Emergent Low Dimensional
Behavior in Large Systems of
Coupled Phase Oscillators
Edward Ott
University of Maryland
1
References
• Main Refs.: E. Ott and T.M. Antonsen,
“Low Dimensional Behavior of Large Systems
of Globally Coupled Oscillators,”
arXiv:0806.0004 and Chaos 18 ,037113 (‘08).
“Long Time Behavior of Phase Oscillator
Systems”, arXiv:0902.2773(‘09) and Chaos 19,
023117 (‘09).
Our other related work that is referred to in
this talk can be found at:
http://www-chaos.umd.edu/umdsyncnets.htm 2
Examples of synchronized oscillators
• Cellular clocks in the brain.
• Pacemaker cells in the heart.
• Pedestrians on a bridge.
• Josephson junction circuits.
• Laser arrays.
• Oscillating chemical reactions.
• Bubbly fluids.
3
Incoherent
Coherent
Cellular clocks in the brain (day-night cycle).
Yamaguchi et al., Science 302, 1408 (‘03).
4
Synchrony in the brain
5
Coupled phase oscillators
q
Change of variables
Limit cycle in
phase space
Many such ‘phase oscillators’:
; i=1,2,…,N »1
Couple them:
Kuramoto:
Global
coupling
6
Framework
• N oscillators described only by
their phase q. N is very large.
n
• The oscillator frequencies are randomly chosen
from a distribution g( ) with a single local
maximum.
g()
qn

7
Kuramoto model (1975)
dq n
k
 n 
sin(q m  q n )

dt
N m 1
N
n = 1, 2, …., N
k= (coupling constant)
• Macroscopic coherence of the system is
characterized by
N
1
r
exp(iq m ) = “order parameter”

N m 1
8
Order parameter measures the coherence
r 1
r 0
9
Results for the Kuramoto model
There is a transition to synchrony at a
critical value of the coupling constant.
r
kc 
gmax
2
gmax

Synchronization
Incoherence
kc
k
10
Generalizations of the Kuramoto Model
External Drive:
N
dq i / dt   n  ( k / N )  sin(q j  q i )  M 0 sin(  0 t  q i )
E.g., circadian rhythm. j 1
drive
Ref.: Sakaguchi, ProgTheorPhys(‘88); Antonsen, Faghih, Girvan,
Ott, Platig,Chaos 18 (‘08); Childs, Strogatz, Chaos 18 (‘08).
Communities of Oscillators:
A = # of communities; σ = community (σ = 1,2,.., s);
Nσ = # of individuals in community σ.


dq i / dt   i 
s
N '
 '1
j 1
 (k ' / N ') 
' 
 '
sin(q j  q i   )
E.g., Barreto, et al., PhysRevE 77 (’08);Martens, et al., PhysRevE
(‘09)(two humped g(ω), s=2); Abrams,et al.,PhysRevLett 101(‘08);
11 101
Laing, Chaos19(’09); and Pikovsky&Rosenblum, PhysRevLett
(‘08); Sethia, Strogatz, Sen, SIAM meeting (‘09).
Generalizations (continued)
Millennium Bridge Problem:
1
d y / dt   dy / dt   y   f i (Bridge mode)
M i
2
2
2
f i (t )  f i 0 cos( q i (t )) (Walker force on bridge)
2
2
dq i / dt   i  bd y / dt cos( q i   ) (Walker phase)
Ref.: Eckhardt,Ott,Strogatz,Abrams,McRobie, PhysRevE
75, 021110(‘07); Abdulrehem and Ott, Chaos 19, 013129
12
(‘09).
Generalizations (continued)
• Time varying coupling: The strengths of the
interaction between oscillators depends on t.
Ref.: So, Cotton, Barreto, Chaos 18 (‘08).
• Time delays in the connections between
oscillators: Each link can have a different delay
with the collection of delays characterized by a
distribution function.
Ref.: W.S.Lee, E.Ott, T.M.Antonsen,
arXiv:0903.1372 (PhysRevLett, to be published).
• Josephson junction circuits:
Ref.: S.A.Marvel, S.H.Strogatz, Chaos 18,
013132 (2009).
• But there was a puzzle.
13
The ‘Order Parameter’ Description
dq i
k N
 i 
sin(q j  q i )

dt
N j 1


  iq 
N iq  
1

j 
i ( q )

