Search Results for “kuramoto model”

Source: Kuramoto model

The Kuramoto model (or Kuramoto–Daido model), first proposed by Yoshiki Kuramoto (蔵本由紀, Kuramoto Yoshiki), is a mathematical model used in describing synchronization. More specifically, it is a model for the behavior of a large set of coupled oscillators. Its formulation was motivated by the behavior of systems of chemical and biological oscillators, and it has found widespread applications in areas such as neuroscience and oscillating flame dynamics. Kuramoto was quite surprised when the behavior of some physical systems, namely coupled arrays of Josephson junctions, followed his model.
The model makes several assumptions, including that there is weak coupling, that the oscillators are identical or nearly identical, and that interactions depend sinusoidally on the phase difference between each pair of objects.

Definition

Kuramoto

model

Transformation

model

Large N limit

Solutions for the large N limit

.
When

g

{\displaystyle g}

is unimodal and symmetric, then a stable state solution for the system is

r
=
r
K

∫

−
π

/

2

π

/

2

cos

2

⁡
θ
g
(
K
r
sin
⁡
θ
)
d
θ

{\displaystyle r=rK\int _{-\pi /2}^{\pi /2}\cos ^{2}\theta g(Kr\sin \theta )d\theta }

As coupling increases, there is a critical value

K

c

=
2

/

π
g
(
0
)

{\displaystyle K_{c}=2/\pi g(0)}

such that when

K
<

K

c

{\displaystyle K
, the long-term average of

r
=
0

{\displaystyle r=0}

, but when

K
=

K

c

(
1
+
μ
)

{\displaystyle K=K_{c}(1+\mu )}

, where

μ
>
0

{\displaystyle \mu >0}

is small, then

r
≈

16

π

K

c

3

μ

−

g

′
′

(
0
)

{\displaystyle r\approx {\sqrt {\frac {16}{\pi K_{\mathrm {c} }^{3}}}}{\sqrt {\frac {\mu }{-g^{\prime \prime }(0)}}}}

.

Small N cases

When N is small, the solutions given above breaks down, as the continuum approximation cannot be used.
The N=2 case is trivial. In the rotating frame

ω

1

=
−

ω

2

{\displaystyle \omega _{1}=-\omega _{2}}

, and so the system is described exactly by the angle between the two oscillators:

Δ
θ
=

θ

1

−

θ

2

{\displaystyle \Delta \theta =\theta _{1}-\theta _{2}}

. When

K
<

K

c

=
2

|

ω

1

|

{\displaystyle K
, the angle cycles around the circle (that is, the fast oscillator keeps lapping around the slow oscillator). When

K
>

K

c

{\displaystyle K>K_{c}}

, the angle falls into a stable attractor (that is, the two oscillators lock in phase). Similarly, the state space of the N=3 case is a 2-dimensional torus, and so the system evolves as a flow on the 2-torus, which cannot be chaotic.
Chaos first occurs when N=4. For some settings of

ω

1

,

ω

2

,

ω

3

,
K

{\displaystyle \omega _{1},\omega _{2},\omega _{3},K}

, the system has a strange attractor.

Connection to Hamiltonian systems

The dissipative Kuramoto model is contained in certain conservative Hamiltonian systems with Hamiltonian of the form

H

(

q

1

,
…
,

q

N

,

p

1

,
…
,

p

N

)
=

∑

i
=
1

N

ω

i

2

(

q

i

2

+

p

i

2

)
+

K

4
N

∑

i
,
j
=
1

N

(

q

i

p

j

−

q

j

p

i

)
(

q

j

2

+

p

j

2

−

q

i

2

−

p

i

2

)

{\displaystyle {\mathcal {H}}(q_{1},\ldots ,q_{N},p_{1},\ldots ,p_{N})=\sum _{i=1}^{N}{\frac {\omega _{i}}{2}}(q_{i}^{2}+p_{i}^{2})+{\frac {K}{4N}}\sum _{i,j=1}^{N}(q_{i}p_{j}-q_{j}p_{i})(q_{j}^{2}+p_{j}^{2}-q_{i}^{2}-p_{i}^{2})}

After a canonical transformation to action-angle variables with actions

I

i

=

(

q

i

2

+

p

i

2

)

/

2

{\displaystyle I_{i}=\left(q_{i}^{2}+p_{i}^{2}\right)/2}

and angles (phases)

ϕ

i

=

a
r
c
t
a
n

(

q

i

/

p

i

)

{\displaystyle \phi _{i}=\mathrm {arctan} \left(q_{i}/p_{i}\right)}

, exact Kuramoto dynamics emerges on invariant manifolds of constant

I

i

≡
I

{\displaystyle I_{i}\equiv I}

. With the transformed Hamiltonian

H
′

(

I

1

,
…

I

N

,

ϕ

1

…
,

ϕ

N

)
=

∑

i
=
1

N

ω

i

I

i

−

K
N

∑

i
=
1

N

∑

j
=
1

N

I

j

I

i

(

I

j

−

I

i

)
sin
⁡
(

ϕ

j

−

ϕ

i

)
,

{\displaystyle {\mathcal {H'}}(I_{1},\ldots I_{N},\phi _{1}\ldots ,\phi _{N})=\sum _{i=1}^{N}\omega _{i}I_{i}-{\frac {K}{N}}\sum _{i=1}^{N}\sum _{j=1}^{N}{\sqrt {I_{j}I_{i}}}(I_{j}-I_{i})\sin(\phi _{j}-\phi _{i}),}

