arXiv:1202.3865v1 [cond-mat.soft] 17 Feb 2012
Hydrodynamic synchronisation of non-linear oscillators at low Reynolds number.
M. Leoni1and T. B. Liverpool1,2
1Department of Mathematics, University of Bristol, Clifton, Bristol BS8 1TW, U.K.
2The Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA
(Dated: February 20, 2012)
We introduce a generic model of weakly non-linear self-sustained oscillator as a simplified tool
to study synchronisation in a fluid at low Reynolds number. By averaging over the fast degrees of
freedom, we examine the effect of hydrodynamic interactions on the slow dynamics of two oscillators
and show that they can lead to synchronisation.
strongly enhanced when the oscillators are non-isochronous, which on the limit cycle means the
oscillations have an amplitude-dependent frequency. Non-isochronity is determined by a nonlinear
coupling α being non-zero. We find that its (α) sign determines if they synchronise in- or anti-
phase. We then study an infinite array of oscillators in the long wavelength limit, in presence
of noise. For α > 0, hydrodynamic interactions can lead to a homogeneous synchronised state.
Numerical simulations for a finite number of oscillators confirm this and, when α < 0, show the
propagation of waves, reminiscent of metachronal coordination.
Furthermore, we find that synchronisation is
Collections of cilia and flagella are examples of systems
that display synchronisation . They are microscopic
active filaments attached to the membrane of pro- and
eukaryote cells  whose synchronisation is thought to aid
the efficiency of transport at the cellular scale. Typically
arrays of cilia generate fluid flows along tissues but can
also be used, like flagella, for the self-propulsion of swim-
ming cells. Due to their tiny size, the Reynolds number
associated with these flows is negligible. The coordinated
beating of cilia is also thought to have important devel-
opmental implications, such as the left-right symmetry
breaking in the arrangement of the internal organs in the
early embryo . A precise understanding of the role hy-
drodynamics plays in their synchronised motion, is still
Both cilium and flagellum are made of complex sub-
units, microtubules driven by molecular motors, and
their modelling can be done at many levels.
chronisation takes place on length-scales larger than the
individual filaments, to a first approximation the fine
details of their internal structure can be ignored. This
coarse-grained approach has led to model studies of self-
sustained oscillators , rotating beads [5–7]; beating
filaments , as well as rigid rotating helices, [9, 10].
More recent work has focused on the conditions for hy-
drodynamic synchronisations for two oscillators  and
the phase dynamics of oscillators with long range inter-
actions . Related experiments investigating the dy-
namics of micro-systems have been performed in vivo on
algae, [13, 14] and on simple model systems , and even
a macroscopic scale model of rotating paddles . All
these studies suggest that simple forms of active forces,
e.g. as prescribed functions of time, are not enough
to guarantee synchronisation. Rather, a complex, non-
linear relation between forces and velocities is necessary.
Important questions therefore are what aspects of hydro-
dynamic interactions aid synchronisation and what fea-
tures of oscillators make them good hydrodynamic syn-
The dynamics of a system close to an oscillatory in-
stability can be conveniently described by weakly non-
linear oscillators whose averaged equations are univer-
sal . This implies that the long time behaviour of many
systems with simple spontaneous oscillations can be cap-
tured by a generic model with a few parameters. Using
this insight, in this paper we introduce a minimal model
of an oscillator at low Reynolds number. To simplify our
presentation, we study our model in one-dimension. At a
coarse grained level, this degree of freedom can be inter-
preted as the centre of a filament beating in a plane .
The slow dynamics of the oscillator is naturally char-
acterised using of two variables: the amplitude and the
phase. Under arbitrary initial conditions, the trajecto-
ries of an isolated oscillator on long timescales converge
to a closed curve, the limit cycle . While the ampli-
tude is tightly constrained to the limit cycle curve, the
phase can vary more freely. Hence many model studies
of synchronisation have focused only on the phase dy-
namics [5–7, 11, 12]. Our goal in this paper is to anal-
yse the role played by both the amplitude and phase dy-
namics on phase synchronisation mediated by hydrody-
namics. We first study a pair of well separated deter-
ministic oscillators and find that hydrodynamic interac-
tions strongly enhance phase locking, if the oscillations
are non-isochronous, which on the limit cycle means that
the frequency of oscillations depends on the amplitude.
