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Low Dimensional Behavior in Large

Systems of Coupled OscillatorsEdward Ott

University of Maryland

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References

• Main Ref.: E. Ott and T.M. Antonsen,

“Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators,”

Chaos 18 (to be published in 9/08).

• Related Ref.: Antonsen, Faghih, Girvan, Ott and Platig,

“External Periodic Driving of Large Systems of Globally Coupled Phase Oscillators,”

arXiv: 0711.4135 and Chaos 18 (to be published in 9/08).

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• Cellular clocks in the brain.• Pacemaker cells in the heart.• Pedestrians on a bridge.• Electric circuits.• Laser arrays.• Oscillating chemical reactions.•Bubbly fluids.•Neutrino oscillations.

Examples of synchronized oscillators

4Cellular clocks in the brain (day-night cycle).

Yamaguchi et al., Science, vol.302, p.1408 (2003).

Incoherent

Coherent

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Synchrony in the brain

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Coupled phase oscillators

Change of variables

Limit cycle in phase space

Many such ‘phase oscillators’:Couple them:

Kuramoto:

; i=1,2,…,N »1

Global coupling

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Framework• N oscillators described only by their phase . N is very large.

g()

• The oscillator frequencies are randomly chosen from a distribution g( ) with a single local maximum.

(We assume the mean frequency is zero)

n

n

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Kuramoto model (1975)

N

mnmn

n

N

k

dt

d

1

)sin(

• Macroscopic coherence of the system is characterized by

N

mmi

Nr

1

)exp(1

n = 1, 2, …., N k= (coupling constant)

= “order parameter”

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Order parameter measures the coherence

1r 0r

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Results for the Kuramoto model

There is a transition to synchrony at a critical value of the coupling constant.

)0(

2

gkc

ck

r

k

Incoherence

Synchronization

g(0)

g()

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External Drive:

Some Generalizations of The Kuramoto Model:

)sin()sin()/(/ 001

i

N

jijni tMNkdtd

driveE.g., circadian rhythm.

Ref.: Antonsen, Faghih, Girvan, Ott, Platig, arXiv: 0711.4135, and Chaos (to be published in 9/08).Communities of Oscillators:

A = # of communities; σ = community (σ = 1,2,.., s);

Nσ = # of individuals in community σ.

'

1

''

1''' )sin()/(/

N

jij

s

ii Nkdtd

E.g., chimera states, s = 2 [Abrams, Mirollo, Strogatz, Wiley] .

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Generalizations (continued)

Time delay: Replace j(t) by j(t-in the abovegeneralizations.

Millenium Bridge Problem:

i

ifM

ydtdydtyd1

// 222 (Bridge mode)

))(cos()( 0 tftf iii (Walker force on bridge)

(Walker phase)

Ref.: Eckhardt, Ott, Strogatz, Abrams, McRobie,

Phys. Rev. E (2007).

)cos(// 22 iii dtybddtd

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The ‘Order Parameter’ Description

N

jiji

i

N

k

dt

d

1

)sin(

)sin(Im1

Im )(

1i

iN

j

ji

ii

rereN

e

ier

N

j

jii

iii

eN

er

rkdtd

1

1

)sin(/

“The order parameter”

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N→∞

Introduce the distribution function f(,t)

[the fraction of oscillators with phases in the range (+d) and frequencies in the range (+d) ]

ddtf ),,(

)(2

0

gfd Conservation of number of oscillators:

0

fdt

df

dt

d

t

f

0

0)sin(

frkt

f

ddefre ii 2

0

and similar formulations for generalizations

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The Main Result* Considering the Kuramoto model and its generalizations,

for i.c.’s f( ,0 ) [or f ,0 ) in the case of oscillator groups], lying on a submanifold M (specified later) of the space of all possible distribution functions f,

• f(, 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, Chaos (to be published 9/08).

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Comments • M is an invariant submanifold.

• ODE’s give ‘macroscopic’ evolution of the order parameter.

• Evolution of f(,t ) is infinite dimensional even though macroscopic evolution is finite dimensional.

