Physica D 143 (2000) 1–20
From Kuramoto to Crawford: exploring the onset of
synchronization in populations of coupled oscillators
Steven H. Strogatz
Center for Applied Mathematics and Department of Theoretical and Applied Mechanics, Kimball Hall, Cornell University,
Ithaca, NY 14853, USA
Abstract
The Kuramoto model describes a large population of coupled limit-cycle oscillators whose natural frequencies are drawn
from some prescribed distribution. If the coupling strength exceeds a certain threshold, the system exhibits a phase transition:
some of the oscillators spontaneously synchronize, while others remain incoherent. The mathematical analysis of this bifur-
cation has proved both problematic and fascinating. We review 25 years of research on the Kuramoto model, highlighting
the false turns as well as the successes, but mainly following the trail leading from Kuramoto’s work to Crawford’s recent
contributions. It is a lovely winding road, with excursions through mathematical biology, statistical physics, kinetic theory,
bifurcation theory, and plasma physics. © 2000 Elsevier Science B.V. All rights reserved.
Keywords: Kuramoto model; Coupled oscillators; Kinetic theory; Plasma physics
1. Introduction
In the 1990s, Crawford wrote a series of papers about the Kuramoto model of coupled oscillators [1–3]. At first
glance, the papers look technical, maybe even a bit intimidating.
Forinstance, take a look at “Amplitudeexpansionsfor instabilities in populations ofglobally coupled oscillators”,
his first paper on the subject [1]. Here, Crawford racks up 200 numbered equations as he calmly plows through a
center manifold calculation for a nonlinear partial integro-differential equation.
Technical, yes, but a technical tour de force. Beneath the surface, there is a lot at stake. In his modest, methodical
way, Crawford illuminated some problems that had appeared murky for about two decades.
My goal here is to set Crawford’s work in context and to give a sense of what he accomplished. The larger setting
is the story of the Kuramoto model [4–9]. It is an ongoing tale full of twists and turns, starting with Kuramoto’s
ingenious analysis in 1975 (which raised more questions than it answered) and culminating with Crawford’s in-
sights. Along the way, I will point out some problems that remain unsolved to this day, and tell a few stories
about the various people who have worked on the Kuramoto model, including how Crawford himself got hooked
on it.
E-mail address: shs7@cornell.edu (S.H. Strogatz)
0167-2789/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved.
PII: S0167-2789(00)00094-4
2 S.H. Strogatz/ Physica D 143 (2000) 1–20
2. Background
The Kuramoto model was originally motivated by the phenomenon of collective synchronization, in which an
enormous system of oscillators spontaneously locks to a common frequency, despite the inevitable differences in the
naturalfrequencies of the individualoscillators[10–13].Biologicalexamplesincludenetworksofpacemakercellsin
the heart [14,15]; circadianpacemakercells in the suprachiasmaticnucleus of the brain(where the individualcellular
frequencies have recently been measured for the first time [16]); metabolic synchrony in yeast cell suspensions
[17,18]; congregations of synchronously flashing fireflies [19,20]; and crickets that chirp in unison [21]. There are
also many examples in physics and engineering, from arrays of lasers [22,23] and microwave oscillators [24] to
superconducting Josephson junctions [25,26].
Collectivesynchronization was first studied mathematically by Wiener[27,28], who recognized its ubiquity in the
natural world, and who speculated that it was involved in the generation of alpha rhythms in the brain. Unfortunately
Wiener’s mathematical approach based on Fourier integrals [27] has turned out to be a dead end.
A more fruitful approach was pioneered by Winfree [10] in his first paper, just before he entered graduate
school. He formulated the problem in terms of a huge population of interacting limit-cycle oscillators. As stated,
the problem would be intractable, but Winfree intuitively recognized that simplifications would occur if the cou-
pling were weak and the oscillators nearly identical. Then one can exploit a separation of timescales: on a fast
timescale, the oscillators relax to their limit cycles, and so can be characterized solely by their phases; on a
long timescale, these phases evolve because of the interplay of weak coupling and slight frequency differences
among the oscillators. In a further simplification, Winfree supposed that each oscillator was coupled to the collec-
tive rhythm generated by the whole population, analogous to a mean-field approximation in physics. His model
is
˙
θ
i
= ω
i
+
N
X
j=1
X(θ
j
)
Z(θ
i
), i = 1,... ,N,
where θ
i
denotes the phase of oscillator i and ω
i
its natural frequency. Each oscillator j exerts a phase-dependent
influence X(θ
j
) on all the others; the corresponding response of oscillator i depends on its phase θ
i
, through the
sensitivity function Z(θ
i
).
