LOCAL SENSITIVITY ANALYSIS FOR THE KURAMOTO MODEL 1 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, 4 and the Kuramoto model serves as a prototype synchronization model for phase-coupled oscillators. 5 In this paper, we provide local sensitivity analysis for the Kuramoto model with random inputs in 6 initial data, distributed natural frequencies and coupling strengths, which exhibits the interplay be- 7 tween random effects and synchronization nonlinearity. Our local sensitivity analysis provides some 8 understanding on the robustness of emergent dynamics of the random Kuramoto model in a large 9 coupling regime, including “propagation and vanishment of uncertainties” and “continuous depen- 10 dence” of phase and frequency variations in random parameter space with respect to the variations 11 on the initial data. 12 Key words. Kuramoto model, synchronization, local sensitivity analysis, random communica- 13 tion, uncertainty quantification 14 AMS subject classifications. 15B48, 92D25 15 1. Introduction. Complex oscillatory systems often exhibit collective coherent 16 behaviors, e.g., flashing of fireflies, chorusing of crickets, synchronous firing of cardiac 17 pacemaker and metabolic synchrony in yeast cell suspension etc [1, 7, 34, 43]. The 18 rigorous mathematical treatment of such problems began from the pioneering works 19 [30, 43] by Kuramoto and Winfree about half century ago. They introduced simple 20 phase models for weakly coupled limit-cycle oscillators, and showed how collective 21 coherent behavior can emerge from the interplay between intrinsic randomness in 22 natural frequency and nonlinear attractive phase couplings. This coherent motion is 23 often called ”synchronziation” which means the adjustment of rhythms in an ensemble 24 of weakly coupled oscillators. Recently, the synchronization of oscillators on networks 25 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. We 29 first briefly introduce the Kuramoto model (see Section 2 for its basic mathematical 30 structures). 31 Let θ i = θ i (t) be the phase of the i-th limit-cycle oscillator, and we assume 32 that the Kuramoto oscillators are located on a symmetric network whose interac- 33 tion(connection) topology is denoted by the coupling matrix K =(κ ij ). In this 34 setting, the evolution of phases is governed by the first-order system of ordinary dif- 35 ferential equations [31, 30]: 36 (1.1) ˙ θ i = ν i + 1 N N X j=1 κ ij sin(θ j - θ i ), 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.
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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) ([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
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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