t, z) = θi(0, z)− θj(0, z), t ≥ 0, 1 ≤ i, j ≤ N.184

2. Θ exhibits asymptotic phase-locking (complete synchronization) if the relative185

frequencies tend to zero asymptotically: for z ∈ Ω,186

limt→∞

|∂tθi(t, z)− ∂tθj(t, z)| = 0, 1 ≤ i, j ≤ N.187

Note that the random Kuramoto model (1.2) can also recast as a gradient flow along188

the sample path. We define a random potential in [23, 25, 39]: for a given random189

phase vector Θ(t, z) = (θ1(t, z), · · · , θN (t, z)),190

(2.4) V (Θ(t, z)) := −N∑k=1

νk(z)θk(t, z) +1

2

N∑k,l=1

κik(z)(1− cos(θk(t, z)− θl(t, z))

).191

Then, it is easy to see that the random Kuramoto model (1.2) can be rewritten as a192

gradient flow: for each z ∈ Ω,193

(2.5) ∂tΘ = −∇ΘV (Θ), t > 0.194

For the deterministic case, the gradient flow formulation (2.4) and (2.5) is useful to195

derive the complete synchronization estimates for generic initial phase configuration196

in [23] without decay rate. Since the following analysis requires a detailed exponential197

decay, we will not employ the gradient flow and instead, we will use the framework198

in [11] where the explicit relaxation rate toward the phase-locked states and uniform199

`1-stability have been studied.200

2.3. Basic estimates. In this subsection, we provide some basic estimates in-201

cluding pathwise emergent dynamics of (1.2) which are useful in the following sections.202

First, we state a special case of Gronwall lemma in [18].203

Lemma 2.4. Let y : R+ ∪ 0 → R+ ∪ 0 be a differentiable function satisfying204

(2.6) y′ ≤ −αy + Ce−βt, t > 0, y(0) = y0,205

where α > β and C are non-negative constants. Then y satisfies206

y(t) ≤ y0e−αt +

C

α− β(e−βt − e−αt)207

Proof. We multiply (2.6) by eαt and integrate the resulting relation over (0, t] to208

derive209

y(t)eαt ≤ y0 +C

α− β(e(α−β)t − 1).210

This yields the desired result.211

Next, we provide pathwise estimates for (1.2).212

Proposition 2.5. Let Θ = Θ(t, z) be a phase vector whose dynamics is governed213

by the random Kuramoto model (1.2). Then, for a given z ∈ Ω, the following asser-214

tions hold.215



1. (Identical oscillators) Suppose that the coupling strength, natural frequencies216

and initial phases satisfy217

D(V(z)) = 0, κm(z) > 0, 0 < D(Θ0(z)) < π.218

Then, there exists κm > 0 such that219

D(Θ0(z))e−‖κ(z)‖∞t ≤ D(Θ(t, z)) ≤ D(Θ0(z))e−κm(z)γt, t ≥ 0,220

where γ(z) := sinD(Θ0(z))D(Θ0(z)) ∈ (0, 1).221

2. (Nonidentical oscillators) Suppose that the coupling strength, natural frequen-222

cies and initial phases satisfy223

κm(z) >D(V(z))

sinD(Θ0(z))> 0, 0 < D(Θ0(z)) <

π

2.224

Then, we have225

D(Θ(t, z)) ≤ D(Θ0(z)), D(V (t, z)) ≤ D(V 0(z))e−κm(z) cosD(Θ0(z))t.226

Proof. The proof is almost the same as in [19] with a slight modification. However,227

for readers’ convenience, we briefly sketch a proof. Note that the Kuramoto model228

employed in [19] is the special case of (1.2) when κij = κ for all i and j. Once we229

choose extremal indices M = M(t, z) and m = m(t, z) such that230

θM (t, z) := maxiθi(t, z), θm(t, z) := min

iθi(t, z),231

we note that for every z ∈ Ω, θM (t, z) and θm(t, z) are piecewise differentiable and232

Lipschitz with respect to t. We also have233

D(Θ(t, z)) = θM (t, z)− θm(t, z).234

Then, initially,235

∂tθM (t)

∣∣∣∣t=0+

≤ νM +1

N

N∑j=1

κm sin(θ0j − θ0

M ) ≤ νM +1

N

N∑j=1

κmsinD(Θ0)

D(Θ0)(θ0j − θ0

M )

= νM − κmsinD(Θ0)

D(Θ0)θ0M .

236

237

Similarly,238

∂tθm(t)

∣∣∣∣t=0+

≥ νm − κmsinD(Θ0)

D(Θ0)θ0m.239

240

Hence, one has241

∂tD(Θ(t, z))

∣∣∣∣t=0+

≤ D(V)− κmsinD(Θ0)

D(Θ0)D(Θ(t, z))

∣∣∣∣t=0

.242

We use contradiction argument and Gronwall’s lemma to show that the above in-243

equality holds for all t > 0. This gives an upper bound for the diameter of phase244



process.245

246

For frequency process, we set ωi := ∂tθi, and choose extremal indices M ′ = M ′(t, z)247

and m′ = m′(t, z) such that248

ωM ′(t, z) := maxiωi(t, z), ωm′(t, z) := min

iωi(t, z).249

We again note that for every z ∈ Ω, ωM ′(t, z) and ωm′(t, z) are piecewise differentiable250

and Lipschitz with respect to t. Moreover, we get251

D(V (t, z)) = ωM ′(t, z)− ωm′(t, z).252

Similar to phase process, one can obtain253

∂tD(V (t, z))

∣∣∣∣t=0+

≤ −κm cosD(Θ0)D(V (t, z))

∣∣∣∣t=0

.254

Again, we use contradiction argument and Gronwall’s lemma to get the desired result.255

256

As a direct application of Proposition 2.5, we obtain statistical estimate for ex-257

pectation of random phase and frequency configurations.258

Corollary 2.6. Suppose that initial data, natural frequencies and coupling stren-259

gth satisfy260

0 < supz∈Ω

D(Θ0(z)) ≤ π

2− ε, for some ε > 0,

N∑i=1

θ0i = 0,

N∑i=1

νi = 0,261

supz∈Ω

D(V(z)) <∞, κm(z) >D(V(z))

sinD(Θ0(z))and inf

z∈Ωκm(z) ≥ η > 0,262

263

and let Θ = Θ(t, z) be a solution to system (1.2). Then, we have264

E[D(Θ(t))] ≤ E[D(Θ0)] and E[D(V (t))] ≤ E[D(V 0)]e−η sin(ε)t.265

Proof. The estimates directly follow from Proposition 2.5.266

Before we close this section, we quote the formula for the chain rules for higher267

derivatives of a composition function from [28]. Its proof can be made using the268

mathematical induction. We first introduce an index set: for given positive integer n,269

Λ(n) := (k1, · · · , kn) ∈ (Z+ ∪ 0)n : k1 + 2k2 + · · ·+ nkn = n.270

Note that (0, · · · , 0, 1) is an element of Λ(n). Then, n-th derivative of f(g(x)) is given271

by the following formula:272

dn

dxnf(g(x))

=∑

(k1,··· ,kn)∈Λ(n)

n!

k1! · · · kn!f (k)(g(x))

(g′(x)

1!

)k1(g′′(x)

2!

)k2

· · ·(g(n)(x)

n!

)kn,

(2.7)

273

274

where k := k1 + · · ·+ kn.275

276



3. A uniform diameter bound for phase process. In this section, we present277

a uniform bound for sensitivity vectors ∂lzΘ with l ≥ 1.278

279

Note that the diameter D(∂lzΘ)) is given by the relation:280

D(∂lzΘ)) = maxi∂lzθi −min

i∂lzθi.281

For l = 0, we have already studied the decay and uniform bound estimates of D(Θ) in282

Proposition 2.5. For l ≥ 1, we will use mathematical induction together with modified283

Gronwall’s lemma to derive bound and decay estimates of D(∂lzΘ).284

285

Consider the equation for ∂lzθi by differentiating (1.2) with respect to z:286

(3.1)

∂t(∂lzθi(t, z)

)= ∂lz(νi(z)) +

1

N

∑1≤j≤N0≤r≤l

(l

r

)∂l−rz

(κij(z)

)∂rz

(sin(θj(t, z)− θi(t, z))

).287

Note that for each z ∈ Ω and l ∈ N ∪ 0, we have a real-analytic solution ∂lzθi(·, z)288

to (3.1).289

3.0.1. Nonidentical oscillators. Now, we state a sufficient framework (F) for290

the local sensitivity analysis for phase process:291

292

• (F1): Initial phase processes are confined in a quarter arc and have zero mean293

and bounded z-variations:294

0 < D(Θ0(z)) <π

2, sup

0≤r≤lD(∂rzΘ0(z)) <∞,

∑i

θ0i (z) = 0.295

• (F2): Natural frequencies satisfy uniform bound and have zero mean:296

sup0≤r≤l

D(∂rzV(z)) <∞,∑i

νi(z) = 0.297

• (F3): Mutual coupling strengths are sufficiently large such that298

κm(z) >D(V(z))

sinD(Θ0(z)), max

0≤r≤l‖∂rzκ(z)‖∞ ≤ κ∞ <∞.299

Note that the large coupling condition (F3) is necessary for the uniform bound-300

edness of diameters. For example, in a low coupling regime which is close to zero, the301

uniform bound for diameter does not hold. Consider the random Kuramoto model302

(1.2) with κij = 0:303

∂tθi(t, z) = νi(z), 1 ≤ i ≤ N.304

Thus, θi is completely integrable:305

θi(t, z) = θ0i (z) + tνi(z), t ≥ 0.306

For a pair of oscillators with νi 6= νj ,307

|θi(t, z)− θj(t, z)| ≥ t|νi(z)− νj(z)| − |θ0i (z)− θ0

j (z)| → ∞, as t→∞,308



which means the unboundedness of D(Θ). The same argument can be applied for309

∂rzΘ to derive unboundedness for κij = 0. By the structural stability of (1.2), this310

unboundedness of diameter also works for low coupling regime κij 1.311

312

We now return to the uniform bound estimate for D(∂lzΘ). As the first step of313

the induction, we study the estimate for D(∂1zΘ) in the following lemma.314

Lemma 3.1. (Uniform bound for D(∂zΘ)) Suppose that the framework (F) with315

l = 1 holds, and let Θ = Θ(t, z) be a solution to system (1.2). Then, for each z ∈ Ω316

and any t ≥ 0, we have317

D(∂zΘ(t, z)) ≤ D(∂zΘ0(z))e−κm(z) cosD(Θ0(z))t + C1(z)(1− e−κm(z) cosD(Θ0(z))t),318

where the random variable C1(z) depends on D(∂rzV(z)), ∂rzκij and D(Θ0(z)) for319

r = 0, 1.320

Proof. Let Θ = Θ(t, z) be a solution to system (1.2) with zero sum conditions in321

(F1) and (F2). Then, it follows from (3.1) that for any l ∈ N,322 ∑i

∂zθi(t, z) = 0, t ≥ 0, z ∈ Ω,323

and ∂zθi(t, z) satisfies324

∂t∂zθi = ∂zνi(z) +1

N

∑1≤j≤N

[(∂zκij(z)) sin(θj − θi)

]+

1

N

∑1≤j≤N

[κij(z) cos(θj − θi)(∂zθj − ∂zθi)

].

