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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) (syha@snu.ac.kr).‡School of Mathematical Sciences, MOE-LSC, and Institute of Natural Sciences, Shanghai Jiao
Tong University, Shanghai 200240, China (shijin-m@sjtu.edu.cn).§Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea (Republic
of) (warp100@snu.ac.kr).
1
This manuscript is for review purposes only.
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
This manuscript is for review purposes only.
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
This manuscript is for review purposes only.
4 S.-Y. HA, S. JIN AND J. JUNG
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
This manuscript is for review purposes only.
LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 5
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
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6 S.-Y. HA, S. JIN AND J. JUNG
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
This manuscript is for review purposes only.
LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 7
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
This manuscript is for review purposes only.
8 S.-Y. HA, S. JIN AND J. JUNG
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
This manuscript is for review purposes only.
LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 9
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
This manuscript is for review purposes only.
10 S.-Y. HA, S. JIN AND J. JUNG
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
This manuscript is for review purposes only.
LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 11
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
This manuscript is for review purposes only.
12 S.-Y. HA, S. JIN AND J. JUNG
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
This manuscript is for review purposes only.
LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 13
• 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
This manuscript is for review purposes only.
14 S.-Y. HA, S. JIN AND J. JUNG
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
This manuscript is for review purposes only.
LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 15
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
This manuscript is for review purposes only.
16 S.-Y. HA, S. JIN AND J. JUNG
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
This manuscript is for review purposes only.
LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 17
+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!
)kp e−κm cosD(Θ0(z))t
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
This manuscript is for review purposes only.
18 S.-Y. HA, S. JIN AND J. JUNG
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!
)kp e−κm cosD(Θ0(z))t
2 ,
I43 ≥ −
∑
1≤r≤l−1(k1, ··· , kr)∈Λ(r)
(l
r
)‖∂l−rz κ(z)‖∞
r!
k1! · · · kr!
(Dp(z)p!
)kp e−κm cosD(Θ0(z))t
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
This manuscript is for review purposes only.
LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 19
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
This manuscript is for review purposes only.
20 S.-Y. HA, S. JIN AND J. JUNG
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
This manuscript is for review purposes only.
LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 21
∑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
This manuscript is for review purposes only.
22 S.-Y. HA, S. JIN AND J. JUNG
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
This manuscript is for review purposes only.
LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 23
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
Then for all t ≥ 0 and each z ∈ Ω,633
‖∂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
This manuscript is for review purposes only.
24 S.-Y. HA, S. JIN AND J. JUNG
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
Then for all t ≥ 0 and each z ∈ Ω,658
∂
∂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
(∂pzθj − ∂pzθi
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
This manuscript is for review purposes only.
LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 25
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
This manuscript is for review purposes only.
26 S.-Y. HA, S. JIN AND J. JUNG
× 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
(∂pzθj − ∂pzθi
p!
)kp
+(
sin(k)(θj − θi)) r∏
p=1
(∂pzθj − ∂pzθi
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−
(∂pz θj − ∂pz θi
p!
)kp ∣∣∣∣∣∣707
≤
0 if kp = 0,
kp
(Cp(z)
)kp−1
|∂pzθj − ∂pzθi − ∂pz θj + ∂pz θi| if kp > 0,708
709
This manuscript is for review purposes only.
LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 27
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
×
(∂pzθj − ∂pzθi
p!
)kp−
(∂pz θj − ∂pz θi
p!
)kp ∣∣∣∣∣717
≤r∑p=1
[ ∏0≤µ≤r+1µ6=p
(Cµ(z)
µ!
)kµ ∣∣∣∣∣(∂pzθj − ∂pzθi
p!
)kp−
(∂pz θj − ∂pz θi
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
(|∂pzθj − ∂pz θj |+ |∂pzθi − ∂pz θi|
),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
This manuscript is for review purposes only.
28 S.-Y. HA, S. JIN AND J. JUNG
×∑
1≤i,j≤N0≤p≤l−1
(|∂pzθj − ∂pz θj |+ |∂pzθi − ∂pz θi|
)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
(|∂pzθj − ∂pz θj |+ |∂pzθi − ∂pz θi|
)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
Then for all t ≥ 0 and each z ∈ Ω,752
‖∂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
This manuscript is for review purposes only.
LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 29
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
This manuscript is for review purposes only.
30 S.-Y. HA, S. JIN AND J. JUNG
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
This manuscript is for review purposes only.
LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 31
• 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
This manuscript is for review purposes only.
32 S.-Y. HA, S. JIN AND J. JUNG
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
This manuscript is for review purposes only.
LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 33
+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
This manuscript is for review purposes only.
34 S.-Y. HA, S. JIN AND J. JUNG
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
This manuscript is for review purposes only.
LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 35
• 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
This manuscript is for review purposes only.
36 S.-Y. HA, S. JIN AND J. JUNG
(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
This manuscript is for review purposes only.
LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 37
(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
This manuscript is for review purposes only.
38 S.-Y. HA, S. JIN AND J. JUNG
(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
This manuscript is for review purposes only.
LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 39
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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