Synchronism%20in%20Networks%20of%20Coupled%20Heterogeneous%20Chaotic%20(and%20Periodic)%20Systems

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Edward Ott
Generalizations of the Kuramoto Model
External driving and interactions
Complex (e.g., chaotic) node dynamics with global
coupling
Complex node dynamics network coupling
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Review of the Onset of Synchronyin the Kuramoto
Model (1975)
N coupled periodic oscillators whose states are
described by phase angle qi , i 1, 2, , N.
All-to-all sinusoidal coupling
Order Parameter
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Typical Behavior
System specified by wis and k. Consider N gtgt
1. g(w)dw fraction of oscillation freqs.
between w and wdw.
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N g 8
fraction of
oscillators whose phases and frequencies lie in
the range q to q dq and w to w dw
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Linear Stability
Incoherent state
This is a steady state solution. Is it
stable? Linear perturbation Laplace
transform ? ODE in q for f ? D(s,k) 0 for
given g(w), Re(s) gt 0 implies instability Results
Critical coupling kc. Growth rates. Freqs.
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A model of circadian rhythm
Ref. Antonsen, Fahih, Girvan, Ott, Platig,
arXiv0711.4135, Chaos (to be published in
9/08)
Bridge
People
Refs. Eckhardt, Ott, Strogatz, Abrams, McRobie,
Phys. Rev. E 75 021110 (2007) Strogatz, et
al., Nature (2006).
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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 day June 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
Modal expansion for bridge plus phase oscillator
for pedestrians
Bridge motion forcing phase oscillator
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Dynamical simulation
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Coupling complex e.g., chaotic systems
All-to-all Network.
Coupled phase oscillators (simple dynamics).
Kuramoto model (Kuramoto, 1975)
All-to-all Network.
More general network.
More general dynamics.
Coupled phase oscillators.
Ichinomiya, Phys. Rev. E 04 Restrepo et al.,
Phys. Rev E 04 Chaos06
Ott et al.,02 Pikovsky et al.96 Baek et al.,04
Topaj et al.01
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
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Formulation
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Stability of the Incoherent State
Goal Obtain stability of coupled system from
dynamics of the uncoupled component
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Convergence
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Decay of
Mixing Chaotic Attractors
kth column
Mixing ? perturbation decays to zero.
(Typically exponentially.)
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Analytic Continuation

• Reasonable assumption

? Analytic continuation of
Im(s)
Re(s)
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Networks
All-to-all
Network
? 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.

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Conclusion

• Framework for the study of networks of
  heterogeneous dynamical systems coupled on a
  network. (N gtgt 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
Uncoupled node dynamics
Could be periodic or chaotic. Kuramoto is a
special case
Main result Separation of the problem into two
parts
Q depends on the collection of node dynamical
behaviors (not on network topology). l Max.
eigenvalue of A depends on network topology (not
on node dynamics).
Restrepo, Hunt, Ott, PRL 06 Physica D 06
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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 gtgt1) 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