(3.2)325

326

We choose extremal indices M1 = M1(t, z),m1 = m1(t, z) such that327

∂zθM1(t, z) := maxi∂zθi(t, z), ∂zθm1(t, z) := min

i∂zθi(t, z).328

Note that for every z ∈ Ω, ∂zθM1(·, z) and ∂zθm1

(·, z) are piecewise differentiable and329

Lipschitz with respect to t. Then, we have330

(3.3) D(∂zΘ(t, z)) := ∂zθM1(t, z)− ∂zθm1

(t, z), t ≥ 0, z ∈ Ω.331

For the estimate of D(∂zΘ), we estimate time-evolution of ∂zθM1and ∂zθm1

as follows.332

333

• (Upper bound estimate of ∂zθM1): For a.e. t > 0, we have334

∂t∂zθM1≤ ∂zνM1

(z) + ‖∂zκ(z)‖∞ sinD(Θ)

+1

N

N∑j=1

κM1,j(z) cos(θj − θM1)(∂zθj − ∂zθM1)

≤ ∂zνM1(z) + ‖∂zκ(z)‖∞ sinD(Θ)

+1

N

N∑j=1

κm(z) cosD(Θ0(z))(∂zθj − ∂zθM1)

= ∂zνM1(z) + ‖∂zκ(z)‖∞ sinD(Θ)− κm(z) cosD(Θ0(z))∂zθM1 ,

(3.4)335

336



where we used D(Θ(t, z)) ≤ D(Θ0(z)) from Proposition 2.5.337

338

• (Lower bound estimate of ∂zθm1): Similarly for a.e. t > 0, we have339

∂t∂zθm1 ≥ ∂zνm1(z)− ‖∂zκ(z)‖∞ sinD(Θ)

+1

N

N∑j=1

κm1,j(z) cos(θj − θm1)(∂zθj − ∂zθm1

)

≥ ∂zνm1(z)− ‖∂zκ(z)‖∞ sinD(Θ)

+1

N

N∑j=1

κm(z) cosD(Θ0(z))(∂zθj − ∂zθm1)

= ∂zνm1(z)− ‖∂zκ(z)‖∞ sinD(Θ)− κm(z) cosD(Θ0(z))∂zθm1

.

(3.5)340

341

We use the relations (3.3), (3.4) and (3.5) to yield the following Gronwall type in-342

equality: for a.e. t > 0,343

∂tD(∂zΘ) ≤ −κm(z) cosD(Θ0)D(∂zΘ) +D(∂zV(z))

+ 2‖∂zκ(z)‖∞ sinD(Θ(t, z))

≤ −κm(z) cosD(Θ0(z))D(∂zΘ) +D(∂zV(z))

+ 2‖∂zκ(z)‖∞ sinD(Θ0(z)).

(3.6)344

345

where we used (2) of Proposition 2.5. Then, Gronwall’s lemma in Lemma 2.4 and346

continuity of D(∂zΘ(·, z)) with respect to t, can be used to obtain the following desired347

estimate: for z ∈ Ω,348

D(∂zΘ(t, z)) ≤ D(∂zΘ0(z))e−κm(z) cosD(Θ0(z))t + C1(z)

(1− e−κm(z) cosD(Θ0(z))t

).349

350

351

Next, we use induction and provide a local sensitivity analysis for the diameter of352

higher-order z-derivatives of phases.353

Theorem 3.2. Suppose that the framework (F) holds, and let Θ = Θ(t, z) be a354

solution to the system (1.2). Then, for z ∈ Ω, we have355

D(∂lzΘ(t, z)) ≤ D(∂lzΘ0(z))e−κm(z) cosD(Θ0(z))t

+ Cl(z)(1− e−κm(z) cosD(Θ0(z))t) for all t ≥ 0,(3.7)356

357

where the random variable Cl(z) (l ≥ 2) depends on D(∂rzV(z)), ∂rzκij and D(∂rzΘ0(z))358

for r = 0, 1, · · · , l.359

Proof. We use the mathematical induction together with initial step in Lemma360

3.1.361

362

• (Initial step): For l = 1, the estimate (3.7) is already established in Lemma 3.1.363

364

• (Inductive step): For l ≥ 2, assume that the estimate (3.7) for D(∂kzΘ(t, z)) with365

k ≤ l− 1 hold, and we will show that the estimate (3.7) holds for D(∂lzΘ(t, z)) below.366

For (k1, · · · , kr) ∈ (N∪0)r, 1 ≤ i, j ≤ N and r ∈ N, we setM(r, k1, · · · , kr,Θ, i, j)367



as follows:368

M(r, k1, · · · , kr,Θ, i, j) := sin(k)(θj − θi)r∏p=1

(∂pzθj − ∂pzθi

p!

)kp,369

where sin(k)(θj − θi) := dk

dxk(sin(x))

∣∣x=θj−θi

and k = k1 + · · · kr.370

We use the chain rule for higher order derivatives at the end of Section 2, and371

deduce that (3.1) becomes as follows:372

∂t(∂lzθi)(t, z)373

= ∂lzνi(z) +1

N

N∑j=1

κij(z) cos(θj − θi)(∂lzθj − ∂lzθi)374

+1

N

∑1≤j≤N

(k1, ··· , kl)∈Λ(l)kl=0

κij(z)l!

k1! · · · kl−1!M(l − 1, k1, · · · , kl−1,Θ, i, j)375

+1

N

∑1≤j≤N

1≤r≤l−1(k1, ··· , kr)∈Λ(r)

(l

r

)∂l−rz κij(z)

r!

k1! · · · kr!M(r, k1, · · · , kr,Θ, i, j)376

+1

N

∑1≤j≤N

∂lzκij(z) sin(θj − θi).377

378

For the l-th phase variations ∂lzθi(t, z), we choose extremal indices Ml = Ml(t, z),379

ml = ml(t, z) such that380

∂zθMl(t, z) := max

i∂lzθi(t, z), ∂zθml(t, z) := min

i∂lzθi(t, z),

D(∂lzΘ(t, z)) := ∂lzθMl(t, z)− ∂lzθml(t, z), t ≥ 0, z ∈ Ω.381

Note that for z ∈ Ω, ∂lzθMl(·, z) and ∂lzθml(·, z) are piecewise differentiable and Lip-382

schitz with respect to t. Our claim is to show that the following inequality holds, as383

we did in Lemma 3.1:384

(3.8) ∂tD(∂lzΘ(t, z)) ≤ −κm(z) cosD(Θ0(z))D(∂lzΘ(t, z)) + C(z),385

where C = C(z) is a nonnegative random function depending on D(∂rzV(z)), ∂rzκij386

and D(∂rzΘ0(z)) for r = 0, 1, · · · , l. For this, we estimate ∂lzθMland ∂lzθml separately387

as follows.388

389



• Case A (Estimate for ∂t∂lzθMl

): For a.e. t ≥ 0, we have390

∂t(∂lzθMl

)(t, z)

≤

∂lzνMl(z) +

1

N

N∑j=1

κMl,j(z) cos(θj − θMl)(∂lzθj − ∂lzθMl

)

+

1

N

∑1≤j≤N

(k1, ··· , kl)∈Λ(l)kl=0

κMl,j(z)l!

k1! · · · kl−1!M(l − 1, k1, · · · , kl−1,Θ,Ml, j)

+1

N

∑1≤j≤N

1≤r≤l−1(k1, ··· , kr)∈Λ(r)

(l

r

)∂l−rz κMl,j(z)

r!

k1! · · · kr!M(r, k1, · · · , kr,Θ,Ml, j)

+1

N

∑1≤j≤N

∂lzκMl,j(z) sin(θj − θMl)

=:

4∑k=1

I1k.

(3.9)

391

392

Case A.1 (Estimate of I11): By direct estimate, we have393

I11 = ∂lzνMl+

1

N

N∑j=1

κMl,j cos(θj − θMl)(∂lzθj − ∂lzθMl

)

≤ ∂lzνMl− κm cosD(Θ0)∂lzθMl

.

394

395

Case A.2 (Estimate of I12 and I13): for r ≤ l − 1, use the induction hypothesis to396

get397

D(∂rzΘ(t, z)) ≤ D(∂rzΘ0(z))e−κm cosD(Θ0)t+Cr(z)(1−e−κm cosD(Θ0)t), for each z ∈ Ω,398

which can be changed to399

D(∂rzΘ(t, z)) ≤ maxD(∂rzΘ0(z)), Cr(z)

=: Cr(z), for each z ∈ Ω,400

Hence we obtain401

|M(r, k1, · · · , kr,Θ,Ml, j)| ≤r∏p=1

(D(∂pzΘ(t, z))

p!

)kp≤

r∏p=1

(Cp(z)

p!

)kp402

for r = 1, · · · , l − 1. Thus,403

I12 ≤ ‖κ(z)‖∞∑

(k1, ··· , kl)∈Λ(l)kl=0

l!

k1! · · · kl−1!

l−1∏p=1

(Cp(z)

p!

