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arXiv:nlin/0509022v3 [nlin.AO] 24 Feb 2006

Coarse-graining the dynamics of coupled oscillators

Sung Joon Moon1, R. Ghanem2, and I. G. Kevrekidis1∗

1Department of Chemical Engineering & Program in Applied and Computational Mathematics (PACM),

Princeton University, Princeton, NJ 08544, USA

2Department of Civil Engineering, University of Southern California, Los Angeles, CA 90089, USA

(Dated: February 5, 2008)

We present an equation-free computational approach to the study of the coarse-grained dynamics

of finite assemblies of non-identical coupled oscillators at and near full synchronization. We use

coarse-grained observables which account for the (rapidly developing) correlations between phase

angles and oscillator natural frequencies. Exploiting short bursts of appropriately initialized detailed

simulations, we circumvent the derivation of closures for the long-term dynamics of the assembly

statistics.

PACS numbers: 05.45.Xt,05.10.-a,02.70.Dh,87.10.+e

Since Winfree’s pioneering work in 1960’s [1], cou-

pled oscillator models have been investigated exten-

sively. Some exact results on the collective dynamics for

an infinite number of coupled oscillators (the so-called

continuum-limit) [2, 3, 4] have shed light on synchro-

nization phenomena in biological [1, 2, 5, 6, 7, 8], chemi-

cal [9, 10], and physical systems [11, 12]. However, even

in this ideal limit, some basic questions including global,

quantitative stability of asymptotic states, still remain

open [13, 14, 15, 16]. Many real-world systems consist

of a large, finite number of non-identical entities, where

statistical techniques for the continuum-limit are not di-

rectly applicable. Exploring and understanding the dy-

namics of such finite oscillator assemblies is an important

topic (e.g., see Ref. [17]).

We present a computer-assisted approach to modeling

the coarse-grained dynamics of such large, finite oscillator

assemblies at and near full synchronization. The premise

is that there exist a small number of coarse-grained vari-

ables (observables) adequately describing the long-term

dynamics, and that a closed evolution equation for these

observables exists, but is not explicitly available. To ac-

count for oscillator variability within the assembly, we

treat both the variable oscillator properties (here, natu-

ral frequencies ω) and the oscillator states (here, phase

angles θ) as random variables. Recognizing a quick de-

velopment of correlations between ω and θ, we express

the latter as a polynomial expansion of the former (bor-

rowing Wiener polynomial chaos (PC) tools [18]); the PC

expansion coefficients are our coarse observables.

Availability of the governing equations for the variables

of interest is a prerequisite to modeling and computation.

We circumvent this step using the recently-developed

equation-free (EF) framework for complex, multiscale

systems modeling [19, 20, 21].

can perform system-level computational tasks without

explicit knowledge of the coarse-grained equations; these

unavailable equations are solved by designing, performing

and processing the results of short bursts of appropriately

initialized detailed (fine-scale, microscopic) simulations.

In this framework we

We consider a paradigmatic model of coupled oscil-

lators, the Kuramoto model, consisting of a population

of N all-to-all, phase-coupled limit-cycle oscillators with

i.i.d. ω with distribution function g(ω). This model has

been extensively studied because of its simplicity and cer-

tain mathematical tractability, yet it is not merely a toy

model. It appears as a normal form for general systems

of coupled oscillators (e.g. Refs. [10, 11]).

We choose a Gaussian with standard deviation σω =

0.1 for g(ω); however, our approach is not limited to this

particular choice, nor to the Kuramoto model. Due to

rotational symmetry, the mean frequency Ω =?

can be set to 0 without loss of generality. The governing

equation for the phase angle of the ith oscillator θiis

iωi/N

dθi

dt

= ωi+K

N

N

?

j=1

sin(θj− θi),1 ≤ i ≤ N,(1)

where K ≥ 0 is the coupling strength. Spontaneous syn-

chronization (phase-locking) occurs at sufficiently large

K. As K decreases across a critical value Ktp, more and

more oscillators desynchronize until they all essentially

evolve with their own frequencies below another critical

value Kc[3, 13, 15]. Kuramoto [3] introduced a complex

order parameter reiψ=

N

?N

time states; the effective radius r(t) measures the phase

coherence; see also Ref. [22] for an order function. The

asymptotic value of r (t → ∞) in the continuum-limit

(N → ∞) exhibits a temporal analog of phase transition

at Kc[3].

