Coupled Phase Oscillators: N → с

Coupled Phase Oscillators: N → ∞

M. Lakshmanan

Centre for Nonlinear Dynamics School of Physics Bharathidasan University Tiruchirappalli – 620024

India

SERC School on ”Nonlinear Dynamics”

Panjab University, Chandigarh

27-30, January 2014

Coupled Phase Oscillators: N → ∞

Kuromoto model of coupled phase oscillators:

dθi

dt =ωi+ K N

N

X

j=1

sin(θj(t)−θi(t))

In the N→ ∞ limit, the state of the oscillator system can be described by a continuous distribution functionf(ω, θ,t)

Coupled Phase Oscillators: N → ∞

dθ_i

dt = ω_i +K

N X

j

h

e^i(θj−θi)−e^{−(θj−θi)}i

= ω_i +K 2i

1 N

Xe^iθj

e^−iθi −K 2i

1 N

Xe^−iθj

e^iθi

= ω_i +K

2i h

re^−iθi −r^∗e^iθii

r = _N¹ PN j=1e^iθj

=⇒ order parameter

Coupled Phase Oscillators: N → ∞

Continuity equation

∂f

∂t + ∂

∂t(vf) = 0

=⇒

∂f

∂t + ∂

∂θ

ω+ K 2i

re^−iθ−r^∗e^iθ f = 0 where

r(t) = Z 2π

0

dθ Z ∞

−∞

dω e^iθf(θ, ω,t) (|r(t)| ≤1)

Ott - Antonsen Ansatz:

Expanding f(θ, ω,t) as a Fourier series in θ f(θ, ω,t) =

∞

X

m=−∞

c_m(ω,t)e^imθ

Ott - Antonsen Ansatz:

Consider a restricted class of f_n(ω,t):

f_n(ω,t) = [α(ω,t)]ⁿ,|α(ω,t)| ≤1 Then

r = Z 2π

0

dθ Z ∞

−∞

dω f e^iθ

= Z ∞

−∞

dω Z _2π

0

dθ e^iθ+

∞

X

n=1

αⁿe^i(n+1)θ+

∞

X

n=1

(α^∗)ⁿe^{−i(n−1)θ}

!g(ω) π

= Z ∞

−∞

dω g(ω) α^∗(ω, t)

Ott - Antonsen Ansatz:

Then the continuity equation becomes

∞

X

n=1

nαⁿ⁻¹∂α

∂t e^inθ+c.c

+[−K 2

r e^−iθ+r^∗ e^iθ 1 +

∞

X

n=1

αⁿ e^inθ+c.c

! +ω

∞

X

n=1

inαⁿ e^inθ+c.c

! + K

2i

∞

X

n=1

n rαⁿ e^i(n−1)θ+c.c

−K 2i

∞

X

n=1

r^∗ n α^∗n e^i(n+1)θ+c.c ]

Macroscopic Equations:

Equating coefficients of powers of e^inθ and c.c:

∂α

∂t +K

2(r α²−r^∗) +iω α= 0 But r(t) =R∞

−∞dω g(ω) α^∗(ω,t) Let

Lorentzian g(ω) =(∆

π) 1

[(ω−ω0)²+ ∆²]

=1 π

1

(ω−ω0−i∆) − 1 (ω−ω0+i∆)

Macroscopic Equations:

(ω0 − i∆)

Then

r^∗ = Z ∞

−∞

dω α(ω,t) ∆ 2π

1

[(ω−ω0)²+ ∆²]

=−1 2πi

I

c

dω α(ω,t) (ω−ω₀+i∆)

=α(ω₀−i∆, t)

By changing the variables (θ, ω)−→

θ−ω0(t), ^ω−ω_∆⁰ we can set ω₀= 0, ∆ = 1

Macroscopic Equations:

r(t) =α^∗(−i, t)

=⇒ ^dr_dt + ^K₂(r^∗r²−r) +r = 0 with r(t) =ρ(t) e^iφ

=⇒ ^dρ_dt + ^K₂(ρ²−1)ρ+ρ= 0

dρ dt +

1−1

2K

ρ+ K 2ρ³ =0

φ˙_t =0

Synchronized/ desynchronized states

ρ(t)

R =

1 +{

R ρ(0)

−1} e^{(1−12K)t −1}₂

whereR = 1−_K²¹₂

For K <Kc = 2, r −→0 as t → ∞ For K >2, r → 1− _K²¹₂

Synchronized/ Desynchronized states

Linear Stability

ρ=ρ0+ξ(t) =⇒ ^dξ_dt + 1−¹₂K ξ= 0 ξ =c⁰ e^−(1−K2)t

(i)ρ= 0: Stable forK <2, Unstable forK >2.

(ii)ρ=ρ_c = 1−_K²¹₂

: ^dξ_dt + 1−^K₂

ξ+ 3ρ²_e ξ= 0

=⇒ ^dξ_dt + (K −2)ξ= 0

=⇒ ξ =c e^−(K−2)t

References

E. Ott and T. M. Antonsen, Low dimensional behavior of large systems of globally coupled oscillators, Chaos18, 037113 (2008)

V. K. Chandrasekar, Jane H. Sheeba and M. Lakshmanan, Mass synchronization: occurrence and its control with possible applications to brain dynamics, Chaos20, 045106 (2010).