)kp,

I13 ≤∑

1≤r≤l−1(k1, ··· , kr)∈Λ(r)

(l

r

)r!

k1! · · · kr!‖∂l−rz κ(z)‖∞

r∏p=1

(Cp(z)

p!

)kp404



405

Case A.3 (Estimate of I14): We deduce from Proposition 2.5 that406

I14 ≤‖∂lzκ(z)‖∞

N

∑1≤j≤N

D(Θ(t, z)) ≤ ‖∂lzκ(z)‖∞D(Θ0(z)).407

In (3.9), we combine all results in Case A.1 - Case A.3 to get the following: for a.e.408

t ≥ 0,409

∂t∂lzθMl

(t, z) ≤ −κm cosD(Θ0)∂lzθMl+ C(z),(3.10)410411

where C = C(z) is a nonnegative random function depending on D(∂rzV(z)), ∂rzκij412

and D(∂rzΘ0(z)) for r = 0, 1, · · · , l.413

• Case B (Estimate for ∂t∂lz θml): Next, we estimate ∂t∂

lzθml as follows: for a.e. t ≥ 0,414

∂t(∂lzθml)(t, z)

≥

∂lzνml(z) +1

N

N∑j=1

κml,j(z) cos(θj − θml)(∂lzθj − ∂lzθml)

− 1

N

∑1≤j≤N

(k1, ··· , kl)∈Λ(l)kl=0

κml,j(z)l!

k1! · · · kl−1!|M(l − 1, k1, · · · , kl−1,Θ, kl, j)|

− 1

N

∑1≤j≤N

1≤r≤l−1(k1, ··· , kr)∈Λ(r)

(l

r

)r!

k1! · · · kr!|∂l−rz κml,j(z)M(r, k1, · · · , kr,Θ,ml, j)|

− 1

N

∑1≤j≤N

|∂lzκml,j(z) sin(θj − θml)|

=:

4∑k=1

I2k.

(3.11)

415

416

Case B.1 (Estimate of I21): In this case, we have417

I21 ≥ ∂lzνml − κm cosD(Θ0)∂lzθml .418

Case B.2 (Estimate of I22 and I23): As in Case A.2, we have419

I22 ≥ −‖κ(z)‖∞∑

(k1, ··· , kl)∈Λ(l)kl=0

l!

k1! · · · kl−1!

l−1∏p=1

(Cp(z)

p!

)kp,420

I23 ≥ −∑

1≤r≤l−1(k1, ··· , kr)∈Λ(r)

(l

r

)r!

k1! · · · kr!‖∂l−rz κ(z)‖∞

r∏p=1

(Cp(z)

p!

)kp.421

422



Case B.3 (Estimate of I24): By direct estimate, we have423

I24 ≥ −‖∂lzκ(z)‖∞

N

∑1≤j≤N

D(Θ(t, z)) ≥ −‖∂lzκ(z)‖∞D(Θ0(z)).424

In (3.11), we combine all results in Case B.1 - Case B.3 to get the following: for a.e.425

t ≥ 0,426

∂t∂lzθml(t, z) ≥ −κm cosD(Θ0)∂lzθml − C(z),(3.12)427428

where C = C(z) is the same random function from (3.10). Therefore, we combine429

(3.10) and (3.12) to obtain the desired inequality (3.8) and we set430

y := D(∂lzΘ), α = κm cosD(Θ0), β = 0, C = C(z).431

Finally, we use Lemma 2.4 to derive the desired estimate.432

As a direct application of Theorem 3.2, we have the local sensitivity estimate for the433

phase diameter of ∂lzθi.434

Corollary 3.3. Suppose that the framework (F) holds, and let Θ = Θ(t, z) be a435

solution to system (1.2). Then, for z ∈ Ω, we have436

(3.13) E[D(∂lzΘ(t, ·))] ≤ EmaxD(∂lzΘ0), Cl(z) t ≥ 0.437

Proof. The result of Theorem 3.2 implies438

D(∂lzΘ(t, z)) ≤ maxD(∂lzΘ0(z)), Cl(z).439

Note that the R.H.S. depend only on given data and parameters, i.e., Θ0,V, κij(z).440

3.1. Identical oscillators. In this subsection, we consider an ensemble of iden-441

tical oscillators, and provide more refined local sensitivity estimates than those in442

Theorem 3.2. For this, we improve estimates in Lemma 3.1 and Theorem 3.2.443

Corollary 3.4. (First-order estimate) Suppose that the framework (F) with l =444

1 and445

νi = 0, i = 1, · · · , N446

hold, and let Θ = Θ(t, z) be a solution to system (1.2). Then, for z ∈ Ω,447

(3.14) D(∂zΘ(t, z)) ≤ D1(z)e−κm(z) cosD(Θ0(z))t

2 .448

where the random variable D1(z) depends on ∂rzκij and D(∂rzΘ0(z)) for r = 0, 1.449

Proof. We use (3.6) with D(∂zV) = 0 and Proposition 2.5 that for every t ≥ 0450

and z ∈ Ω,451

∂tD(∂zΘ(t, z))

≤ −κm(z) cosD(Θ0(z))D(∂zΘ(t, z)) + 2‖∂zκ(z)‖∞D(Θ(t, z))

≤ −κm(z) cosD(Θ0(z))D(∂zΘ(t, z)) + 2‖∂zκ(z)‖∞D(Θ0(z))e−κm(z) cosD(Θ0(z))t

2 .

(3.15)

452

453



We set454

y = D(∂zΘ), α = κm cosD(Θ0), β =α

2and C = 2‖∂zκ‖∞D(Θ0),455

and apply Lemma 2.4 in (3.15) to derive the exponential decay estimate:456

D(∂zΘ(t, z)) ≤ 4‖∂zκ(z)‖∞D(Θ0(z))

κm(z) cosD(Θ0(z))e−

κm(z) cosD(Θ0(z))t2457

+D(∂zΘ0(z))e−κm(z) cosD(Θ0(z))t.458459

This implies our desired result.460

Remark 3.5. For the constant couplings and initial data that are strictly confined461

in a quarter arc, there exists a small positive constant ε ∈ (0, π2 ) such that462

κm = constant, ∂zκij = 0, θ0i (z) <

π

2− ε,463

the estimate in (3.14) implies464

D(∂zΘ(t, z)) ≤ D(∂zΘ0(z))e−

κm cos(π2−ε)t

2 .465

Hence, we have466

ED(∂zΘ(t, ·)) ≤ ED(∂zΘ0(·))e−

κm cos(π2−ε)t

2 .467

Next, we provide local sensitivity analysis for the identical case with higher order468

z-derivatives.469

Corollary 3.6. (Higher-order estimates) Suppose that the framework (F) with470

νi = 0, i = 1, · · · , N471

hold, and let Θ = Θ(t, z) be a solution to system (1.2). Then, for z ∈ Ω,472

D(∂lzΘ(t, z)) ≤ Dl(z)e−κm(z) cosD(Θ0(z))t

2 ,473

where the random variable Dl(z) (l ≥ 2) depends on ∂rzκij and D(∂rzΘ0(z)) for r =474

0, 1, · · · , l.475

Proof. We will proceed by induction. Note that we have already proved l = 1476

case in Corollary 3.4. Then for the induction step, we aim to obtain the following477

inequality, as we did in Corollary 3.4:478

(3.16) ∂tD(∂lzΘ(t, z)) ≤ −κm(z) cosD(Θ0(z))D(∂lzΘ(t, z))+C(z)e−κm(z) cosD(Θ0(z))t

2 ,479

where C = C(z) is a nonnegative random function. To get this, we estimate ∂t∂lzθMl

480

and ∂t∂lzθml as we did in Theorem 3.2.481

482

• Case C (Estimate for ∂t∂lzθMl

): For ∂t∂lzθMl

,483

∂t(∂lzθMl

)(t, z)484

≤ 1

N

N∑j=1

κMl,j(z) cos(θj − θMl)(∂lzθj − ∂lzθMl

)485



+1

N

∑1≤j≤N

(k1, ··· , kl)∈Λ(l)kl=0

κMl,j(z)l!

k1! · · · kl−1!M(l − 1, k1, · · · , kl−1,Θ,Ml, j)486

+1

N

∑1≤j≤N

1≤r≤l−1(k1, ··· , kr)∈Λ(r)

(l

r

)∂l−rz κMl,j(z)

r!

k1! · · · kr!M(r, k1, · · · , kr,Θ,Ml, j)487

+1

N

∑1≤j≤N

∂lzκMl,j(z) sin(θj − θMl)488

=:

4∑k=1

I3k.489

490

Case C.1 (Estimate of I31): for I31,491

I31 =1

N

N∑j=1

κMl,j cos(θj − θMl)(∂lzθj − ∂lzθMl

) ≤ −κm cosD(Θ0)∂lzθMl.492

Case C.2 (Estimate of I32 and I33): By induction hypothesis, for each z ∈ Ω,493

D(∂rzΘ(t, z)) ≤ Dr(z)e−κm cosD(Θ0(z))t

2 .494

Thus,495

M(r, k1, · · · , kr,Θ,Ml, j) ≤r∏p=1

(D(∂pzΘ(t, z))

p!

)kp≤ e−

κm cosD(Θ0(z))t2

r∏p=1

(Dp(z)p!

)kp,

(3.17)496

where we used k1 + · · ·+ kr ≥ 1. Thus, (3.17) yields497

I32 ≤

‖κ(z)‖∞∑

(k1, ··· , kl)∈Λ(l)kl=0

l!

k1! · · · kl−1!

l−1∏p=1

(Dp(z)p!

)kp e−κm cosD(Θ0(z))t

2 ,

I33 ≤

∑

1≤r≤l−1(k1, ··· , kr)∈Λ(r)

(l

r

)‖∂l−rz κ(z)‖∞

r!

k1! · · · kr!

(Dp(z)p!


2 .

