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Edward Ott

Synchronism in Large Networks of Coupled Heterogeneous

Dynamical Systems:Lecture II

Generalizations of the Kuramoto Model:External driving and interactionsComplex (e.g., chaotic) node dynamics with global couplingComplex node dynamics & network coupling

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Review of the Onset of Synchrony

in the Kuramoto Model (1975)

)θ(θKωdtdθ ijj ijii )sinθNK() θ (K ij

N coupled periodic oscillators whose states are described by

phase angle i , i =1, 2, …, N.

All-to-all sinusoidal coupling:

j jiN

1ii )θsin(θk{ωdtdθ

Order Parameter; ]}

iψre

e[Im{e})θsin(θ{j

iθ

N1iθ

j iiN1 ji

N

1jN1iψ

iii

)exp(ire

)θkrsin(ψωdtdθ

j

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Typical BehaviorSystem specified by i’s and k.

Consider N >> 1.

g()d= fraction of oscillation freqs. between and +d.

rrr ckk

ckk )

N1O(

N

kt ck

4

N ∞ = fraction of oscillators whose phases and frequencies lie in the range to d and to d

ddtF ),,(

0)()(

FF

dtd

dtd

tF

2

0

0]))sin([(

ddFere

Fkr

ii

tF

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Linear StabilityIncoherent state: 2πθ0,2π) ω g(F This is a steady state solution. Is it stable? Linear perturbation:

Laplace transform ODE in for f

D(s,k) = 0 for given g(), Re(s) > 0 implies instability

Results: Critical coupling kc. Growth rates. Freqs.

Fff;FF

kck

r

k ck

Re(s)

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Examples of generalizations:Interaction with external world

A model of circadian rhythm

Crowd synchrony on the Millennium Bridge

Refs.: Eckhardt, Ott, Strogatz, Abrams, McRobie, Phys. Rev. E 75 021110 (2007); Strogatz, et al., Nature (2006).

Ref.: Antonsen, Fahih, Girvan, Ott, Platig, arXiv:0711.4135, Chaos (to be published in 9/08)

N

1ji0iji

i )θtsin(ΩM)θsin(θN

kω

dt

dθ

N

1jj

2B2

2)cos(θkyω

dt

dy2γ

dt

yd

)θ cos( dt

ydbω

dt

dθi2

2

ii

Bridge

People

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Crowd synchronization on the London Millennium bridge

Bridge opened in June 2000

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The phenomenon:

London,Millennium bridge:Opening dayJune 10, 2000

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Tacoma narrows bridge

Tacoma,

Pudget Sound

Nov. 7, 1940

See

KY Billahm, RH Scanlan, Am J Phys 59, 188 (1991)

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Differences between MB and TB:

• No resonance near vortex shedding frequency and

• no vibrations of empty bridge

• No swaying with few people

• nor with people standing still

• but onset above a critical number of people in motion

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Studies by Arup:

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Forces during walking:

• Downward: mg, about 800 N

• forward/backward: about mg

• sideways, about 25 N

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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)

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The model

)cos(

))(cos()(

)(2

0

2

iii

iii

ii

yb

tftf

tfyMyMyM

Bridge motion:

forcing:

phase oscillator:

Modal expansion for bridge plus phase oscillator for pedestrians:

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Dynamical simulation

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Coupling complex [e.g., chaotic] systems

Kuramoto model (Kuramoto, 1975)

All-to-all Network.

Coupled phase oscillators (simple dynamics).

Ott et al.,02; Pikovsky et al.96Baek et al.,04; Topaj et al.01

All-to-all Network.

More general dynamics.

Ichinomiya, Phys. Rev. E ‘04Restrepo et al., Phys. Rev E ‘04; Chaos‘06

More general network.

Coupled phase oscillators.

More general Network.

More general dynamics.

Restrepo et al. Physica D ‘06

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A Potentially Significant Result

Even when the coupled units are

chaotic systems that are individually

not in any way oscillatory (e.g., 2x mod

1 maps or logistic maps), the global

average behavior can have a transition

from incoherence to oscillatory

behavior (i.e., a supercritical Hopf

bifurcation).

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The activity/inactivity cycle of an individual ant is ‘chaotic’, but it is periodic for may

ants.

Cole, Proc.Roy. Soc. B, Vol. 224, p. 253 (1991).

