1 edward ott university of maryland emergence of collective behavior in large networks of coupled...
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Edward Ott University of Maryland
Emergence of Collective Behavior In Large Networks of Coupled Heterogeneous Dynamical Systems
(Second lecture on network sync)
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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
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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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Generalizations of the Kuramoto Model
• General coupling function: Daido, PRL(‘94); sin(θj -θi ) f(θi - θj ).
• Time delay: θj(t) θj(t- τ ).
• Noise: Increases kc.• ‘Networks of networks’: Communities of phase
oscillators are uniformly coupled within communities but have different coupling strengths between communities. Refs.:Barretto, Hunt, Ott, So, Phys.Rev.E 77 03107(2008);Chimera states model of Abrams, Mirollo, Strogatz, Wiley, arXiv0806.0594.
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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.: Sakaguchi,Prog.Theor.Phys(’88);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
Generalizations (Continued)
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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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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:
(Walkers feel the bridge acceleration through its acceleration.)
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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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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
29useful)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 many heterogeneous dynamical systems coupled on a network (N >> 1).
Applies to periodic, chaotic and ‘mixed’ ensembles.
Our papers can be obtained from :http://www.chaos.umd.edu/umdsyncnets.html
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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