498

499

Case C.3 (Estimate of I34): In this case,500

I34 ≤‖∂lzκ‖∞N

∑1≤j≤N

D(Θ(t, z)) ≤ D(Θ0(z))‖∂lzκ(z)‖∞e−κm cosD(Θ0(z))t.501

We combine all estimates in Case C.1 - Case C.3 to get502

(3.18) ∂t∂lzθMl

(t, z) ≤ −κm cosD(Θ0)∂lzθMl+ C(z)e−

κm cosD(Θ0(z))t2 ,503



where C = C(z) is a random function depending on ∂rzκij and D(∂rzΘ0(z)) for r =504

0, 1, · · · , l.505

• Case D (Estimate for ∂t∂lzθml): Now we evaluate ∂t∂

lzθml as follows.506

∂t(∂lzθml)(t, z)507

≥ 1

N

N∑j=1

κml,j cos(θj − θml)(∂lzθj − ∂lzθml)508

− 1

N

∑1≤j≤N

(k1, ··· , kl)∈Λ(l)kl=0

κml,jl!

k1! · · · kl−1!|M(l − 1, k1, · · · , kl−1,Θ, kl, j)|509

− 1

N

∑1≤j≤N

1≤r≤l−1(k1, ··· , kr)∈Λ(r)

(l

r

)r!

k1! · · · kr!|∂l−rz κml,j | |M(r, k1, · · · , kr,Θ,ml, j)|510

− 1

N

∑1≤j≤N

|∂lzκml,j sin(θj − θml)|511

=:

4∑k=1

I4k.512

513

Case D.1 (Estimate of I41): In this case,514

I41 ≥ −κm cosD(Θ0)∂lzθml .515

Case D.2 (Estimate of I42 and I43): Similar to estimates I32 and I33,516

I42 ≥ −

‖κ(z)‖∞∑

(k1, ··· , kl)∈Λ(l)kl=0

l!

k1! · · · kl−1!

l−1∏p=1

(Dp(z)p!


2 ,

I43 ≥ −

∑

1≤r≤l−1(k1, ··· , kr)∈Λ(r)

(l

r

)‖∂l−rz κ(z)‖∞

r!

k1! · · · kr!

(Dp(z)p!


2 .

517

518

Case D.3 (Estimate of I44): We have519

I44 ≥ −‖∂lzκ‖∞N

∑1≤j≤N

D(Θ(t, z)) ≥ −D(Θ0(z))‖∂lzκ(z)‖∞e−κm cosD(Θ0(z))t.520

Here, we combine all esitmates in Case D.1 - Case D.3 to get521

(3.19) ∂t∂lzθml(t, z) ≥ −κm cosD(Θ0)∂lzθml − C(z)e−

κm(z) cosD(Θ0(z))t2 ,522

where C = C(z) is the same random function in (3.18).523

524

Finally, we combine (3.18) and (3.19) to obtain the desired inequality (3.16) and apply525

Gronwall’s lemma in Lemma 2.4 to get the desired result.526



4. Uniform `1-stability estimate for the phase process. In this subsection,527

we provide `1-stability estimates for phase variations ∂lzΘ with respect to initial phase528

variations in random space. Let Θ and Θ be solutions to (1.2) with sufficiently regular529

initial data Θ0 and Θ0 in random space, respectively. In the sequel, we will derive an530

estimate like531

∂

∂t‖∂lz(Θ− Θ)(t, z)‖1 + Λl(t, z) ≤ El(z)

l−1∑p=0

‖∂pz (Θ− Θ)(t, z)‖1,532

where l ∈ N, the nonnegative random variable El(z) is independent of t and Λl(t, z)533

is a non-negative functional.534

535

Let Θ and Θ be two random Kuramoto flows whose dynamics are governed by536

(1.2). Then, for r ∈ N ∪ 0 and z ∈ Ω, we set537

I0r (t, z) := 1 ≤ i ≤ N | ∂rzθi(t, z)− ∂rz θi(t, z) = 0,I+r (t, z) := 1 ≤ i ≤ N | ∂rzθi(t, z)− ∂rz θi(t, z) > 0,I−r (t, z) := 1 ≤ i ≤ N | ∂rzθi(t, z)− ∂rz θi(t, z) < 0,I0(t, z) := I0

0 (t, z), I+(t, z) := I+0 (t, z), I−(t, z) := I−0 (t, z).

(4.1)538

539

First, we provide pathwise `1- stability for (1.2) following the arguments line by line540

in [11].541

Proposition 4.1. (Pathwise `1-stability) Suppose that the framework (F) with542

l = 0 hold, and let Θ = Θ(t, z) and Θ = Θ(t, z) be two solutions to system (1.2) with543

initial data Θ0 and Θ0, respectively. Then, for z ∈ Ω, the following assertions hold:544

1. If θc(0) 6= θc(0), then545

∂

∂t‖(Θ− Θ)(t, z)‖1 + Λ(t, z) ≤ 0,546

for all t ≥ 0 and each z ∈ Ω, where the non-negative functional Λ(s, z) is547

defined by548

Λ(t, z) :=κm(z)γ(z) cosD(Θ0(z))

N

[ (|I0(t)|+ 2|I−(t)|

) ∑i∈I+(t)

|(θi − θi)(t)|

+(|I0(t)|+ 2|I+(t)|

) ∑i∈I−(t)

|(θi − θi)(t)|],

549

550

where γ(z) := sinD(Θ0(z))D(Θ0(z)) .551

2. If θc(0, z) = θc(0, z) for every z ∈ Ω, then552

‖(Θ− Θ)(t, z)‖1 ≤ ‖(Θ0 − Θ0)(z)‖1e−κm(z) cosD(Θ0(z))γ(z)t,553

for all t ≥ 0 and each z ∈ Ω.554

Proof. The proof is almost the same as in the deterministic case in [11]. So we555

omit the proof.556



For A(t, z) = α(t, z)Ni=1 ∈ RN , and define J0, J± and ∆ij as in (4.1): for t ≥ 0, z ∈557

Ω,558

J0(t, z) := 1 ≤ i ≤ N | αi(t, z) = 0,J+(t, z) := 1 ≤ i ≤ N | αi(t, z) > 0,J−(t, z) := 1 ≤ i ≤ N | αi(t, z) < 0,∆ij(t, z) := (sgn(αi(t, z))− sgn(αj(t, z)))(αj(t, z)− αi(t, z)).

559

560

561

Lemma 4.2. The following assertions hold.562

1. We have563

N∑i,j=1

∆ij = −2

(|J0|+ 2|J−|

) ∑i∈J+

|αi|+(|J0|+ 2|J+|

) ∑i∈J−

|αi|

.564

2. Moreover, if

N∑i=1

αi = 0, then565

N∑i,j=1

∆ij = −2N

N∑i=1

|αi|.566

Proof. (i) Note that if αiαj > 0, then ∆ij = 0. So we consider the other cases567

for evaluating ∆ij as follows:568

αi > 0, αj = 0 =⇒ ∆ij = −|αi|; αi < 0, αj = 0 =⇒ ∆ij = −|αi|,αi = 0, αj > 0 =⇒ ∆ij = −|αj |; αi = 0, αj < 0 =⇒ ∆ij = −|αj |,αi > 0, αj < 0 =⇒ ∆ij = −2(|αi|+ |αj |),αi < 0, αj > 0 =⇒ ∆ij = −2(|αi|+ |αj |).569

These imply570 ∑(i,j)∈J+×J0

∆ij = −|J0|∑i∈J+

|αi|,∑

(i,j)∈J−×J0

∆ij = −|J0|∑i∈J−

|αi|,

∑(i,j)∈J0×J+

∆ij = −|J0|∑i∈J+

|αi|,∑

(i,j)∈J0×J−∆ij = −|J0|

∑i∈J−

|αi|,

∑(i,j)∈J+×J−

∆ij = −2|J−|∑i∈J+

|αi| − 2|J+|∑i∈J−

|αi|,

∑(i,j)∈J+×J0

∆ij = −2|J−|∑i∈J+

|αi| − 2|J+|∑i∈J−

|αi|.

(4.2)571

572

We combine all estimates in (4.2) to derive the estimate (i).573

(ii) Now, assume that∑Ni=1 αi = 0. Then574 ∑

i∈J−|αi| = −

∑i∈J−

αi =∑i∈J+

αi =∑i∈J+

|αi|.575

Thus,576



∑i∈J−

|αi| =∑i∈J+

|αi| =1

2

N∑i=1

|αi|.577

This yields578

N∑i,j=1

∆ij = −2

(|J0|+ 2|J−|

) ∑i∈J+

|αi|+(|J0|+ 2|J+|

) ∑i∈J−

|αi|

579

= −2

(|J0|+ |J−|+ |J+|

) N∑i=1

|αi|

= −2N

N∑i=1

|αi|.580

581

Now we ready to prove the `1-stability result. First we provide l = 1 case.582

Proposition 4.3. Suppose that the framework (F) with l = 1 for initial data Θ0583

and Θ0 hold, and let Θ := Θ(t, z) and Θ := Θ(t, z) be two solutions (1.2) with initial584

data Θ0 and Θ0, respectively satisfying585

D(∂rzΘ0(z)) ≤ D(∂rzΘ0(z)), for r = 0, 1, and each z ∈ Ω.586

Then for all t ≥ 0 and each z ∈ Ω,587

∂

∂t‖∂z(Θ− Θ)(t, z)‖1 + Λ1(t, z) ≤ E1(z)‖(Θ− Θ)(t, z)‖1,588

where the non-negative functional Λ1(s, z) is defined by589

Λ1(s, z) :=κm(z) cosD(Θ0(z))

N

[ (|I0

1 (s)|+ 2|I−1 (s)|) ∑i∈I+

1 (s)

|(∂zθi − ∂z θi)(s)|590

+(|I0

1 (s)|+ 2|I+1 (s)|

) ∑i∈I−1 (s)

|(∂zθi − ∂z θi)(s)|],591

592

and the random variable E1(z) depends on ∂rzκij and D(∂rzΘ0(z)) for r = 0, 1.593

Proof. For each z ∈ Ω, it follows from (3.2) that594

∂

∂t

N∑i=1

|(∂zθi − ∂z θi)(t)|595

=1

N

N∑i,j=1

sgn(∂zθi − ∂z θi)∂zκij(z)(sin(θj − θi)− sin(θj − θi))596

+1

N

N∑i,j=1

sgn(∂zθi − ∂z θi)κij(z)(cos(θj − θi)− cos(θj − θi))(∂zθj − ∂zθi)597

+1

N

N∑i,j=1

sgn(∂zθi − ∂z θi)κij(z) cos(θj − θi)(∂zθj − ∂zθi − ∂z θj + ∂z θi)598

=:

3∑k=1

I5k.599



600

Now we estimate I5k (k =1, 2, 3) separately.601

602

• Case E.1 (Estimate for I51) : In this case,603

I51 =2

N

N∑i,j=1

sgn(∂zθi − ∂z θi)∂zκij cos

(θj − θi

2+θj − θi

2

)sin

(θj − θi

2− θj − θi

2

)604

≤ 2

N

N∑i,j=1

|∂zκij | cosD(Θ0)

(∣∣∣∣∣θj − θj2

∣∣∣∣∣+

∣∣∣∣∣θi − θi2

∣∣∣∣∣)

605

≤ 2‖∂zκ‖∞ cosD(Θ0)‖(Θ− Θ)(t, z)‖1.606607

• Case E.2 (Estimate for I52): Similarly, we have608

I52 ≤2

N

N∑i,j=1

κij sin

∣∣∣∣∣θj − θi2+θj − θi

2

∣∣∣∣∣ sin∣∣∣∣∣θj − θi2

− θj − θi2

∣∣∣∣∣ |∂zθj − ∂zθi|609

≤ 2‖κ‖∞ sinD(Θ0)D(∂zΘ(t, z))

N

N∑i,j=1

sin

∣∣∣∣∣θj − θj2− θi − θi

2

∣∣∣∣∣610

≤ 2‖κ‖∞ sinD(Θ0)D(∂zΘ(t, z))

N

N∑i,j=1

(∣∣∣∣∣θj − θj2

∣∣∣∣∣+

∣∣∣∣∣θi − θi2

∣∣∣∣∣)

611

≤ 2‖κ‖∞ sinD(Θ0)C1(z)‖(Θ− Θ)(t, z)‖1.612613

• Case E.3 (Estimate for I53) : We get614

I53 =1

N

N∑i,j=1

sgn(∂zθi − ∂z θi)κij cos(θj − θi)(∂zθj − ∂zθi − ∂z θj + ∂z θi)615

=1

2N

N∑i,j=1

[(sgn(∂zθi − ∂z θi)− sgn(∂zθj − ∂z θj))616

× κij cos(θj − θi)(∂zθj − ∂zθi − ∂z θj + ∂z θi)]

617

≤ κm cosD(Θ0)

2N

N∑i,j=1

[(sgn(∂zθi − ∂z θi)− sgn(∂zθj − ∂z θj))618

× (∂zθj − ∂zθi − ∂z θj + ∂z θi)],619

620

where we used621

(4.3) (sgn(α)− sgn(β))(β − α) ≤ 0, α, β ∈ R,622

with the choice of α = ∂zθi − ∂z θi and β = ∂zθj − ∂z θj . Now we let αi = ∂zθi − ∂z θi.623

Then we correspond I01 , I+

1 and I−1 to J0, J+ and J−, respectively, and apply Lemma624

4.2 to obtain625

I53 ≤ −Λ1(z),626

which completes the proof.627



For the case θc(0, z) = θc(0, z), we have the exponential decay estimate.628

Corollary 4.4. Suppose that the framework (F) with l = 1 for initial data Θ0629

and Θ0 hold, and further assume that θc(0, z) := θc(0, z). Let Θ := Θ(t, z) and630

Θ := Θ(t, z) be two solutions (1.2) with initial data Θ0 and Θ0, respectively satisfying631

D(∂rzΘ0(z)) ≤ D(∂rzΘ0(z)), for r = 0, 1, and each z ∈ Ω.632


‖∂z(Θ− Θ)(t, z)‖1 ≤ E1(z)(z)e−κm(z) cosD(Θ0(z))γ(z)t1∑p=0

‖∂pz (Θ− Θ)(t, z)‖1,634

where γ(z) := sinD(Θ0(z))D(Θ0(z)) ∈ (0, 1) and the random variable E1(z) is given by635

E1(z) := max

1,

E1(z)

κm(z) cosD(Θ0(z))(1− γ(z))

.636

Proof. We set637

αi = ∂zθi − ∂z θi.638

Since θc(0, z) = θc(0, z),639

N∑i=1

αi = 0.640

On the other hand, in the course of proof of Proposition 4.1,641

I53 ≤ −κm cosD(Θ0)‖∂z(Θ− Θ)(t, z)‖1.642

Hence643

∂

∂t‖∂z(Θ− Θ)(t, z)‖1

≤ −κm cosD(Θ0)‖∂z(Θ− Θ)(t, z)‖1 + E1(z)‖(Θ− Θ)(t, z)‖1≤ −κm cosD(Θ0)‖∂z(Θ− Θ)(t, z)‖1 + E1(z)|(Θ0 − Θ0)(z)‖1e−κmγ cosD(Θ0)t,

(4.4)

644

645

where we used the estimate (ii) of Proposition 4.1:646

‖(Θ− Θ)(t, z)‖1 ≤ ‖(Θ0 − Θ0)(z)‖1e−κmγ cosD(Θ0)t.647

We now apply Gronwall’s lemma in Lemma 2.4 to (4.4) to obtain648

‖∂z(Θ− Θ)(t, z)‖1 ≤ ‖∂z(Θ0 − Θ0)(z)‖1e−κm cosD(Θ0)t649

+E1(z)e−κm cosD(Θ0)γt

κm cosD(Θ0)(1− γ)‖(Θ0 − Θ0)(z)‖1(1− e−κm cosD(Θ0)(1−γ)t),650

651

which yields our desired result.652

Finally, we provide the stability result for higher-order derivatives.653



Theorem 4.5. (Higher-order estimates) Suppose that the framework (F) for ini-654

tial data Θ0 and Θ0 hold, and let Θ := Θ(t, z) and Θ := Θ(t, z) be two solutions (1.2)655

with initial data Θ0 and Θ0, respectively satisfying656

D(∂rzΘ0(z)) ≤ D(∂rzΘ0(z)), for any r = 0, 1, · · · , l and each z ∈ Ω.657


∂

∂t‖∂lz(Θ− Θ)(t, z)‖1 + Λl(t, z) ≤ El(z)

l−1∑p=0

‖∂pz (Θ− Θ)(t, z)‖1,659

where the non-negative functional Λl(s, z) is defined by660

Λl(s, z) :=κm(z) cosD(Θ0(z))

N

[ (|I0l (s)|+ 2|I−l (s)|

) ∑i∈I+

l (s)

|(∂lzθi − ∂lz θi)(s)|

+(|I0l (s)|+ 2|I+

l (s)|) ∑i∈I−l (s)

|(∂lzθi − ∂lz θi)(s)|],

661

662

and the random variable El(z) depends on ∂rzκij and D(∂rzΘ0(z)) for r = 0, 1, · · · , l.663

Proof. It follows from (2.7) that664

∂t(∂lzθi)(t, z)

= ∂lzνi(z) +1

N

N∑j=1

κij(z) cos(θj − θi)(∂lzθj − ∂lzθi)

+1

N

∑1≤j≤N

(k1, ··· , kl)∈Λ(l)kl=0

κij(z)l!

k1! · · · kl−1!M(l − 1, k1, · · · , kl−1,Θ, i, j)

+1

N

∑1≤j≤N

1≤r≤l−1(k1, ··· , kr)∈Λ(r)

(l

r

)∂l−rz κij(z)

r!

k1! · · · kr!M(r, k1, · · · , kr,Θ, i, j)

+1

N

∑1≤j≤N

∂lzκij(z) sin(θj − θi),

665

666

whereM(r, k1, · · · , kr,Θ, i, j) is defined in the proof of Theorem 3.2. Recall that the667

functional M(r, k1, · · · , kr,Θ, i, j) has the following form:668

M(r, k1, · · · , kr,Θ, i, j) := sin(k)(θj − θi)r∏p=1


p!

)kp,669

where (k1, · · · , kr) ∈ (N ∪ 0)r, 1 ≤ i, j ≤ N , r ∈ N and k = k1 + · · · + kr. And670

also for more simplicity, we let671

Mr :=M(r, k1, · · · , kr,Θ, i, j), Mr :=M(r, k1, · · · , kr, Θ, i, j).672



Now we proceed by induction on l. By using above notations, we can get the following673

relation:674

∂

∂t‖∂lz(Θ− Θ)(t, z)‖1

=1

N

N∑i,j=1

sgn(∂lzθi − ∂lz θi)κij(z)(cos(θj − θi)− cos(θj − θi))(∂lzθj − ∂lzθi)

+1

N

N∑i,j=1

sgn(∂lzθi − ∂lz θi)κij(z) cos(θj − θi)(∂lzθj − ∂lzθi − ∂lz θj + ∂lz θi)

+1

N

∑1≤i,j≤N

(k1, ··· , kl)∈Λ(l)kl=0

sgn(∂lzθi − ∂lz θi)κij(z)l!

k1! · · · kl−1!(Ml−1 − Ml−1)

+1

N

∑1≤i,j≤N1≤r≤l−1

(k1, ··· , kr)∈Λ(r)

sgn(∂lzθi − ∂lz θi)(l

r

)∂l−rz κij(z)

r!

k1! · · · kr!(Mr − Mr)

+1

N

N∑i,j=1

sgn(∂lzθi − ∂lz θi)∂lzκij(z)(sin(θj − θi)− sin(θj − θi))

=:

5∑k=1

I6k.

(4.5)675

676

As we did in Proposition 4.3, we estimate each I6k (k =1, 2, 3, 4, 5) separately. Here,677

our aim is to show that678

I62 ≤ −Λl(z),∑k 6=2

I6k ≤ El(z)l−1∑p=0

‖∂pz (Θ− Θ)(t, z)‖1.679

680

• Case F.1 (Estimate for I61):681

I61 ≤2

N

N∑i,j=1

κij sin

∣∣∣∣∣θj − θi2+θj − θi

2

∣∣∣∣∣ sin∣∣∣∣∣θj − θi2

− θj − θi2

∣∣∣∣∣ |∂lzθj − ∂lzθi|≤ 2‖κ‖∞ sinD(Θ0)D(∂lzΘ(t, z))

N

N∑i,j=1

sin

∣∣∣∣∣θj − θj2− θi − θi

2

∣∣∣∣∣≤ 2‖κ‖∞ sinD(Θ0)D(∂lzΘ(t, z))

N

N∑i,j=1

(∣∣∣∣∣θj − θj2

∣∣∣∣∣+

∣∣∣∣∣θi − θi2

∣∣∣∣∣)

≤ 2‖κ‖∞ sinD(Θ0)Cl(z)‖(Θ− Θ)(t, z)‖1.

