Aging transition and clustering in a large population of coupled oscillators Hiroaki DAIDO Department of Mathematical Sciences Graduate School of Engineering Osaka Prefecture University Sakai 599-8531, JAPAN KIAS conference: NSPCS2008 (Seoul, July 2008) emblem of OPU
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Aging transition and clustering in a large population of coupled oscillators
Aging transition and clustering in a large population of coupled oscillators. Hiroaki DAIDO Department of Mathematical Sciences Graduate School of Engineering Osaka Prefecture University Sakai 599-8531, JAPAN. KIAS conference: NSPCS2008 (Seoul, July 2008). - PowerPoint PPT Presentation
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Aging transition and clustering in a large population of coupled oscillators
Hiroaki DAIDO
Department of Mathematical Sciences Graduate School of Engineering Osaka Prefecture University Sakai 599-8531, JAPAN
KIAS conference: NSPCS2008 (Seoul, July 2008)
emblem ofOPU
Contents
1. Background2. Aging in globally coupled oscillators#
Aging transition: Examples Universal scaling Clustering of active oscillators Summary3. Aging in locally coupled oscillators System size dependence of pc
Summary
# in collaboration with Kenji Nakanishi.
1. 1. BackgroundBackground
Coupled oscillators often model biological or physiological systems.
Deterioration due to aging or accidents etc.
self-sustained oscillator damped oscillator
( active oscillator ) ( inactive oscillator )
The problem of aging HD&KN, PRL 93 (2004), 104101.
What happens when the ratio of inactive elements increases ?
( “aging” )active
inactive
How robust is the activity of coupled oscillatorsagainst aging ?
important not only biologically but technologically
effects of bad components
Globally and diffusively coupled oscillators
periodic or chaoticGeneral form
2. Aging in globally coupled oscillators2. Aging in globally coupled oscillators
Examples (laboratory experiments)
Coupled electrochemical reaction systems Kiss et al. Science 296(2002), 176.
Coupled salt-water oscillators Miyakawa et al. Physica D 151(2001), 217.
(1) Coupled Stuart-Landau equations
Active oscillators
Inactive oscillators
Aging transition: examples
a
-b
The behavior of an order parameter
A measure of macroscopic activity
Synchronization withineach group
N=1000vanishes at
( Aging transition )
K=3, p=0.6
Theory
Reduction to a four-dimensional system
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
death
Aging transition
The death stabilized at
q=1-p
(2) Coupled Roessler systems
limit-cycle
fixed point
Parameters (1)
N=100
Parameters (2)
chaos
Reverse period-doubling cascade
Universal Scaling at the aging transition
ExampleCoupled periodic Roessler systemsN=1000
General theory
Assumptions
Reduction
1. The reduction is possible.
2. A fixed point exists.
3. Its becomes Hopf unstable at
active
inactive
Hopf
Universalcrossoverscaling
Clustering of active oscillators
The synchronization within the active group breaks down in a region of the parameter plane.
Coupled SL equations N=4000, p=0
Resonance-like enhancementof inhomogeneity measured by
the bar average over all j
Diffusion-induced inhomogeneitybelow the peak point under thescalar type diffusive coupling:
Dx=Dy=K (x=Re(z), y=Im(z))
< > time average
Analysis of the simplest 2-cluster state
Example of the cluster structure
clusterset of perfectlysynchronized oscillators
fractions of clusters in theorder of the size from aboveexcept the largest
2-cluster states with one cluster much smaller than the otherare abundant near the both ends of the clustering region
1. The problem of aging : Effects of increasing inactive elements
2. Aging transition
Strong coupling: favorable for coherence, but less robust against aging !
3. Universal scaling at the aging transition
4. Clustering and Swing-by mechanism of
Diffusion-induced inhomogeneity
3. Aging in locally coupled oscillators3. Aging in locally coupled oscillators
Effects of aging in locally coupled oscillators
a chain under the periodic boundary condition
(i.e. a ring) as a first step
1
Aging proceeds through random inactivation of oscillators
N >> 1
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1
2
N
N-1
Model & methods
Coupled Stuart-Landau oscillators on a ring
for all active oscillators (a>0)
for all inactive oscillators (b>0)
Number ratio: active: inactive=1-p:p
For K=0
active oscillator → limit-cycle
inactive oscillator → z=0 20
Aging scheme
Randomly choosing some active sites to inactivate at each step of increasing p .
The chosen oscillators remain inactive for all p after this.
Then, results are averaged over many realizations.
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(K,p) phase diagram
Example: a=b=1, c1=1, c2=-0.5
Aging transition boundaries
N = 6400 ( 20) 1600 ( 40) 400 ( 50) 100 (100)
active phase
inactive phase
Δp=0.0122
zj=0 for all j
Number of realizationsK
p
Key features of the phase diagram
(1) Existence of Kc insensitive to changes in N
A linear stability analysis of the inactive state for p=(N-1)/N shows that Kc is given, for N → ∞, by
Example
a=b=1, c1=1 Kc=0.648…23
(2) Vanishing of the inactive region for N → ∞
pc(K,N) → 1 for N → ∞ with K fixed
Simulation results suggest
Absence of the aging transition in the thermodynamic limit
Note: This does not imply unimportance of the AT, because (1) convergence of pc is slow, and (2) system sizes of real coupled oscillators are not always huge.
e.g. Lamprey’s CPG N ~ 100 mammalian circadian clocks N ~ 10000
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Scaling behavior of pc
How does pc approach unity as N grows toward infinity ?
Example a=b=1, c1=1, c2=-0.5
N = 100 ~ 12800K = 1, 1.8, 2.2, 2.6, 3.3, 4
Power laws !
1-pc(N) vs. N
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K dependence of the power law exponent
fit range N=100 ~ 12800
γ takes small valuesand tends to decreasewith K.
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Summary of Part 2
Aging in locally coupled oscillators
a chain of Stuart-Landau oscillators with n. n. interactions (as a first step)