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Globally Coupled Lorenz Systems

],[ in ddistribute uniformly

38,10

)(

)(

11

rrr

b

yxbzdtdz

zxyxrdtdy

txNx

xkxydtdx

i

iiii

iiiiii

Ni i

iii

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21

22

23

24

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Formulation

))Ωρ( ( Ω )μ (x

xxμ

Ωd)dμΩρ(xx

(t)xNlimx

qqK

(t)]x,(t),x(t),[x(t)x

)Ωρ( Ω

N,1,2,i

)xx(K)Ω,x(Gdtxd

Ω

Ω

Ω

i i1

N

T(q)i

(2)i

(1)ii

i

iii

pdfover andviaover average

:State" Incoherent" measure natural

matrix coupling

ondistributi smooth withvector parameter

members ensemble lables

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Stability of the Incoherent State

Goal: Obtain stability of coupled system from dynamics of the uncoupled component

:M

xδKx)δΩ(t),x(GDdtxdδ

(t)xδ(t)x(t)x

i

iiii

iii

system uncoupled the ofmatrix (Lyapunov) lFundamenta

)t(txδ i

)t(xδ i )t(tix

)t(x i

1)Ω);tx(;t(0,M

)t(x)δΩ);t(x;t(t,M)t(txδ

i

ii

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)dT(T,Me(s)M

0}K(s)M1det{D(s)0Δ}K(s)M1{

: τtT ,eΔxδ ,

dτxδK)Ω;(xτ)τ,(tMxδ

1)Ω);t(x;t(0,M

M)Ω),t(tx(GDdtMd

0sT

st

tτiii

i

i

iii

~

~~

where

let andassumeTake

: function Dispersion

:xδfor Solution

28useful)yet (Not

~

average and integral einterchangFor

attractors chaoticfor and

cycleslimit for

exponent Lyapunovlargest

0sT-

Ω,x

dT)(T,Me(s)M

: ΓRe(s)

0 Γ h

0h

)Ω,xh(

hmaxΓRe(s)

(s)M~

0sT )dT(T,Me(s)M ~

Convergence

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Decay of (T)M

Mixing Chaotic Attractors

direction k in vector unit a

aδ(0)x(0)x

k

kk

cloud. of centroid of onperturbati δ]M[ kk

kth column

Mixing perturbation decays to zero. (Typically exponentially.)

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Analytic Continuation

0sT

0sT

0ΓRe(s) fordT(T)Me

(T)dTMe(s)M~

Reasonable assumption

0 γ,κe(t)M γt

Analytic continuation of : (s)M~

Im(s)

Re(s)

γ

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NetworksAll-to-all :

0}K(s)M1det{ ~

Network :

0}K(s)MN

λ1det{ ~

= max. eigenvalue of network adj. matrix

An important point:

Separation of the problem into two parts: A part dependent only on node dynamics

(finding ), but not on the network topology. A part dependent only on the network (finding ) and

not on the properties of the dynamical systems on each node.

M~

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Conclusion

Framework for the study of networks of heterogeneous dynamical systems coupled on a network. (N >> 1)

Applies to periodic, chaotic and ‘mixed’ ensembles.

Our papers can be obtained from :http://www.math.umd.edu/~juanga/umdsyncnets.htm

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Networks With General Node Dynamics

Restrepo, Hunt, Ott, PRL ‘06; Physica D ‘06

Uncoupled node dynamics:

time)(discrete )x(Mx

or

time) (continous (t))x(F(t)/dtxd

(n)ii

1)(ni

iii

Could be periodic or chaotic.Kuramoto is a special case: iiii ωF,θx

Main result: Separation of the problem into two parts

Q: depends on the collection of node dynamical behaviors (not on network topology).: Max. eigenvalue of A; depends on network topology (not on node dynamics).

Q/λkc

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Synchronism in Networks ofCoupled HeterogeneousChaotic (and Periodic)

Systems Edward Ott

University of Maryland

Coworkers:

Paul So Ernie Barreto

Tom Antonsen Seung-Jong Baek

Juan Restrepo Brian Hunt

http://www.math.umd.edu/~juanga/umdsyncnets.htm

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Previous Work

Limit cycle oscillators with a spread of natural frequencies:• Kuramoto• Winfree• + many others

Globally coupled chaotic systems that show a transition from incoherence to coherence:• Pikovsky, Rosenblum, Kurths, Eurph. Lett. ’96• Sakaguchi, Phys. Rev. E ’00• Topaj, Kye, Pikovsky, Phys. Rev. Lett. ’01

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Our Work Analytical theory for the stability of the incoherent

state for large (N >>1) networks for the case of arbitrary node dynamics ( K , oscillation freq. at onset and growth rates).

Examples: numerical exps. testing theory on all-to-all heterogeneous Lorenz systems (r in [r-, r+]).

Extension to network coupling.

References:

Ott, So, Barreto, Antonsen, Physca D ’02. Baek, Ott, Phys. Rev. E ’04 Restrepo, Ott, Hunt (preprint) arXiv ‘06