We then consider an array of many oscillators, still well
separated, in the presence of fluctuations. On long wave-
lengths their slow dynamics can be naturally represented
in terms of a broken symmetry (phase) variable, which
is a non-equilibrium analogue of a Goldstone mode .
Denoting by α ?= 0 the parameter responsible for the non-
isochronity of the oscillations, we find that when α > 0,
hydrodynamic interactions can lead to in-phase synchro-
nisation of the array. These results are confirmed by
numerical simulations, which show also that conversely,
for α < 0, the synchronisation is more subtle and leads
to the propagation of waves.
a.The model oscillator
spontaneous oscillations is provided by the normal form
A universal model for stable
of a dynamical system close to a supercritical Hopf bifur-
cation . To be concrete, we represent the oscillator in
a low Reynolds number fluid as a sphere of radius a sub-
ject to a time-varying force f. The equation of motion
for the sphere, with x its deviation from its equilibrium
˙ x =f
where γ = 6πηa is the Stokes drag. The dynamics is
encoded in the evolution equation for the force f :
˙f = Ψ(f,x) := −k
τx + µf
?1 − σx2?+ αx3. (2)
Here, all the parameters, except α are positive quantities,
The 1st and 3rd term of eq (2) give rise to respectively,
a linear and a non-linear passive oscillator, while the 2nd
term is responsible for active, self-sustained oscillations.
We emphasize that all the terms in eq (2) would emerge
naturally from coarse-graining any friction-dominated
microscopic model oscillator [4, 10, 17].
can be conveniently non-dimensionalised as ˙ x = f; and
˙f = −x + ǫµf(1 − x2) + ǫαx3, choosing units where
k. They correspond to a weakly non-linear Van
der Pol-Duffing oscillator . The parameters ǫµ:=
kσare small quantities. We restrict ourselves
here mainly to cases where ǫα/ǫµ= O(1).
b. Two oscillators coupled hydrodynamically
oscillators are arranged along the x-axis. The forces fi
acting on the spheres, for i = 1,2, are directed along
the same axis and cause sphere 1 to oscillate around the
origin and sphere 2 around position d. We denote by xi
the deviations from these equilibrium positions, see fig 1.
Their equations of motion are
Eqs (1), (2)
where H(r) is a scalar, representing the hydrodynamic in-
teractions, and r := d+x2−x1is the separation between
the sphere centres. We shall consider the limit of large
separation r compared to the sphere radius a. Then, for
an unbounded three-dimensional fluid, interactions are
described by the Oseen tensor  as H(r) =3a
rigid surface with a non-slip boundary condition, placed
at distance h from the oscillators, one obtains effective in-
teractions scaling as H(r) ∼ah2
oscillators arranged on a regular lattice, d can be thought
of as the lattice spacing, see fig 1. We assume that it
is large compared to the amplitude of the oscillations,
d > x2− x1, and that the ratio ǫd:= a/d, characterising
the hydrodynamic coupling, satisfies ǫd ≪ ǫµ,ǫα. The
time evolution of forces is given by˙fi= Ψ(fi,xi), with
Ψ(fi,xi) defined in eq (2), and is entirely local . The
long-range hydrodynamic coupling links the coordinates
xi via eq (3). In the following we denote the nonlinear
parts of Ψ(fi,xi) by Fi(xi,fi) :=µ
2r. For a
r3 . For an assembly of
?1 − σx2
FIG. 1: (color online) One dimensional lattice of oscillators.
The inset illustrates the dynamic variables of a pair.
To proceed, we take the time derivative of both sides
of eq (3) and use, on the rhs, the evolution equation for
the forces and the expression of forces as functions of
velocities ˙ xi obtained by inverting eq (3) as an expan-
sion in a/r. Thus, to leading order, we obtain equations
for oscillators with reactive couplings  (given by
as ¨ x1+ ω2
γF2(x2,γ ˙ x2)+D
the natural frequency of the linear oscillators, defined
We now derive the equations governing the slow dy-
namics of the oscillators .
using a complex amplitude Ak and its complex con-
krelated to position and velocities by xk =
k = 1,2. This requires of course that ˙A∗
Here ω is the (unknown) frequency of the non-linear os-
cillators, determining the period, T =2π
cillations. The (slow) complex amplitudes, on the other
hand, hardly change on this timescale. Writing eq (3)
and the dynamic equations for the forces in terms of Ak
kand averaging over the period T one obtains
γF1(x1,γ ˙ x1) +D
γx1, where D := −H(d)k
γx2 and ¨ x2+ ω2
γτ. Note terms like
dr˙ rfi, of order O(ǫ2
γFi(xi,γ ˙ xi), of order
d) have been neglected.