Using numerical simulations and analytical approximations, Winfree discovered that such oscillator populations
could exhibit the temporal analog of a phase transition. When the spread of natural frequencies is large compared to
the coupling, the system behaves incoherently, with each oscillator running at its natural frequency. As the spread is
decreased, the incoherence persists until a certain threshold is crossed — then a small cluster of oscillators suddenly
freezes into synchrony.
This cooperative phenomenon apparently made a deep impression on Kuramoto. As he wrote in a paper with his
student Nishikawa ([8], p. 570):
“...Prigogine’s concept of time order [29], which refers to the spontaneous emergence of rhythms in nonequi-
librium open systems, found its finest example in this transition phenomenon... It seems that much of fresh
significance beyond physiological relevance could be derived from Winfree’s important finding (in 1967) after
our experience of the great advances in nonlinear dynamics over the last two decades.”
Kuramoto himself began working on collective synchronization in 1975. His first paper on the topic [4] was a
brief note announcing some exact results about what would come to be called the Kuramoto model. In later years, he
would keep wrestling with that analysis, refining and clarifying the presentation each time, but also raising thorny
new questions too [5–9].
S.H. Strogatz/ Physica D 143 (2000) 1–20 3
3. Kuramoto model
3.1. Governing equations
Kuramoto[5] put Winfree’s intuition about phase models on a firmer foundation. He used the perturbative method
of averaging to show that for any system of weakly coupled, nearly identical limit-cycle oscillators, the long-term
dynamics are given by phase equations of the following universal form:
˙
θ
i
= ω
i
+
N
X
j=1
0
ij
(θ
j
− θ
i
), i = 1,... ,N.
The interaction functions 0
ij
can be calculated as integrals involving certain terms from the original limit-cycle
model (see Section 5.2 of [5] for details).
Even though the reduction to a phase model represents a tremendous simplification, these equations are still far
too difficult to analyze in general, since the interaction functions could have arbitrarily many Fourier harmonics
and the connection topology is unspecified — the oscillators could be connected in a chain, a ring, a cubic lattice,
a random graph, or any other topology.
Like Winfree, Kuramoto recognized that the mean-field case should be the most tractable. The Kuramoto model
corresponds to the simplest possible case of equally weighted, all-to-all, purely sinusoidal coupling:
0
ij
(θ
j
− θ
i
) =
K
N
sin (θ
j
− θ
i
),
where K ≥0 is the coupling strength and the factor 1/N ensures that the model is well behaved as N →∞.
The frequencies ω
i
are distributed according to some probability density g(ω). For simplicity, Kuramoto assumed
that g(ω) is unimodal andsymmetric about its meanfrequency,i.e., g( +ω) =g( −ω) forall ω, like a Gaussian
distribution. Actually, thanks to the rotational symmetry in the model, we can set the mean frequency to =0by
redefining θ
i
→θ
i
+t for all i, which corresponds to going into a rotating frame at frequency . This leaves the
governing equations
˙
θ
i
= ω
i
+
K
N
N
X
j=1
sin (θ
j
− θ
i
), i = 1,... ,N (3.1)
invariant, but effectively subtracts from all the ω
i
and therefore shifts the mean of g(ω) to zero. So from now on,
g(ω) = g(−ω)
for all ω, and the ω
i
denote deviations from the mean frequency . We also suppose that g(ω) is nowhere increasing
on [0,∞), in the sense that g(ω) ≥g(v) whenever ω ≤v; this formalizes what we mean by “unimodal”.
3.2. Order parameter
To visualize the dynamics of the phases, it is convenient to imagine a swarm of points running around the unit
circle in the complex plane. The complex order parameter [5]
r e
iψ
=
1
N
N
X
j=1
e
iθ
j
(3.2)
4 S.H. Strogatz/ Physica D 143 (2000) 1–20
Fig. 1. Geometric interpretation of the order parameter (3.2). The phases θ
j
are plotted on the unit circle. Their centroid is given by the complex
number r e
iψ
, shown as an arrow.
is a macroscopic quantity that can be interpreted as the collective rhythm produced by the whole population. It
corresponds to the centroid of the phases. The radius r(t) measures the phase coherence, and ψ(t) is the average
phase (Fig. 1).