Hamilton's equation of motion become

d

I

i

d
t

=
−

∂

H

′

∂

ϕ

i

=
−

2
K

N

∑

k
=
1

N

I

k

I

i

(

I

k

−

I

i

)
cos
⁡
(

ϕ

k

−

ϕ

i

)

{\displaystyle {\frac {dI_{i}}{dt}}=-{\frac {\partial {\mathcal {H}}'}{\partial \phi _{i}}}=-{\frac {2K}{N}}\sum _{k=1}^{N}{\sqrt {I_{k}I_{i}}}(I_{k}-I_{i})\cos(\phi _{k}-\phi _{i})}

and

d

ϕ

i

d
t

=

∂

H

′

∂

I

i

=

ω

i

+

K
N

∑

k
=
1

N

[

2

I

i

I

k

sin
⁡
(

ϕ

k

−

ϕ

i

)

+

I

k

/

I

i

(

I

k

−

I

i

)
sin
⁡
(

ϕ

k

−

ϕ

i

)

]

.

{\displaystyle {\frac {d\phi _{i}}{dt}}={\frac {\partial {\mathcal {H}}'}{\partial I_{i}}}=\omega _{i}+{\frac {K}{N}}\sum _{k=1}^{N}\left[2{\sqrt {I_{i}I_{k}}}\sin(\phi _{k}-\phi _{i})\right.\left.+{\sqrt {I_{k}/I_{i}}}(I_{k}-I_{i})\sin(\phi _{k}-\phi _{i})\right].}

So the manifold with

I

j

=
I

{\displaystyle I_{j}=I}

is invariant because

d

I

i

d
t

=
0

{\displaystyle {\frac {dI_{i}}{dt}}=0}

and the phase dynamics

d

ϕ

i

d
t

{\displaystyle {\frac {d\phi _{i}}{dt}}}

becomes the dynamics of the Kuramoto model (with the same coupling constants for

I
=
1

/

2

{\displaystyle I=1/2}

). The class of Hamiltonian systems characterizes certain quantum-classical systems including Bose–Einstein condensates.

Variations of the models

There are a number of types of variations that can be applied to the original model presented above. Some models change the topological structure, others allow for heterogeneous weights, and other changes are more related to models that are inspired by the Kuramoto model but do not have the same functional form.

= Variations of network topology and network weights: from vehicle coordination to brain synchronization

=

Some works in the control community have focused on the Kuramoto model on networks and with heterogeneous weights (i.e. the interconnection strength between any two oscillators can be arbitrary). The dynamics of this model reads as follows:

d

θ

i

d
t

=

ω

i

+

∑

j
=
1

N

a

i
j

sin
⁡
(

θ

j

−

θ

i

)
,

i
=
1
…
N

{\displaystyle {\frac {d\theta _{i}}{dt}}=\omega _{i}+\sum _{j=1}^{N}a_{ij}\sin(\theta _{j}-\theta _{i}),\qquad i=1\ldots N}

where

a

i
j

{\displaystyle a_{ij}}

is a nonzero positive real number if oscillator

j

{\displaystyle j}

is connected to oscillator

i

{\displaystyle i}

. Such model allows for a more realistic study of, e.g., flocking, schooling, and vehicle coordination. In the work from Dörfler and colleagues, several theorems provide rigorous conditions for phase and frequency synchronization of this model. Further studies, motivated by experimental observations in neuroscience, focus on deriving analytical conditions for cluster synchronization of heterogeneous Kuramoto oscillators on arbitrary network topologies. Since the Kuramoto model seems to play a key role in assessing synchronization phenomena in the brain, theoretical conditions that support empirical findings may pave the way for a deeper understanding of neuronal synchronization phenomena.

= Variations of the phase interaction function

=
Kuramoto approximated the phase interaction between any two oscillators by its first Fourier component, namely

Γ
(
ϕ
)
=
sin
⁡
(
ϕ
)

{\displaystyle \Gamma (\phi )=\sin(\phi )}

, where

ϕ
=

θ

j

−

θ

i

{\displaystyle \phi =\theta _{j}-\theta _{i}}

. Better approximations can be obtained by including higher-order Fourier components,

Γ
(
ϕ
)
=
sin
⁡
(
ϕ
)
+

a

1

sin
⁡
(
2
ϕ
+

b

1

)
+
.
.
.
+

a

n

sin
⁡
(
2
n
ϕ
+

b

n

)

{\displaystyle \Gamma (\phi )=\sin(\phi )+a_{1}\sin(2\phi +b_{1})+...+a_{n}\sin(2n\phi +b_{n})}

,
where parameters

a

i

{\displaystyle a_{i}}

and

b

i

{\displaystyle b_{i}}

must be estimated. For example, synchronization among a network of weakly-coupled Hodgkin–Huxley neurons can be replicated using coupled oscillators that retain the first four Fourier components of the interaction function. The introduction of higher-order phase interaction terms can also induce interesting dynamical phenomena such as partially synchronized states, heteroclinic cycles, and chaotic dynamics.

Availability

pyclustering library includes a Python and C++ implementation of the Kuramoto model and its modifications. Also the library consists of oscillatory networks (for cluster analysis, pattern recognition, graph coloring, image segmentation) that are based on the Kuramoto model and phase oscillator.