i
 Im e
e

Im
r
e
 r sin(  q i )
N 


j 1




“The order parameter”
r e i
dq i / dt   i  k r sin(  q i )
N iq
j
1
i
re 
e

N j 1
14
N →∞
Introduce the distribution function f(q,,t)
f (q ,  , t ) ddq  [the fraction of oscillators
with phases in the range
(q,q+dq) and frequencies in
the range (,+d) ]
2
 fdq  g( )
0
Conservation of number of oscillators:
f
  dq    d 0 

f
f0


t q  dt    dt 
f

    k r sin(  q ) f   0
t q
2
re i  

0
f e i q dq d
and similar
formulations for
generalizations
15
The Main Message of This Lecture*
Considering the Kuramoto model and its generalizations,
for i.c.’s f( ,q,0 ) [or f ( ,q,0 ) in the case of oscillator
groups], lying on a submanifold M (specified later) of the
space of all possible distribution functions f,
• f(,q, t) continues to lie in M,
• for appropriate g( ) the time evolution of r( t )
(or r( t )) satisfies a finite set of ODE’s which we
obtain.
* Ott and Antonsen, arXiv:0806.0004 and Chaos 18,
037113 (‘08); and arXiv:0902.2773 (’09) and Chaos
19, 023119 (‘09).
16
Comments
• M is an invariant submanifold.
M
• ODE’s give ‘macroscopic’ evolution of the order
parameter.
• Evolution of f(,q,t ) is infinite dimensional even
though macroscopic evolution is finite dimensional.
• Is it useful? Yes: we have recently shown that, under
weak conditions, M contains all the attractors and
bifurcations of the order parameter dynamics when
17
there is nonzero spread in the frequency distribution.
Specifying the Submanifold M
The Kuramoto Model as an Example:
f / t   / q     (k / 2i  ( Re iq  R*e  iq   f  0
R  re i 
2

0
dq

iq
d

f
e
,


g( ) 
2
 f ( ,q , t )dq
0
Inputs: k, the coupling strength, and
the initial condition, f (,q,0 (infinite
dimensional).
M is specified by two constraints on f(,q,0): 18
Specifying the Manifold M (continued)
Fourier series for f:




g( ) 


inq
f ( ,q , t ) 
 c .c . 
1    f n ( , t )e
2   n 1
 


 (ω, 0 )  n,  (ω, 0 )  1 .
n
f n ( , t )   ( , t )  ,  (  , t )  1 ?
Constraint #1: f n (ω, 0 )  
Question: For t >0 does


 k

R 2  R*  i  0
t 2

*
R 
  g d

19
Specifying the Manifold M (continued)
Constraint #2: α(ω,0) is analytic for all real ω, and,
when continued into the lower-half
complex ω-plane ( Im(ω)< 0 ) ,
(a) α(ω,0) has no singularities in Im(ω)< 0,
(b) lim α(ω,0) → 0 as Im(ω) → -∞ .
• It can be shown that, if α(ω,0) satisfies constraints 1
and 2, then so does α(ω,t) for all t < ∞.
• The invariant submanifold M is the collection of
distribution functions satisfying constraints 1 and 2.
• The problem for α(ω,t) is still infinite dimensional.
20
If, for Im(ω)< 0, |α(ω,0)|<1, then |α(ω,t)|<1 :


1
 / t  k R 2  R*  i  0
2
Multiply by α* and take the real part:
(

 |  |2 / t  k 1 |  |2 Re R 2 Im(  ) |  |2  0
At |α(ω,t)|=1:  |  |2 / t  2 Im(  ) |  |2  0
|α| starting in |α(ω,0)| < 1 cannot cross into
|α(ω,t)| > 1.
|α(ω,t)| < 1 and the solution exists for all t
( Im(ω)< 0 ) .
21
If α(ω,0) → 0 as Im(ω) → -∞ ,
then so does α(ω,t)

Since |α| < 1, we also have (recall that R* 
| R(t)|< 1 and
( 1 |  |2  Re R  1 .
  g d )

Thus
2
α
2
 K  2Im( ω ) | α |
t
|α| → 0 as Im(ω) → -∞ for all time t.
22
Lorentzian g(ω)

1 
1
1

g( ) 




2
2
 (     
2i    0  i   0  i 
0

1

R* ( t ) 
  ( , t ) g( )d   (0  i , t )
g( )