This is done naturally
ke−iωt) and ˙ xk=iω
ωof the (fast) os-
˙A1= −i∆A1+ λA1− (β + iχ)A1|A1|2+ iδA2
˙A2= −i∆A2+ λA2− (β + iχ)A2|A2|2+ iδA1. (4)
The parameters are defined as ∆ :=
Writing the complex amplitudes Ak in polar form,
Ak = Rkeiφk, eqs (4) become a coupled system for the
amplitudes Rk and the phases φk. Finally, this system
can be reduced to a single equation for the phase dif-
ference . This can be achieved perturbatively, when
the parameter δ, parametrising the hydrodynamic inter-
actions, is small compared to the other terms.
teractions are neglected, Rk have fixed points given by
fixed points can be studied by writing Rk=
for sk≪ 1. One finds that the deviations skrelax quickly
to zero. Setting ˙ sk = 0 we obtain sk as functions of
, λ :=
8γ, χ :=3
γω, and δ(d) :=H(d)
β. The dynamics of small deviations from these
the phase difference ψ := φ2− φ1. The resulting expres-
sions are then substituted in the equations for the phases.
From them one obtains an Adler equation  for ψ,
˙ψ = ˜ ν − 2δχ
Hence, eq (5) illustrates that phase locking is deter-
mined by the hydrodynamic coupling, via δ, provided
the oscillator is nonisochronous, i.e. α ?= 0. Note thatδχ
scales as ∼1
natural frequencies of the oscillators. We choose them to
be identical, so we can set ˜ ν = 0. While for ˜ ν ?= 0 varying
the ratio of ˜ ν andδχ
βcontrols the saddle-node bifurcation
of cycles , for ˜ ν = 0 eq (5) has a stable fixed point
given by one of the zeros of sin(ψ) for ψ ∈ [0,2π]. The
position of the stable point is determined by the sign of
β, which in turn is determined solely by the sign of
the non-isochronism parameter α: when α < 0, then the
equation has a stable fixed point at ψ = π, i.e. the os-
cillators lock in anti-phase; vice-versa, if α > 0 then the
equation has a stable fixed point at ψ = 0 and the os-
cillators lock in-phase. A numerical solution, using the
Euler method, of eq (3) confirms this.
It is also interesting to note that the two flagella of
the microscopic algae C. Reinhardtii are found to alter-
nate between periods of synchronised (with small phase
difference) and non-synchronized beating [13, 14]. This
is well described by a stochastic Adler equation, of the
same form as eq (5) but with an additional fluctuating
term . The estimates of the parameters presented
in , for the flagellar synchronisation, indicate positive
values for α and ˜ ν of our model.
When α = 0, we need to include higher order correc-
tions in deriving eq (5). Upon doing this we find to lead-
ing order˙ψ ≈ −3ǫdǫµ[1+3
the synchronisation is in-phase. Otherwise, both in- and
anti-phase states are possible and synchronization de-
pends on details such as initial conditions (confirmed nu-
merically). These higher order terms also indicate that
the transition from in-phase to anti-phase in general oc-
curs at some αc?= 0. Unsurprisingly when α = 0, syn-
chronisation occurs more slowly (a higher order effect).
c.Many oscillators coupled hydrodynamically
have discussed above, the amplitudes of the oscillators
are tightly constrained to the limit cycle and the long
time behaviour can be reduced to an effective (amplitude
dependent) dynamics of the phases. For a large num-
ber N of oscillators, in the dilute regime, this is done
by introducing the one-particle probability c(ϕ,y,t) =
with slow phase ϕ, at site y at time t, where the brack-
ets ?? indicate the average over noise. The probability
satisfies a Smoluchowski equation
ǫµǫdand ˜ ν is related to the difference of the
µcosψ]sinψ. When ǫd<4
k=1δ(ϕ−φk(t))δ(y−yk(t))? of having an oscillator
∂tc = D∂2
ϕϕc − ∂ϕ([ω1+ Ω]c). (6)
D is the diffusion coefficient resulting from both thermal
and active fluctuations, ω1the deterministic contribution
of an isolated oscillator with ω1= −∆ −χλ
deterministic effect of the hydrodynamic interactions,
βand Ω the
the dynamics of two oscillators, (see eq (5)). It describes
the effect of the interactions on the phase of one oscillator
due to the presence of the another. Here, δ′:= δ(|y2−y|).