The order parameter r conveniently represents statis-

tical behaviors around the critical point K = Kc; how-

ever, r does not uniquely specify the microscopic state,

and it may not adequately describe transient dynamics.

The statistical moments of the phase angle distribution

?N

a positive integer, are a “natural” first choice of coarse-

grained observables (in a kinetic theory-like description).

Due to the symmetry, we consider only even-order mo-

ments, and test whether a closure in terms of M2 and

1

j=1eiθjto describe the long-

function Mn≡

1

N

j=1

?

θj−

1

N

?N

i=1θi

?n

, where n is

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2

0.010.015 0.020.025

0

0.5

1

1.5

2

x 10

−3

M2

M4

05 10

0

1

2

x 10

−3

time

M4

initial

state for

constrained

integration

final

state for

constrained

integration

FIG. 1:

microscopically different initializations (dashed, dotted and

dot-dashed lines; see text) evolve along different trajectories,

to a slow manifold and, ultimately, the same synchronized

state (N = 300; K = 0.7 > Ktp). Constraining the evolution

to M2 = 0.017 (solid line) guides the trajectory directly to

this slow manifold; the inset shows M4 becoming slaved to

M2 during this constrained evolution by t ≈ 2.0.

Three coarsely identical (same M2 and M4) but

M4 is likely for K ≥ Ktp. We prepare several distinct

initial phase angle distributions with identical coarse-

grained values (M2 = 0.017 and M4 = 0.0020); these

phase angles are randomly assigned to oscillators. The

phase portrait in Fig. 1 shows direct simulation [using

Eq. (1)] for three of these initial assemblies; the trajec-

tories are clearly distinct, suggesting that the dynamics

cannot close simply on these two observables. Including

higher order moments (such as M6) as observables does

not remedy the situation. It is also clear, however, that

the long-term dynamics lie on a low-dimensional man-

ifold (ultimately a one-dimensional one) towards which

all trajectories are eventually attracted.

The dynamical differences among the three cases arise

from the microscopic differences of the (macroscopically

identical) initial conditions; this is best seen in the ω-θ

plane (Fig. 2). Correlations between θ and ω develop

(the initial “cloud” in the ω-θ plane quickly evolves to

a “curve”), as all transients initially approach the slow

manifold: The oscillators “sort themselves out”. These

correlations were not accounted for when we assigned an-

gles randomly to oscillators in the assembly.

We now include a “remedial initialization” step, evolv-

ing the dynamics by constraining them on prescribed val-

ues of the moments, as a system of differential algebraic

equations (DAEs) using Lagrange multipliers. The solid

line in Fig. 1 shows this preparatory step with a con-

straint on M2only; constrained evolution brings the as-

sembly down to the right point on the slow manifold,

and the same “sorting” develops as in the aforementioned

freely-evolving cases. Phase angle statistics alone do not,

therefore, constitute good observables [23]; ω-θ correla-

tions should be accounted for in the coarse description.

Motivated by this observation, we explore the long-

−0.40 0.4

−0.5

0

0.5

θ−<θ>

−0.40 0.4

−0.5

0

0.5

ω−<ω>

θ−<θ>

−0.40 0.4

−0.5

0

0.5

ω−<ω>

−0.40 0.4

−0.5

0

0.5

−0.40 0.4

−0.5

0

0.5

−0.40 0.4

−0.5

0

0.5

−0.40 0.4

−0.5

0

0.5

(a) t = 0

(b) t = 1.0

(c) t = 2.0 (d) t = 6.0

FIG. 2:

tion (main panels; dashed line in Fig. 1) and for constrained

evolution (insets; solid line in Fig. 1) respectively. Each dot

represents an oscillator, and (a) to (d) are snapshots at t = 0,

1.0, 2.0, and 6.0, respectively, marked by filled circles in Fig. 1.

Strong correlations develop during the initial transient stages

(“oscillator sorting”).

Time snapshots in the ω − θ plane for free evolu-

term dynamics with a different set of observables, treat-

ing both θ and ω as random variables. The former is

expanded in Hermite polynomials of the latter, Gaus-

sian random variable; Wiener polynomial chaos is the

appropriate choice for Gaussian distributions. General-

ized polynomial chaos (gPC) [24] would be invoked for

different frequency distributions (e.g. we also successfully

used Legendre polynomial expansions for uniform g(ω)).