For instance, if all the oscillators move in a single tight clump, we have r ≈1 and the population acts like a giant
oscillator. On the other hand, if the oscillators are scattered around the circle, then r ≈0; the individual oscillations
add incoherently and no macroscopic rhythm is produced.
Kuramoto noticed that the governing equation
˙
θ
i
= ω
i
+
K
N
N
X
j=1
sin(θ
j
− θ
i
)
can be rewritten neatly in terms of the order parameter, as follows. Multiply both sides of the order parameter
equation by e
−iθ
i
to obtain
r e
i(ψ−θ
i
)
=
1
N
N
X
j=1
e
i(θ
j
−θ
i
)
.
Equating imaginary parts yields
r sin(ψ − θ
i
) =
1
N
N
X
j=1
sin(θ
j
− θ
i
).
Thus (3.1) becomes
˙
θ
i
= ω
i
+ Kr sin(ψ − θ
i
), i = 1,... ,N. (3.3)
In this form, the mean-field character of the model becomes obvious. Each oscillator appears to be uncoupled
from all the others, although of course they are interacting, but only through the mean-field quantities r and ψ .
Specifically, the phase θ
i
is pulled towardthemeanphase ψ, rather thantowardthephase of any individualoscillator.
Moreover, the effective strength of the coupling is proportional to the coherence r. This proportionality sets up a
positive feedback loop between coupling and coherence: as the population becomes more coherent, r grows and so
the effective coupling Kr increases, which tends to recruit even more oscillators into the synchronized pack. If the
coherence is further increased by the new recruits, the process will continue; otherwise, it becomes self-limiting.
Winfree [10] was the first to discover this mechanism underlying spontaneous synchronization, but it stands out
especially clearly in the Kuramoto model.
S.H. Strogatz/ Physica D 143 (2000) 1–20 5
Fig. 2. Schematic illustration of the typical evolution of r(t) seen in numerical simulations of the Kuramoto model (3.1).
3.3. Simulations
If we integrate the model numerically, how does r(t) evolve? For concreteness, suppose we fix g(ω)tobea
Gaussian or some other density with infinite tails, and vary the coupling K. Simulations show that for all K less
than a certain threshold K
c
, the oscillators act as if they were uncoupled: the phases become uniformly distributed
around the circle, starting from any initial condition. Then r(t) decays to a tiny jitter of size O(N
−1/2
), as expected
for any random scatter of N points on a circle (Fig. 2).
But when K exceeds K
c
, this incoherent state becomes unstable and r(t) grows exponentially, reflecting the
nucleation of a small cluster of oscillatorsthat are mutuallysynchronized, thereby generatinga collective oscillation.
Eventually r(t) saturates at some level r
∞
< 1, though still with O(N
−1/2
) fluctuations.
At the level of the individual oscillators, one finds that the population splits into two groups: the oscillators
near the center of the frequency distribution lock together at the mean frequency and co-rotate with the average
phase ψ(t), while those in the tails run near their natural frequencies and drift relative to the synchronized cluster.
This mixed state is often called partially synchronized. With further increases in K, more and more oscillators are
recruited into the synchronized cluster, and r
∞
grows as shown in Fig. 3.
The numerics further suggest that r
∞
depends only on K, and not on the initial condition. In other words, it seems
there is a globally attracting state for each value of K.
3.4. Puzzles
These numerical results cry out for explanation. A good theory should provide formulas for the critical coupling
K
c
and for the coherence r
∞
(K) on the bifurcating branch. The theory should also explain the apparent stability of
the zero branch below threshold and the bifurcating branch above threshold. Ideally, one would like to formulate
and prove global stability results, since the numerical simulations give no hint of any other attractors beyond those
seen here. Even more ambitiously, can one formulate and prove some convergence results as N→∞?
As we will see below, the first few of these problems have been solved, while the rest remain open. Specifically,
Kuramoto derived exact results for K
c
and r
∞
(K), Mirollo and I solved the linear stability problem for the zero
Fig. 3. Dependence of the steady-state coherence r
∞
on the coupling strength K.