The 1-particle probability can be expressed as an ex-
pansion in its moments:
βsin(ϕ2− ϕ) + δ′cos(ϕ2− ϕ) is obtained from
?ρ(y,t) +?e−iϕΦ(y,t) + c.c.?+ ...?
To study synchronization we only need the first two :
dϕeiϕc(ϕ,y,t) ; (1st harmonic) . (9)
The emergence (or not) of a globally synchronized state
is obtained from the homogeneous probability c0(ϕ,t),
with associated moments ρ0(t), Φ0(t) representing spa-
tially homogeneous dynamical states. The correspond-
ing expression for Ω0is obtained by evaluating the
space integral in eq (7) with c ≡ c0.
namic interactions scaling as H(r) ∼a
from the integral depends both on the lattice spacing
d, and the total length L of the array. Hence, Ω0(t) =
scaling as H(r) ∼ah2
r3, the leading term in the integral
depends only on the lattice spacing d. Consequently, the
term aln(L/d) is replaced by one ∼ah2
tions for the homogeneous moments are derived by tak-
ing the time derivative of both sides of eq (9), inserting
eq (6) and using eq (8) to close the system. Since ρ is a
conserved variable, ∂tρ0= 0, while Φ0satisfies
rthe leading term
β+ 1]e−iϕ1Φ0(t) + c.c. For interactions
d2. Dynamic equa-
∂tΦ0= ΓΦ0. (10)
It is worth noting that in the absence of noise
the order parameter introduced by Kuramoto, Φ0(t) =
over a population of oscillators [1, 12, 22].
It is useful to express Φ0(t) = P0(t)eiQ0(t)in polar
form (reflecting the U(1) symmetry). We obtain equa-
tions for its amplitude and phase as ∂tP0= Re[Γ]P0
and ∂tQ0= Im[Γ]. Re(Γ) = −(D −χ
is the real part of Γ. Here, the first term is due to
noise, whereas the second term encodes the effect of
two body interactions. The imaginary part is Im(Γ) =
As in the Kuramoto model [1, 22], order (synchronisa-
tion) is determined by a non-zero, constant value of P0.
k=1δ(ϕ − ϕk(t)) and Φ0(t) reduces to
k=1eiϕk(t), representing the (mean field) average
N = 100 deterministic oscillators, (D = 0). (a), (b) describe
respectively the case for α > 0 and α < 0, after long time.
The initial conditions of the oscillators are the same for both
values of α: identical amplitudes, close to the maximum value,
and random, Gaussian distributed, phases. The parameters of
the model are γ = 10−3Pa s µm;
σ = 1(µm)−2; |α| = 0.05
(µm)3sand a/d ≈ 0.005.
(color online) Space-time plots of the positions for
µm s; µ = 0.05pN
Here, the dynamic equation for P0shows that the onset
of order is controlled by the sign of Re[Γ]. If Re[Γ] < 0,
order is suppressed. On the contrary, when Re[Γ] > 0, or-
der is enhanced. A stabilising term of the type ∼ Φ0|Φ0|2
in eq (10) is needed for P0to stop unbounded growth and
attain a finite value at long times. Such a term could be
generated for instance by taking into account three-body
interactions. Finally, the condition Re[Γ] = 0 defines
a transition line in the space of parameters . Cru-
cially, from these considerations, homogeneous synchro-
nization is possible only when α > 0: (i) in presence of
noise (D ?= 0) and by keeping all the parameters fixed,
synchronisation occurs only above a particular value of
density; (ii) neglecting noise (D = 0), instead, synchroni-
sation occurs for any (finite) value of the density. On the
contrary, when α < 0 both terms in Re(Γ) are negative
and homogeneous order is prohibited. This behaviour
suggests a spin analogy, where α > 0 (ferromagnet) pro-
motes alignment of neighbouring oscillator phase (spins)
while α < 0 (antiferromagnet) promotes anti-alignment.
