A Frequency-weighted
Kuramoto Model
Lecturer: 王瀚清
Adaptive Network and Control Lab
Electronic Engineering Department of
Fudan University
Outlines
Background
 Our Model
 Simulations
 Analysis
 Conclusion

Outlines
Background
 Our Model
 Simulations
 Analysis
 Conclusion

The Original Kuramoto Model
A classical and useful tool to analyze networks of
coupled oscillators
 Drawbacks: Idealized assumptions and constraints
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•
•

All-to-all, Equal-weighted
The distribution of natural frequencies should be unimodal
and symmetric
Extension
•
•
Practically, the couplings among the oscillators should be
influenced by their own charateristics, i.e. Power grid
Unimodal distributions are not universal, especially in
human dynamics.
Outlines
Background
 Our Model
 Simulations
 Analysis
 Conclusion

Description and Definition
Governing equations:
i
K  N
 i  i  sin( j  i )
t
N
j 1
Frequency-distribution
(1)
The order parameter
1 N i j
r e  e
N j 1
With this definition, Eq.1 becomes
i
(2)

 i  i  Ki  r sin(  i )
(3)
When N   , Eq.2 becomes
i
r e 
 2

i
e
  (, , t )d dt
 0
(4)
1
1

 2 (  1)  ,   1/ 
g ( )  
 1 (  1)( )  ,   1/ 
 2
(5)
Outlines
Background
 Our Model
 Simulations
 Analysis
 Conclusion

Instructions



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In most of the simulations: we set
N=1000,   2.5 ,   100 .
The left figures shows the results
when   0 .
In the bottom figure, we illustrate
how will the final value of r varies
with the coupling strength K .
The oscillatory part in the r-K
figure indicates that the final value
of r will oscillate instead of
converging to a steady value, as
shown in the top figure.
We averaged the results when the
final value of r is converged.
The Cases of Odd Beta:   1



A threshold of the coupling
strength K c exists.
Below the threshold, the
final value will oscillate.
Exceeding the threshold,
the final value will converge
to a steady value. However
with the coupling strength
increasing, r decreases.
 1
  1
A Microscopic View


The oscillators will spontaneously split into two
clusters of different synchronization when K  Kc .
And these two clusters locate roughly at the
opposite sides on the unit-circle.
With coupling strength increasing, more oscillators
will be locked to the two clusters, and run at a
common frequency   Kr / 
  1, K  100
  1, K  100, t  1000
  1, K  5000, t  150
  1, K  5000
The Cases of Even Beta:   2



These also exists a
threshold Kc .
Below the threshold, the
final values oscillate
significantly.
Above the threshold ,the
final values will come to a
steady value.
 2
  2
A Microscopic View


The oscillators would spontaneously split into
two clusters even when K  Kc .In this condition,
however, the two clusters running in opposite
directions at the same frequency.
When K  Kc , the two locked clusters move
closer to each other, and will finally stop near 0,
with a tiny difference between their average
phases.
  2, K  1
  2, K  300, t  2000
  2, K  1000, t  400
  2, K  1000
Others Distributions
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Uniform Distribution

Gaussian Distribution
Outlines
Background
 Our Model
 Simulations
 Analysis
 Conclusion

Two Questions
What kind of oscillators compose the two clusters we
observed previously？
 Why significant synchronization is only possible for
even beta, and why increasing coupling strength will
decrease the order parameter when beta is odd？

Even Beta: illustrating with the case   2

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Suppose there is a oscillator i with its
natural frequencyi , and its phase is  i .
Meanwhile, there is another oscillator j ,
whose natural frequency and phase
satisfy  j  i , j  i . Then we
proved that this condition will always be
satisfied.
We assume that the locked positive
cluster and locked negative cluster run
at frequencies  respectively, then
we can decide which oscillator can be
locked.
Consider the case where the coupling
strength is large enough to lock all
oscillators, then we can get the solution
of  and finally get the following:
2 K 2 r 2 2 ( K 2 r 2 2  1)2/3
r

2
3
3Kr
Odd beta: illustrating with the case   1



The symmetry of network lose when beta
is odd, which can be also proved.
We suppose that the locked clusters,
positive and negative run in the same
direction with frequency  . With these,
we derived which oscillator can be locked
and find some can’t be locked no matter
how large the coupling strength is.
We get  for the locked oscillators, and
only consider their contribution to the
order parameter, because the
unsynchronized oscillator’s influence on
order parameter is relatively small.
4( Kr  1)3/2 ( K 2 r 2  1)3/2
r

3 3
K r
K 3r 3
Chimera States when beta is odd
  1, K  10
  1, K  10
  1, K  10, t  14000
  1, K  10
Chimera States
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The so-called Chimera States were first found by Kuramoto in
a reaction and diffusion system, where identical nodes
behave quite differently. And Strogatz named this phenomena
as ‘Chimera States’
Recently, the Chimera States in a heterogeneous network
have also been studied. However, Our model is different from
the previous work in the following aspects:
•
•
In previous work , the Kuramoto oscillator networks with observed
Chimera States are usually phase-delayed.
In previous work , oscillators were deliberately divided into several
groups, and the coupling strengths in each group are different.
Outlines
Background
 Our Model
 Simulations
 Analysis
 Conclusion

Conclusion
Proposed a frequency-weighted Kuramoto model
 Investigated its dynamics by numeric simulations
 Analyzed the observations with mathematical
method.

Thank you for listening!
Q&A