• Is it useful? Yes, if the dynamics of r(t) found in M is attracting in some sense. Ref.: Antonsen, Faghih, Girvan, Ott, Platig, Chaos (9/08), arXiv: 0711.4135.

M

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Specifying the Submanifold M The Kuramoto Model as an Example:

dtfgefddreR

feRReiktf

ii

ii

),,()( ,

02///

2

0

2

0

*

Inputs: k, the coupling strength, and

the initial condition, f ( (infinite dimensional).

M is specified by two constraints on f(0):

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Specifying the Manifold M (continued) Fourier series for f:

1

..),(12

)(),,(

n

inn ccetf

gtf

Constraint #1: 1),( ,),(),(

.1)( ,)()(

tttf

0ω,0ω,0ω,f

nn

nn

?

dgR

iRRk

t

*

*2 02

Question: For t >0 does

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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.

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If, for Im(ω)< 0, |α(ω,0)|<1, then |α(ω,t)|<1 : 0

2

1/ *2 iRRkt

Multiply by α* and take the real part:

0||)Im(2Re||1/|| 222 Rkt

At |α(ω,t)|=1: |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 ) .

0||)Im(2/|| 22 t

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If α(ω,0) → 0 as Im(ω) → -∞ , then so does α(ω,t)

Since |α| < 1, we also have (recall that )

| R(t)|< 1 and .

Thus

dgR*

1Re||1 2 R

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|α | ) ω2Im(Kt

α

|α| → 0 as Im(ω) → -∞ for all time t.

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Lorentzian g(ω)

),)( 0* ()(),(

0

1

0

121

220

1)(

titR dgt

iiig

i0

i0

)Im(

)Re(

Set ω = ω0 –iΔ in 0)*2)(2/(/ iRRkt

0)()12|(|2

RiRRk

dtdR

0

)(g

0

2

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Solution for |R(t)|=r(t)

2ckk

)(tr

)(r

ckk

)(r

ck

)/(1 kkc

k

tt

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Circadian Rhythm Problem

Antonsen et al.

)sin()sin(// 00 iij

jii tMNkdtd Observed behaviors depending on parameters:

A. One globally attracting state in which the drive entrains the oscillator system.

B. An unentrained state is the attractor (bad sleep pattern).

C. Same as in A, but there are also two additional unstable entrained solutions.

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Parameter Space

M0 = driving strengthΩ = frequency mismatch between oscillator average and drivek = 5 coupling strength

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Schematic Blow-up Around T

• A↔B: Hopf bifurcation• A↔C: Saddle-node bifurcation of 2 and 3• C↔B: Saddle-node bifurcation on a periodic orbit (1

and 2 created as B→C)

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Low Dim. ODE Reduction of the Circadian Rhythm Model

• The above results were obtained from solution of the full problem for f(ω,θ,t) (without restricting the dynamics to M), e.g., partly by numerical solution of N ≥ 103 ODE’s,• The ODE for evolution on the manifold M is

• The results from solution of this equation for give the same picture (quantitatively!) as obtained from solving the N ≥ 103 ODE’s,• Thus all the observed attractors and bifurcations of the original system occur on M.

.../ dtd i

012

10

2*0 RiMkRRMkR

dt

dR

)(tR

.../ dtd i

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Further Discussion • A qualitatively similar parameter space diagram applies for Gaussian g(ω). Also, our method can treat certain other g()’s, e.g., g() ~ [()4 + 4 ]-1.

• Numerical studies of other generalizations of the Kuramoto model (e.g., chimera states [Abrams, Mirollo, Strogatz, Wiley]) also show all the interesting dynamics taking place on M.

• For generalizations of the Kuramoto problem in which s interacting groups are treated (e.g., s=2 for the chimera problem), our method yields a set of s coupled complex ODE’s for s complex order parameters describing the system’s state.

• For the Millenium Bridge model we get a 2nd order ODE for the bridge driven by a complex order parameter describing the collective state of the walkers, plus an ODE for the walker order parameter driven by the bridge.

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Conclusion

The macroscopic behavior of large systems of globally coupled oscillators have been demonstrated (at least in some cases) to be low dimensional.

Thanks:Tom AntonsenMichelle GirvanRose FaghihJohn PlatigBrian Hunt