682

683

• Case F.2 (Estimate for I62) :684

I62 =1

N

N∑i,j=1

sgn(∂lzθi − ∂lz θi)κij cos(θj − θi)(∂lzθj − ∂lzθi − ∂lz θj + ∂lz θi)685

=1

2N

N∑i,j=1

[(sgn(∂lzθi − ∂lz θi)− sgn(∂lzθj − ∂lz θj))686



× κij cos(θj − θi)(∂lzθj − ∂lzθi − ∂lz θj + ∂lz θi)]

687

≤ κm cosD(Θ0)

2N

N∑i,j=1

[(sgn(∂lzθi − ∂lz θi)− sgn(∂lzθj − ∂lz θj))688

× (∂lzθj − ∂lzθi − ∂lz θj + ∂lz θi)],689

690

where we used (4.3). Now we set αi := ∂lzθi − ∂lz θi, and apply Lemma 4.2 to get691

I62 ≤ −Λl(t, z),692

and here, note that J0, J+ and J− in Lemma 4.2 correspond to I0l , I+

l and I−l , re-693

spectively.694

695

• Case F.3 (Estimates for I63 and I64) : Here, it suffices to estimate Mr − Mr:696

Mr − Mr

=(

sin(k)(θj − θi)− sin(k)(θj − θi)) r∏p=1


p!

)kp

+(

sin(k)(θj − θi)) r∏

p=1


p!

)kp−

r∏p=1

(∂pz θj − ∂pz θi

p!

)kp=: K1 +K2,

697

698

where we set k0 = kr+1 = 0. For K1, note that699

∣∣∣sin(k)(θj − θi)− sin(k)(θj − θi)∣∣∣ ≤ 2 sin

∣∣∣∣∣θj − θi2− θj − θi

2

∣∣∣∣∣ .700

Hence, for K1,701

K1 ≤ 2 sin

∣∣∣∣∣θj − θi2− θj − θi

2

∣∣∣∣∣r∏p=1

∣∣∣∣∂pzθj − ∂pzθip!

∣∣∣∣kp702

≤ 2

(∣∣∣∣∣θj − θj2

∣∣∣∣∣+

∣∣∣∣∣θi − θi2

∣∣∣∣∣)

r∏p=1

∣∣∣∣D(∂pzΘ)

p!

∣∣∣∣kp703

≤(∣∣∣θj − θj∣∣∣+

∣∣∣θi − θi∣∣∣) r∏p=1

∣∣∣∣∣ Cp(z)p!

∣∣∣∣∣kp

.704

705

For the term K2,706 ∣∣∣∣∣∣(∂pzθj − ∂pzθi

p!

)kp−


p!

)kp ∣∣∣∣∣∣707

≤

0 if kp = 0,

kp

(Cp(z)

)kp−1

|∂pzθj − ∂pzθi − ∂pz θj + ∂pz θi| if kp > 0,708

709



where we used the relation: for n ∈ N and 0 ≤ p ≤ l710

an − bn = (a− b)n−1∑k=0

akbn−1−k, D(∂pzΘ(t, z)), D(∂pz Θ(t, z)) ≤ Cp(z).711

Note that the second relation follows from the assumption D(∂rzΘ0) ≤ D(∂rzΘ0) for712

r = 0, 1, · · · , l. Together with the following identity713

r∏p=1

ap −r∏p=1

bp =

r∑p=1

[(r+1∏

µ=p+1

aµ

)(p−1∏µ=0

bµ

)(ap − bp)

],714

where a0 = b0 = ap+1 = bp+1 = 0 is assumed, we obtain715

K2 ≤r∑p=1

∣∣∣∣∣

p−1∏µ1=0

(∂µ1z θj − ∂µ1

z θiµ1!

)kµ1

r+1∏µ2=p+1

(∂µ2z θj − ∂µ2

z θiµ2!

)kµ2

716

×


p!

)kp−


p!

)kp ∣∣∣∣∣717

≤r∑p=1

[ ∏0≤µ≤r+1µ6=p

(Cµ(z)

µ!

)kµ ∣∣∣∣∣(∂pzθj − ∂pzθi

p!

)kp−


p!

)kp ∣∣∣∣∣]

718

≤∑

1≤p≤rkp 6=0

[ ∏0≤µ≤r+1µ6=p

(Cµ(z)

µ!

)kµkp

(Cp(z)

)kp−1(|∂pzθj − ∂pz θj |+ |∂pzθi − ∂pz θi|

)]719

≤

[ ∑1≤p≤rkp 6=0

kp

(Cp(z)

)kp−1 ∏

0≤µ≤r+1µ 6=p

(Cµ(z)

µ!

)kµ ]720

×r∑p=1

(|∂pzθj − ∂pz θj |+ |∂pzθi − ∂pz θi|

).721

722

We combine estimates for K1 and K2 to get723

|Mr − Mr| ≤ C(r, z)

r∑p=0


),724

where C = C(r, z) depends on ∂pzκij and D(∂pzΘ0(z)) for r = 0, 1, · · · , r.725

726

Now, the terms I63 and I64 can be estimated as follows.727

I63 ≤1

N

∑1≤i,j≤N

(k1, ··· , kl)∈Λ(l)kl=0

κijl!

k1! · · · kl−1!|Ml−1 − Ml−1|728

≤ ‖κ‖∞N

∑(k1, ··· , kl)∈Λ(l)

kl=0

l!

k1! · · · kl−1!C(l − 1, z)

729



×∑

1≤i,j≤N0≤p≤l−1


)730

≤ C(z)∑

0≤p≤l−1

‖∂pzΘ− ∂pz Θ‖1,731

732733

where C = C(z) depends on ∂rzκij and D(∂rzΘ0(z)) for r = 0, 1, · · · , l − 1and734

I64 ≤1

N

∑1≤i,j≤N1≤r≤l−1

(k1, ··· , kr)∈Λ(r)

(l

r

)‖∂l−rz κ‖∞

r!

k1! · · · kr!|Mr − Mr|735

≤ 1

N

∑1≤r≤l−1

(k1, ··· , kr)∈Λ(r)

(l

r

)‖∂l−rz κ‖∞

r!

k1! · · · kr!C(r, z)736

×N∑

i,j=1

l−1∑r=1

r∑p=0


)737

≤ C(z)

l−1∑p=0

‖∂pz (Θ− Θ)(t, z)‖1.738

739

740

• Case F.4 (Estimate for I65) : Finally, we can estimate741

I65 =2

N

N∑i,j=1

sgn(∂lzθi − ∂lz θi)∂lzκij cos

(θj − θi

2+θj − θi

2

)sin

(θj − θi

2− θj − θi

2

)742

≤ 2

N

N∑i,j=1

|∂lzκij | cosD(Θ0)

(∣∣∣∣∣θj − θj2

∣∣∣∣∣+

∣∣∣∣∣θi − θi2

∣∣∣∣∣)

743

≤ 2‖∂lzκ‖∞ cosD(Θ0)‖(Θ− Θ)(t, z)‖1.744745

In (4.5), we combine all estimates for I6k to obtain the desired estimate.746

Corollary 4.6. Suppose that the framework (F) with r = l ∈ N for initial747

data Θ0 and Θ0 hold, and further assume θc(0, z) = θc(0, z) for every z ∈ Ω. Let748

Θ := Θ(t, z) and Θ := Θ(t, z) be two solutions (1.2) with initial data Θ0 and Θ0,749

respectively satisfying750

D(∂rzΘ0(z)) ≤ D(∂rzΘ0(z)), for any r = 0, 1, · · · , l and each z ∈ Ω.751


‖∂lz(Θ− Θ)(t, z)‖1 ≤ El(z)e−κm(z)γ(cosD(Θ0(z)))tl∑

p=0

‖∂pz (Θ0 − Θ0)(z)‖1,753

where the random variable El(z) (l ≥ 2) depends on ∂rzκij and D(∂rzΘ0(z)) for r =754

0, 1, · · · , l.755



Proof. The proof will be done inductively on l.756

757

• (Initial step): In this case, we have already proved this case in Corollary 4.4.758

759

• (Inductive step): Set760

αi = ∂lzθi − ∂lz θi.761

Then, we have∑Ni=1 αi = 0, since we assumed θc(0, z) = θc(0, z). Thus, it follows762

from Proposition 4.3 that763

Λl(t, z) = −κm cosD(Θ0)‖∂lz(Θ− Θ)(t, z)‖1.764

On the other hand, it follows from Theorem 4.5 that765

∂

∂t‖∂lz(Θ− Θ)(t, z)‖1

≤ −κm cosD(Θ0)‖∂lz(Θ− Θ)(t, z)‖1 + El(z)l−1∑p=0

‖∂pz (Θ− Θ)(t, z)‖1.(4.6)766

767

Note that from (ii) of Proposition 4.3 and induction hypothesis,768

‖∂pz (Θ− Θ)(t, z)‖1 ≤ Ep(z)e−κm cosD(Θ0)γt

p∑s=0

‖∂sz(Θ0 − Θ0)(z)‖1769

for each p ∈ N ∪ 0. We apply this to (4.6) and for every l ≥ 1, to get770

∂

∂t‖∂lz(Θ− Θ)(t, z)‖1

≤ −κm cosD(Θ0)‖∂lz(Θ− Θ)(t, z)‖1

+ El(z)e−κm cosD(Θ0)γtl−1∑p=0

Ep(z)p∑s=0

‖∂sz(Θ0 − Θ0)(z)‖1

≤ −κm cosD(Θ0)‖∂lz(Θ− Θ)(t, z)‖1

+

(lEl(z)

l−1∑p=0

Ep(z)

)l−1∑p=0

‖∂pz (Θ0 − Θ0)(z)‖1e−κm cosD(Θ0)γt.