We compared these results with numerical simulation
for a large but finite number of deterministic oscillators
(D = 0). In fig 2 we show typical space-time plots for the
positions of N = 100 oscillators and compare the effects
of different signs of α. For α > 0, see fig 2(a), the sys-
tem displays spatially homogeneous order, i.e. in-phase
synchronised state. Interestingly, when α < 0, although
homogeneous order is lacking, fig 2(b) still shows a co-
herent motion of the oscillators, with propagating waves.
As suggested by the antiferromagnetic analogy, the oscil-
lators self-organise into a dynamical state which is close
to the anti-phase synchronised state, but deviates from
it at long wavelengths.
In conclusion, we have presented a simple, one-
dimensional model (that can be generalised to higher di-
mensions ) to investigate analytically the role of hy-
drodynamic interactions on the synchronisation dynam-
ics of oscillators at low Reynolds number. We studied the
case of two oscillators and found that synchronisation, ei-
ther in- or anti-phase, was determined to leading order by
both hydrodynamic interactions and non-isochronism of
the oscillations (α ?= 0). We then derived a coarsegrained
description for an infinite array of oscillators and found
that spatially homogeneous order, corresponding to the
in-phase synchronisation of the array, can occur only for
α > 0. Systems of cilia are known to display metachronal
waves . Our analysis suggests that these could be ob-
tained in two different ways: either as slow hydrodynamic
(phase) modes, like spin waves, when α > 0; or alterna-
tively, for α < 0, as a spatially inhomogeneous, approxi-
mately anti-phase synchronised state, as indicated by the
numerics. A more extensive investigation of these issues
is left for the future.
We acknowledge the support of the EPSRC, Grant
EP/G026440/1 (ML & TBL); the NSF, Grant PHY05-
51164 (TBL); and the University of Bristol (ML).
 A. Pikovsky, M. Rosenblum, and J. Kurths, Synchroniza-
tion: A Universal Concept in Nonlinear Science (Cam-
bridge University Press, 2002).
 D. Bray, Cell Movements: From Molecules to Motility
(Garland Science, New York, 2000).
 S. Nonaka et al., Cell 95, 829 (1998).
 M. C. Lagomarsino, P. Jona, and B. Bassetti, Phys. Rev.
E 68, 021908 (2003).
 A. Vilfan and F. Julicher, Phys. Rev. Lett. 96, 058102
 T. Niedermayer, B. Eckhardt, and P. Lenz, Chaos 18,
 N. Uchida and R. Golestanian, Phys. Rev. Lett. 104,
 B. Guirao and J. F. Joanny, Biophys. J. 92, 1900 (2007).
 M. Kim and T. R. Powers, Phys. Rev. E 69, 061910
 M. Reichert and H. Stark, Eur. Phys. J. E 17, 493 (2005).
 N. Uchida and R. Golestanian, Phys. Rev. Lett. 106,
 N. Uchida, Phys. Rev. Lett. 106, 064101 (2011).
 M. Polin et al., Science 325, 487 (2009).
 R. E. Goldstein, M. Polin, and I. Tuval, Phys. Rev. Lett.
103, 168103 (2009).
 J. Kotar et al., PNAS 107, 7669 (2010).
 B. Qian et al., Phys. Rev. E 80, 061919 (2009).
 S. Camalet, F. Julicher, and J. Prost, Phys. Rev. Lett.
82, 1590 (1999).
 S. Strogatz, Nonlinear Dynamics And Chaos (Westview
 M. Leoni and T. B. Liverpool, Phys Rev Lett 105, 238102
 M. Doi and S. Edwards, The Theory of Polymer Dynam-
ics (Oxford University Press, 1986).
5 Download full-text
 J. R. Blake, Proc. Cambridge Philos. Soc. 70, 303 (1971).
 J. A. Acebr´ on et al., Rev. Mod. Phys. 77, 137 (2005).
 M. Leoni and T. B. Liverpool, unpublished (2012).
 A. Hamel et al., Proc Natl Acad Sci USA 108, 7290