For convenience, we introduce the normalized random

variable ξ ≡ ω/σω:

θ(ω,t) =

p

?

i=0

αi(t)Hi(ξ) =

p

?

i=0

?θ,Hi?

?Hi,Hi?Hi(ξ), (2)

where p is the highest order retained in the truncated

series, ?· ,·? denotes the inner product with respect to

the Gaussian measure, and Hi is the ith Hermite poly-

nomial [H0(x) = 1,H1(x) = x,H2(x) = x2− 1,H3(x) =

x3− 3x,···]. Only odd-order αi’s are considered, due to

symmetry. We will see that here the first two nonvan-

ishing coefficients α1 and α3 provide an adequate rep-

resentation. Given a particular detailed realization of

the oscillator state, its PC coefficients αi’s are estimated

through a least squares fitting algorithm, interpreting θ

as an empirical function f(ξ) ≡ α1ξ + α3

minimizing the residual R2≡

This procedure corresponds to the restriction (fine to

coarse) step in the EF framework, described below.

In the EF approach, appropriately initialized short

bursts of detailed, fine-scale simulation are used to esti-

mate quantities pertaining to the evolution of the coarse-

grained variables (observables).

coarse-grained model in terms of the first few PC coeffi-

cients, we estimate the quantities necessary for scientific

computation with it (time derivatives, action of Jaco-

?ξ3− 3ξ?

and

?

j[θj− f(ξj;α1,α3)]2.

Lacking an explicit

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3

0 102030

0

0.1

0.2

0.3

time

(a)

projective

integration

direct

integ−

ration

α1

α3× 10

tn−1 tn tn+M

δ

Mδ

α1,n+M

α1,n

α1,n−1

projection

0 20 4060 80

−0.4

−0.2

0

0.2

0.4

time

α1

α3× 10

θfree/ 40

(b)

02040 6080

−0.4

−0.2

0

0.2

0.4

time

α1

α3× 10

θ1

free/ 40

(c)

θ2

free/ 40

FIG. 3: (color online) Coarse projective integration (dots) and detailed coupled oscillator dynamics (lines); N = 300. (a) Two

PC coefficients (K = 0.4; full synchronization). (b) Two PC coefficients and a single “free” oscillator (K = 0.31). (c) Two

PC coefficients and two “free” oscillators (K = 0.31). Natural frequencies are newly drawn from g(ω) at each lifting step (see

text). Inset in (a): Schematic of a projective integration step: The last part (last two dots, at tn−1 and tn) of a short burst of

direct integration (five dots) is used to estimate the local time derivative (solid line). PC values at a future time t = tn+M are

“projected” through forward Euler, i.e., α1|t=tn+M= α1|t=tn+

α1,n−α1,n−1

tn−tn−1

(tn+M − tn).

bians, residuals) through on demand numerical experi-

mentation with the detailed, fine-scale model [Eq. (1)].

The general procedure consists of (i) identifying good

observables that sufficiently describe the coarse-grained

dynamical state (here, a few αi’s), (ii) constructing a lift-

ing operator, mapping the coarse description to one (or

more, for variance reduction purposes) consistent fine-

scale realization(s) [randomly drawing ω from g(ω) and

assigning θ, using Eq. (2) and given αivalues], (iii) evolv-

ing the lifted, fine-scale initial conditions for certain time

horizon, (iv) restricting the resulting fine-scale descrip-

tion to the coarse observables [finding the PC coefficients

of the final state], and (v) repeating the procedure as nec-

essary to perform specific scientific computation steps.

This is a general approach that has been combined with

various fine-scale models [19, 25]; see Refs. [20, 21].

We first demonstrate coarse projective integration [26].

Each group of five dots in Fig. 3 represents the time hori-

zon during which the detailed equations are integrated to

enable the projective step; the local time derivatives of

the observables are estimated here simply from the last

two dots in each group. Coarse variable values at a pro-

jected, future time are estimated using these derivatives

and (for projective forward Euler) linear extrapolation

in time [see the inset in Fig. 3 (a)]. After the projection

step we lift the coarse variables to consistent fine-scale

realizations, and use these as the initial condition for an-

other short burst of direct detailed integration [steps (ii)

and (iii) above]. Depending on the relative lengths of

the projection step (Mδ) and the short run required to

estimate the coarse time derivatives (nδ), this procedure

may significantly accelerate the computational evolution

of the oscillator statistics; the cost of the lifting step (here

negligible) must also be considered. At each lifting step,

ω was newly drawn from g(ω), and the full integration

(lines) and projective integration (dots) agree on the level

of fluctuation among realizations. The PC coefficients

display smooth behavior, nearly independently of partic-

ular random draws; for the same random draw at every

step, results would be even better. Projective integra-

tion in Fig. 3 (a) reduced the computational effort in our

illustrative direct integration by a factor of four. The nu-

merical analysis of projective integration (stability, accu-

racy, stepsize selection and estimation issues) is a topic

of current research (see e.g., Refs. [20, 26, 27]); here we

simply demonstrated the procedure and its potential.

