Title: Synchronism%20in%20Networks%20of%20Coupled%20Heterogeneous%20Chaotic%20(and%20Periodic)%20Systems
1 Emergence of Collective Behavior In
Large Networks of Coupled Heterogeneous
Dynamical Systems (Second lecture on network
sync)
Edward Ott University of Maryland
2Review 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
3Typical Behavior
System specified by wis and k. Consider N gtgt
1. g(w)dw fraction of oscillation freqs.
between w and wdw.
4N g 8
fraction of
oscillators whose phases and frequencies lie in
the range q to q dq and w to w dw
5Linear 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.
6Generalizations of the Kuramoto Model
- General coupling function Daido, PRL(94)
sin(?j -?i ) f(?i - ?j ). - Time delay ?j(t) ?j(t- t ).
- 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.
7 Generalizations (Continued)
A model of circadian rhythm
Ref. Sakaguchi,Prog.Theor.Phys(88)Antonsen,
Fahih, Girvan, Ott, Platig, arXiv0711.4135Chaos
(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).
8Crowd Synchronization on the London Millennium
Bridge
- Bridge opened in June 2000
9The Phenomenon
London, Millennium bridge Opening day June 10,
2000
10Tacoma Narrows Bridge
- Tacoma,
- Pudget Sound
- Nov. 7, 1940
See KY Billahm, RH Scanlan, Am J Phys 59, 188
(1991)
11Differences 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
12Studies by Arup
13Forces During Walking
- Downward mg, about 800 N
- forward/backward about mg
- sideways, about 25 N
14The 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)
15The Model
Modal expansion for bridge plus phase oscillator
for pedestrians
Bridge motion forcing phase oscillator
(Walkers feel the bridge acceleration through its
acceleration.)
16Dynamical Simulation
17Coupling 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
18A 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).
19The 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).
20Globally Coupled Lorenz Systems
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26Formulation
27Stability of the Incoherent State
Goal Obtain stability of coupled system from
dynamics of the uncoupled component
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29Convergence
30Decay of
Mixing Chaotic Attractors
kth column
Mixing ? perturbation decays to zero.
(Typically exponentially.)
31Analytic Continuation
? Analytic continuation of
Im(s)
Re(s)
32Networks
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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34Conclusion
- Framework for the study of networks of many
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.chaos.umd.edu/umdsyncnets.html
35Networks 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
36Synchronism 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
37Previous 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
38Our 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