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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
0.010.0150.02 0.025
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-
−0.4 00.4
−0.5
0
0.5
θ−<θ>
−0.40 0.4
−0.5
0
0.5
ω−<ω>
θ−<θ>
−0.400.4
−0.5
0
0.5
ω−<ω>
−0.400.4
−0.5
0
0.5
−0.400.4
−0.5
0
0.5
−0.400.4
−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
010 2030
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 204060 80
−0.4
−0.2
0
0.2
0.4
time
α1
α3× 10
θfree/ 40
(b)
020406080
−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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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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