(4.7)771

772

For (4.7), we use Gronwall’s inquality in Lemma 2.4 to yield773

‖∂lz(Θ− Θ)(t, z)‖1≤ ‖∂lz(Θ0 − Θ0)(z)‖1e−κm cosD(Θ0)t

+

(lEl(z)

∑l−1p=0 Ep(z)

)e−κm cosD(Θ0)γt

κm cosD(Θ0)(1− γ)

l−1∑p=0

‖∂pz (Θ0 − Θ0)(z)‖1(1− e−κm cosD(Θ0)t),

≤ El(z)e−κm(z)γ cosD(Θ0)tl∑

p=0

‖∂pz (Θ0 − Θ0)(z)‖1.

774

775

This yields our desired result.776



5. Local sensitivity analysis for frequency process. In this section, we777

present a synchronizing property of frequency variations in a random space in a large778

coupling regime and discuss a generalization of local sensitivity on the abstract con-779

sensus system.780

5.1. Local sensitivity analysis. As noticed in Proposition 2.5, under some781

conditions on natural frequencies, coupling strengths and initial data, we can find a782

positively invariant set and ”vanishing of uncertainty” for frequency processes which783

are consistent with emergent dynamics of the deterministic Kuramoto model. Below,784

we will show that the variations ∂lzV = (∂lzω1, · · · , ∂lzωN ) will also enjoy a synchroniz-785

ing property in a large coupling regime, which clearly exhibits vanishing of uncertainty.786

Lemma 5.1. Suppose that the framework (F) in Section 3 with l = 1 hold, and787

let Θ = Θ(t, z) be a solution to system (1.2). Then, for z ∈ Ω,788

D(∂zV (t, z)) ≤ F1(z)e−κm(z)(cosD(Θ0))t

2 ,789

where F1(z) is a nonnegative random function depending on D(∂rzV0(z)), ∂rzκij and790

D(∂rzΘ0(z)) for r = 0, 1.791

Proof. First, we differentiate (1.2) with respect to t to obtain792

(5.1) ∂tωi(t, z) =1

N

N∑j=1

κij(z) cos(θj(t, z)− θi(t, z))(ωj(t, z)− ωi(t, z)).793

We again differentiate the above relation with respect to z to get794

∂t∂zωi(t, z)

=1

N

N∑j=1

∂zκij(z) cos(θj(t, z)− θi(t, z))(ωj(t, z)− ωi(t, z))

− 1

N

N∑j=1

κij(z) sin(θj(t, z)− θi(t, z))(∂zθj(t, z)− ∂zθi(t, z))(ωj(t, z)− ωi(t, z))

+1

N

N∑j=1

κij(z) cos(θj(t, z)− θi(t, z))(∂zωj(t, z)− ∂zωi(t, z)).

(5.2)

795

796

We choose indices M ′1 and m′1 such that for t > 0 and z ∈ Ω,797

∂zωM ′1(t, z) := maxi∂zωi(t, z), ∂zωm′1(t, z) := min

i∂zωi(t, z).798

Note that for every z ∈ Ω, ∂zωM ′1(·, z) and ∂zωm′1

(·, z) are piecewise differentiable799

and Lipschitz continuous with respect to t. Then our aim is to obtain the following800

inequality:801

(5.3) ∂tD(∂zV (t, z)) ≤ −κm cosD(Θ0(z))D(∂zV (t, z)) + C(z)e−κm(z) cosD(Θ0(z))t

2 ,802

where C = C(z) is a nonnegative random function.803

804



• Case F (Estimate for ∂zωM ′1): We use (5.2) to obtain, for a.e. t ≥ 0,805

∂t∂zωM ′1(t)

=1

N

N∑j=1

∂zκM ′1,j(z) cos(θj(t)− θM ′1(t))(ωj(t)− ωM ′1(t))

− 1

N

N∑j=1

κM ′1,j(z) sin(θj(t)− θM ′1(t))(∂zθj(t)− ∂zθM ′1(t))(ωj(t)− ωM ′1(t))

+1

N

N∑j=1

κM ′1,j(z) cos(θj(t)− θM ′1(t))(∂zωj(t)− ∂zωM ′1(t))

=:

3∑k=1

I7k.

(5.4)

806

807

We use the result and same arguments in Proposition 2.5 and Lemma 3.1 to obtain808

I71 ≤ ‖∂zκ‖∞D(V (t, z)) ≤ ‖∂zκ‖∞D(V 0(z))e−κm cosD(Θ0)t,

I72 ≤‖κ‖∞N

N∑j=1

|∂zθj(t, z)− ∂zθM ′1(t, z)||ωj(t, z)− ωM ′1(t, z)|

≤ ‖κ‖∞D(∂zΘ(t, z))D(V (t, z))

≤ ‖κ‖∞C1(z)D(V 0(z))e−κm cosD(Θ0)t,

I73 ≤1

N

N∑j=1

κm cosD(Θ0)(∂zωj(t, z)− ∂zωM ′1(t, z))

≤ −κm cosD(Θ0)∂zωM ′1(t, z).

(5.5)809

810

In (5.4), we combine all estimates in (5.5) to obtain811

∂t∂zωM ′1(t, z) ≤ −κm cosD(Θ0)∂zωM ′1

+ C(z)e−κm cosD(Θ0)t,(5.6)812813

where C = C(z) depends on D(∂rzV0(z)), ∂rzκij and D(∂rzΘ0(z)) for r = 0, 1 Similarly,814

∂t∂zωm′1(t, z) ≥ −κm cosD(Θ0)∂zωm′1

− C(z)e−κm cosD(Θ0)t.(5.7)815816

Now, we combine (5.6) and (5.7) to get the inequality (5.3) and set817

y := D(∂zV (t, z)), α := κm cosD(Θ0(z)) β :=α

2andC = C(z).818

Finally, we apply Lemma 2.4 to get the desired estimate.819

Next, we provide synchronizing property of ∂lzV as follows.820

Theorem 5.2. Suppose that the framework (F) in Section 3 hold, and let Θ =821

Θ(t, z) be a solution to system (1.2). Then, for all t ≥ 0 and each z ∈ Ω,822

D(∂lzV (t, z)) ≤ Fl(z)e−κm cosD(Θ0)t

2 ,823

where the nonnegative random variable Fl(z) depends on D(∂rzV0(z)), ∂rzκij and824

D(∂rzΘ0(z)) for r = 0, 1, · · · , l.825



Proof. We apply ∂lz to (5.1) to obtain826

∂t(∂lzωi)(t, z)827

=1

N

N∑j=1

κij(z) cos(θj − θi)(∂lzωj − ∂lzωi)828

− 1

N

N∑j=1

κij(z) sin(θj − θi)(∂lzθj − ∂lzθi)(ωj − ωi)829

+l

N

∑1≤j≤N

(k1, ··· , kl)∈Λ(l)kl=0

κij(z)(l − 1)!

k1! · · · kl−1!

∂

∂tM(l − 1, k1, · · · , kl−1,Θ, i, j)830

+1

N

∑1≤j≤N

1≤r≤l−1(k1, ··· , kr)∈Λ(r)

(l

r

)∂l−rz κij(z)

r!

k1! · · · kr!∂

∂tM(r, k1, · · · , kr,Θ, i, j)831

+1

N

∑1≤j≤N

∂lzκij(z) cos(θj − θi)(ωj − ωi).832

833

As in the proof of Theorem 3.2, we will proceed by induction. First we choose indices834

M ′l and m′l such that for t > 0 and z ∈ Ω,835

∂lzωM ′l (t, z) := maxi∂lzωi(t, z), ∂lzωm′l(t, z) := min

i∂lzωi(t, z).836

Note that for every z ∈ Ω, ∂lzωM ′l(·, z) and ∂lzωm′l

(·, z) are piecewise differentiable837

and Lipschitz continuous with respect to t. As we previously did, our objective is to838

derive the following inequality:839

(5.8) ∂tD(∂lzV (t, z)) ≤ −κm cosD(Θ0(z))D(∂lzV (t, z)) + C(z)e−κm(z) cosD(Θ0(z))t

2 ,840

where C = C(z) is a nonnegative random function.841

842

Now we consider the estimate ∂lzωM ′las follows: for a.e. t ≥ 0,843

∂t(∂lzωM ′l

)(t, z)844

=1

N

N∑j=1

κM ′l ,j(z) cos(θj − θM ′l )(∂

lzωj − ∂lzωM ′l )845

− 1

N

N∑j=1

κM ′l ,j(z) sin(θj − θM ′l )(∂

lzθj − ∂lzθM ′l )(ωj − ωM ′l )846

+l

N

∑1≤j≤N

(k1, ··· , kl)∈Λ(l)kl=0

κM ′l ,j(z)

(l − 1)!

k1! · · · kl−1!

∂

∂tM(l − 1, k1, · · · , kl−1,Θ,M

′

l , j)847

+1

N

∑1≤j≤N

1≤r≤l−1(k1, ··· , kr)∈Λ(r)

(l

r

)∂l−rz κM ′l ,j

(z)r!

k1! · · · kr!∂

∂tM(r, k1, · · · , kr,Θ,M

′

l , j)848



+1

N

∑1≤j≤N

∂lzκM ′l j(z) cos(θj − θM ′l )(ωj − ωM ′l )849

=:

5∑k=1

I8k.850

851

Next, we estimate the terms I8k as follows.852

853

• Case G.1 (Estimate of I81 and I82): In this case,854

I81 ≤ −κm cosD(Θ0)∂lzωM ′l,

I82 ≤ ‖κ‖∞D(∂lzΘ(t, z))D(V (t, z)) ≤ ‖κ‖∞Cl(z)D(V 0(z))e−κm cosD(Θ0)t.855

• Case G.2 (Estimate of I83 and I84): We first estimate ∂∂t

[M(r, k1, · · · , kr,Θ,M

′

l , j)]

856

as follows:857

∂

∂t

[M(r, k1, · · · , kr,Θ,M

′

l , j)]

858

=∂

∂t

(sin(k)(θj − θM ′l )

) r∏p=1

(∂pzθj − ∂pzθM ′l

p!

)kp859

+∑

1≤p≤rkp 6=0

kp

∏s6=p

(∂szθj − ∂szθM ′l

s!