Im(  )
0  i
0  i
Re( )
2
0
Set ω = ω0 –iΔ in  /  t  (k / 2)( R 2  R* )  i   0
dR k
2
 (| R | 1) R  (  i 0   ) R  0
23
dt
2
Solution for |R(t)|=r(t)
r (t )
r ( )
k  k c  2
t
t
k  kc
r ( )
1  ( kc / k )
kc
k
Thus this steady solution is nonlinearly stable and globally attracting.
24
Crowd Synchronization on the
London Millennium Bridge
Bridge opened in June 2000
25
The Phenomenon:
London,
Millennium bridge:
Opening day
June 10, 2000
26
Studies by
Arup:
27
The Frequency of Walking:
People walk at
a rate of about
2 steps per
second (one
step with each
foot).
Matsumoto et al., Trans JSCE 5, 50 (1972)
28
MODEL
Model expansion for bridge + phase oscillators for walkers
d2y
dy
1
2

 y
 i f i ( t ) (Bridge)
2
dt
M
dt
f i ( t )  f io cosq i ( t ) (Walker force on bridge)
dq i ( t )
d2y
  i  b 2 cosq i ( t ) (Walker phase)
dt
dt
Ref.:Eckhardt, Ott, Strogatz, Abrams and McRobie,
Phys.Rev.E75, 021110 (‘07)
REDUCED MODEL
F(ω)
d2y
dy
2
M 2  M
 M y  N F Re[ R( t )]
dt
dt
2
dR( t )
d
y 2

 i{(  i ) R   2 [ R ( t )  1]}  0
dt
dt


Ref.: M.M.Abdulrehem and E.Ott, Chaos 19, 013129(‘09), 29
arXiv:0809.0358
ω
NUMERICAL SOLUTIONS OF REDUCED EQS.
30
Further Discussion
• Our method can treat certain other g()’s, e.g.,
g() ~ [(0 )4 + 4 ]-1, or g(ω)=[polynomial]/[polynomial]. Then
there are s coupled ODE’s for s order parameters where s is the
number of poles of g(ω) in Im(ω)<0.
•For generalizations in which s interacting groups are treated our
method yields a set of s coupled complex ODE’s for s complex
order parameters. E.g., s=2 for the chimera problem.
• Numerical and analytical work [e.g., Martens, et al. (‘09); Lee,
et al. (‘09); Laing (‘09)] shows similar results from Lorentzian
and Gaussian distributions of oscillator frequencies, implying
that qualitative behavior does not depend on details of g(ω).
31
ATTRACTION TO M
Ott & Antonsen have recently rigorously shown that, for
the Kuramoto model and its generalizations discussed
above, all attractors of the order parameter dynamics
and their bifurcations occur on M, provided that  > 0,
and certain weak additional conditions are satisfied.
I.e., M is an inertial manifold wrt a proper choice of the
distance metric in the space of distribution functions f.
Ref.: arXiv:0902.2773; and Chaos 19, 023117(‘09).
If =0, long time behavior not on M can occur [e.g.,
Pikovsky & Rosenblum, PhysRevLett (‘08); Marvel &
Strogatz, Chaos 19 (‘09); Sethia, Strogatz & Sen, SIAM
mtg.(‘09)]. The long time behavior of systems of
heterogeneous oscillators is simpler when the oscillator
frequencies are heterogeneous (>0). (Furthermore, for
all the previous generalizations of Kuramoto, >0 is the
more realistic model.)
32
TRANSIENT BEHAVIOR: ECHOES
• The transient behavior that occurs as the orbit relaxes
to M can be nontrivial. An example of this is the ‘echo’
phenomenon studied in Ott, Platig, Antonsen &
Girvan, Chaos 18, 037115 (‘08). [Similar to Landau
echoes in plasmas; e.g., T.M.O’Neil & R.W.Gould,
Phys. Fluids (1968).]
• For the classical Kuramoto model with k below its
critical value and external stimuli:
k < kc
r
Stimulus

Stimulus

Echo
t
33
AN ANALOGY
N eqs. for N>>1 oscillator phases.
Relaxation to M.
ODE description for order parameter.
Hamilton’s eqs. for N>>1 interacting fluid particles.
Relaxation to a local Maxwellian.
Fluid eqs. for moments (density, velocity, temp., …).
34
Conclusion
The long time macroscopic behavior of large systems
of globally coupled oscillators has been demonstrated
to be low dimensional.
Systems of ODE’s describing this low dimensional
behavior can be explicitly obtained and utilized to
discover and analyze all the long time behavior (e.g.,
the attractors and bifurcations) of these systems.
Ref.: Ott & Antonsen, Chaos 18, 037113 (2008). Also,
for the demonstration of attraction to M see Chaos 19,
023117 (2009).
35