Slightly below the transition value Ktp, where only few

oscillators become desynchronized, we consider the sys-

tem as a combination of synchronized “bulk” and a few

“free” oscillators. Good coarse-grained observables then

are a few PC coefficients for the “bulk” synchronized os-

cillators and the phase angle(s) of the (few) desynchro-

nized one(s). The EF approach can be directly “wrapped

around” this alternative representation.

one free and two free oscillator cases, projective inte-

grations on the new observables successfully track (and

accelerate) direct detailed simulations [Figs. 3 (b) and

(c)]. These “good observables”are suggested by direct in-

spection and common sense; for more complicated, high-

dimensional problems, good state parameterizations re-

quire modern data mining algorithms. Diffusion maps on

graphs constructed by the data [29] are a promising tool

for detecting good “reduction coordinates” (observables)

on which to base EF computations.

Direct, long-time simulation is often inefficient in com-

puting long-time (stationary) states. Numerical bifurca-

tion algorithms, more appropriate for stability and para-

metric analysis, can be implemented in an equation-free

framework: The residual and the action of the unavail-

able Jacobian are numerically estimated through short

bursts of appropriately initialized detailed simulations.

Starting from a coarse-grained initial condition, we lift,

and integrate the full model for time ∆T. We then re-

strict to the observables of the final state Φ∆T; this is the

Both for the

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4

0.30.50.7

0

1

2

3

4

K

α1×10

θfree

Coarse−grained variables

FIG. 4: (color online) Coarse bifurcation diagram for the full

synchronization regime (K ≥ Ktp), obtained through coarse

Newton-GMRES method and pseudo arc-length continuation

(N = 300). The same variables as in Fig. 3 (b) are used;

the phase angle of the single “free” oscillator (θfree) is an ex-

tra observable (its natural frequency is positive in this case).

The PC coefficients, obtained by discounting the “free” os-

cillator, exhibit nearly the same values both for the stable

(filled circles) and the unstable (open circles) branch (only α1

is shown here). Only θfreeshows significant variation along

the two branches. Arrows are included to guide the eye.

coarse time-stepper. We now solve for the fixed point sat-

?

θfree

θfree

Newton-GMRES [28], a matrix-free iterative method (to-

gether with the pseudo arc-length continuation); addi-

tional coarse observables θfreeare appended when nec-

essary. We construct bifurcation diagrams like the one in

(Fig. 4) with respect to the parameter K, showing a turn-

ing point (actually, a “sniper”) bifurcation at K = Ktp.

A single oscillator (whose phase angle θfreeis treated as

a separate coarse observable) becoming “free” from the

synchronized “bulk” at that point. For sufficiently large

K values (when r ≈ 1) analytical estimates of certain

elements of the shape of the ω-θ correlation become pos-

sible (e.g., from Eq. (1) one can obtain, at steady state,

α1 ≈ σω/K, in reasonable agreement with our steady

state computations at K >∼ 0.5).

In summary, the EF multiscale approach was success-

fully used for coarse-grained dynamic computations of

finite assemblies of non-identical coupled oscillators; the

derivation of explicit closures at- and close to the syn-

chronization regime was circumvented. Initial transient

“sorting” of the oscillators, establishing correlations be-

tween natural frequencies and phase angles, suggested

Wiener PC coefficients as the appropriate coarse observ-

ables. If the problem dynamics can be coarse-grained,

traditional numerical analysis algorithms can be used as

protocols for the “intelligent” design of short bursts of

computational experiments with the detailed, fine-scale

model. The approach can be directly generalized to an-

alyze the simulation and modeling of more complicated

oscillator dynamics.

This work was supported by DARPA and by the Na-

isfying Φ∆T

α

?

−

?

α

?

= 0, using the coarse

tional Science Foundation.

∗yannis@arnold.princeton.edu

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