)ks (∂pzωj − ∂pzωM ′l )

(∂pzθj − ∂pzθM ′l

p!

)kp−1

860

≤ D(V (t, z))

r∏p=1

(Cp(z)

p!

)kp861

+∑

1≤p≤rkp 6=0

kp

∏s6=p

(Cs(z)

s!

)ksD(∂pzV (t, z))

(Cp(z)

p!

)kp−1

862

≤ D(V 0(z))

r∏p=1

(Cp(z)

p!

)kpe−κm cosD(Θ0)t

863

+

∑1≤p≤rkp 6=0

kp

∏s 6=p

(Cs(z)

s!

)ksFp(z)(Cp(z)

p!

)kp−1

e−κm cosD(Θ0)t2864

≤ C(z)e−κm cosD(Θ0)t

2 .865866

This yields867

I83 + I84 ≤ C(z)e−κm cosD(Θ0)t

2 ,868

where C = C(z) depends on D(∂rzV0(z)), ∂rzκij and D(∂rzΘ0(z)) for r = 0, 1, · · · , l.869

870

• Case G.3 (Estimate of I85): In this case,871

I85 ≤ ‖∂lzκ‖∞D(V (t, z)) ≤ ‖∂lzκ‖∞D(V 0(z))e−κm cosD(Θ0)t.872



Now we combine all results in Case G.1 - Case G.3 to obtain873

(5.9) ∂t(∂lzωM ′l

)(t, z) ≤ −κm cosD(Θ0)∂lzωM ′l(t, z) + C(z)e−

κm cosD(Θ0)t2 ,874

where the nonnegative random variable C(z) depends on D(∂rzV0(z)), ∂rzκij and875

D(∂rzΘ0(z)) for r = 0, 1, · · · , l. Similarly, we have876

(5.10) ∂t(∂lzωm′l

)(t, z) ≥ −κm cosD(Θ0)∂lzωm′l(t, z)− C(z)e−

κm cosD(Θ0)t2 ,877

where C = C(z) is the same function in (5.9). Finally, it follows from (5.9) and (5.10)878

that we have the inequality (5.8) and then, Lemma 2.4 yields the desired result.879

5.2. Remarks on the consensus model. In this subsection, we address how880

our local sensivity argument in previous subsections can be applied to other random881

consensus model as well. Consider the following consensus model [10, 13, 29] with882

random inputs:883

(5.11) ∂txi = Fi(z) +

N∑j=1

aij(z)η(xj − xi), 1 ≤ i ≤ N,884

where xi ∈ R, aij ≥ 0 and η : R→ R is an odd, analytic and bounded function whose885

derivatives η(k) are bounded and there exists Rη > 0 such that886

η(x), η′(x) > 0, x ∈ (0, Rη).887

Note that the above assumption can be satisfied by sine function or arctangent func-888

tion. Then, under the framework below, the same argument in Section 3- Section889

5 would yield the relaxation dynamics of vi = ∂txi’s toward relative equilibria and890

`1-stability for the random dynamical systems of the type (5.11):891

• (Fg1): (Initial boundedness of z-variations and mean-zero condition)892

0 < D(X0(z)) < Rη, sup0≤r≤l

D(∂rzX0(z)) <∞,

∑i

x0i (z) = 0.893

• (Fg2): (Uniform boundedness and mean-zero condition for external forces Fi)894

sup0≤r≤l

D(∂rzF (z)) <∞,∑i

Fi(z) = 0.895

• (Fg3): Large coupling regime:896

am(z) := mini,j

aij(z) >D(F (z))

η(D(X0(z))

) , max0≤r≤l

‖∂rza(z)‖∞ ≤ a∞ <∞.897

6. Numerical simulations. In this section, we provide several numerical sim-898

ulations supporting our theoretical results, and give some insights for the cases in899

the low coupling regime which is beyond our analytical framework. For the simula-900

tions, we used the fourth-order Runge-Kutta method, and we employed the following901

specific setting for the implementation of numerical simulations:902



• For simplicity, we focus on the randomness in the coupling strength so that903

natural frequencies and initial data would be regarded as deterministic data.904

Moreover, we assume the all-to-all coupling condition κij = κ.905

• We consider the dynamics of 100 oscillators and they satisfy the following906

initial conditions:907

D(V) = 0.9588 and D(Θ0) = 0.9915.908

• Coupling strength κ follows the uniform distribution on the interval [Kl, 2Ks],909

where Kl and Ks are given by910

Kl :=ND(V)

2(N − 1), Ks :=

D(V)

sinD(Θ0),911

where N is the number of oscillators. Here, κ > Kl is the necessary condition912

for the emergence of synchronization as discussed in [12].913

Recall that our framework implies that the relaxation dynamics of z-variations of914

frequency process can be observed if κ > Ks. Next, we present numerical simulation915

results.916

First, we provide results supporting our theoretical results. In every graph, Each917

line denotes the value for the diameter of (z-variations of) phase process or frequency918

process for each coupling strength selected from the uniform distribution. Thick line919

denotes the mean value and shaded region denotes the 95% confidence interval.920

If κ is restricted to (Ks, 2Ks], then each κ satisfies our framework for relaxation921

dynamics of z-variations. Thus, we can observe the emergence of uniform bound-922

edness of z-variations of phase process and exponential relaxation of z-variations of923

frequency process for each κ (see Figure 1- Figure 3).924

925

(a) D(Θ(t)) is bounded for each choice of κ. (b) For each case, it exhibits an exponentialrelaxation to the average.

Fig. 1. Zeroth z-variations in a large coupling regime



(a) D(∂zΘ(t)) is bounded, which confirmsLemma 3.1.

(b) D(∂zV (t)) shows an exponential decay,which agrees with Lemma 5.1.

Fig. 2. First z-variations in a large coupling regime

(a) D(∂2zΘ(t)) is bounded, as in Theorem 3.2. (b) For each case, D(∂2zV (t)) exhibits an ex-ponential relaxation, which confirms Theorem5.2.

Fig. 3. Second z-variations in a large coupling regime

Next, we consider the case that the random coupling strength κ follows the original926

uniform distribution on [Kl, 2Ks]. Our main concern is for the case where relaxation927

dynamics is observed. Here, we even investigate the cases for asynchronization and928

unboundedness of diameter of phase process for possible future estimates. Graphs for929

the diameter of the zeroth, first and second z-variations of processes are presented in930

Figure 4-Figure 6, respectively. For the mean value and 95% confidence interval, we931

presented them in Figure 7- Figure 9 for each z-variation.932

933

As we can observe, when D(Θ(t)) is not bounded, it increases almost linearly in934

most cases. On the other hand, when D(V (t)) does not exhibit relaxation dynamics,935

it changes aperiodically and rapidly (see Figure 4). For the first z-variations, it can936

be observed that for the asynchronization case, D(∂zΘ(t)) and D(∂zV (t)) changes937

drastically after some time (see Figure 5). This type of dynamics appears in the938

dynamics of D(∂2zΘ(t)) and D(∂2

zV (t)) as well (see Figure 6). Moreover, we can939

observe that the dynamics of mean values and confidence intervals of each z-variations940

almost follows the dynamics of asynchronization cases (see Figure 7-9).941



(a) For the unbounded cases, D(Θ(t)) increasesalmost linearly.

(b) For asynchronizing cases, D(V (t)) changesrapidly.

Fig. 4. Zeroth z-variations

(a) For the unbounded cases, D(∂zΘ(t)) ex-hibits a drastic change after some time.

(b) Similar to D(∂zΘ(t)), D(∂zV (t)) drasticallychanges after some time.

Fig. 5. First z-variations

(a) Similar dynamics to D(∂zΘ(t)) is observed. (b) Similar dynamics to D(∂zV (t)) is observed.

Fig. 6. Second z-variations



(a) Mean value and 95% confidence interval forD(Θ(t)) increases steadily

(b) Initially, 95% confidence interval is quite nar-row, but as asynchronization happens, it gets abit larger.

Fig. 7. Mean value and confidence interval for the zeroth z-variations

(a) Mean value and 95% confidence intervalchange drastically after some time.

(b) Simillar to D(∂zΘ(t)), data change drasti-cally after some time.

Fig. 8. Mean value and confidence interval for the first z-variations

(a) Similar dynamics to D(∂zΘ(t)) is observed. (b) Similar dynamics to D(∂zV (t)) is observed.

Fig. 9. Mean value and 95% confidence interval for the second z-variations



7. Conclusion. In this paper, we studied local sensitivity analysis for the ran-942

dom Kuramoto model with pairwise symmetric coupling strengths. More precisely, we943

provided a sufficient framework leading to the uniform bound for diameter and uni-944

form stability estimate for phase variations and synchronization property of frequency945

variations. Our framework is explicitly expressed in terms of initial data, distributed946

natural frequencies and coupling strengths. Our results reveal the stochastic robust-947

ness of synchronizing property of the Kuramoto ensemble in a large coupling regime.948

Of course, there are several unresolved problems to be explored. For example, in a949

small coupling regime and intermediate coupling regime, the dynamics of the Ku-950

ramoto model in a deterministic setting is itself not clearly understood at present,951

not to mention uncertainty quantification. More precisely, the phase transition like952

phenomenon from the disordered state to ordered state occurs at a critical coupling953

strength in a mean-field setting. Thus, how does the uncertainty affects in this phase-954

transition like process? Another interesting project is to understand the interplay955

between the mean-field limit and uncertainty, which will be be pursued in a future956

work.957

Acknowledgments. The work of S.-Y. Ha was supported by National Research958

Foundation of Korea(NRF-2017R1A2B2001864), and the work of S. Jin was supported959

by NSFC grant No. 31571071, NSF grants DMS-1522184 and DMS-1107291: RNMS960

KI-Net, and by the Office of the Vice Chancellor for Research and Graduate Education961

at the University of Wisconsin. The work of J. Jung is supported by the German962

Research Foundation (DFG) under the project number IRTG2235.963

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Local sensitivity analysis for the Kuramoto model with ...jin/PS/Kuramoto-2.pdf · 1 local sensitivity analysis for the kuramoto model 2 with random inputs in a large coupling regime