Dynamics of Limit Cycle Oscillators/ Phase Oscillators

Dynamics of Limit Cycle Oscillators/ PhaseOscillators

M. Lakshmanan

Centre for Nonlinear DynamicsSchool of Physics

Bharathidasan UniversityTiruchirappalli – 620024

India

SERC School on ”Nonlinear Dynamics”Panjab University, Chandigarh

27-30, January 2014

Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators

Limit Cycle Oscillators

Consider the van der Pol oscillator exhibiting limit cycle motion:

x + ω2x = εa(1− bx2)x

x + ω2x = ε[−ω2

0x + a(1− bx2)x]

Identify multiple time scales: t0 = t (actual time)t1 = εt (slow time)t2 = ε2t, .....



d

dt=

∂

∂t0+ ε

∂

∂t1+ ε2 ∂

∂t2+ ....

or

D = D0 + εD1 + ε2D2 + ....

and

d2

dt2= D2 = D2

0 + 2εD0D1 + O(ε2)

D20x + 2εD0D1x + ω2

0x = −εω20x + εa(1− bx2)(D0x + εD1x) + O(ε2)

Equating various powers of ε to zero=⇒.



ε0: D20x + ω2x = 0=⇒ x = A(t1)e iωt0 + c .c

ε1 = 2D0D1x = −ω20x + a(1− b2x)D0x

=⇒ [2iω

dA

dt1+ ω2

0A− iωa(1− b|A|2)A

]+iωA3e2iωt0 − iωA?e−2iωt0 = 0

On averaging over the fast variable over a period T = 2πω

A = (a + iω)A

2− ab

2|A|2A

Or in the standard notations:

z =[(a + iω)− |z |2

]z(t)

=⇒ ”Stuart-Landau (SL) Oscillator”


Stuart- Landau Oscillator

z(t) =[(a + iω)− |z |2

]z(t)

=⇒ z(t) = x + iy = r(t)e iθ(t)

=⇒ r = r(a− r2)θ = ω

Equilibrium states: (i) r = 0, (ii) r =√a

Linearized Equation: r = r0 + ξ, ξ << 1

=⇒ ξ = (a− 3r20 )ξ

ξ(t) = ξ(0)e(a−3r20 )t

(i) r = 0, ξ = ξ(0)eat : Stable for a < 0

(ii) r =√a, ξ = ξ(0)e−2at : Stable for a > 0


Stuart- Landau Oscillator

=⇒ Hopf bifurcation

Solution r(t) =√a

(1+e(−at+c))12

θ(t) = ωt + δ


Consider two coupled van der Pol’s Oscillator

x1 + ω21x1 = a1(1− b1x

21 )x1 + k(x1 − x2)

x2 + ω22x2 = a2(1− b2x

22 )x2 + k(x2 − x1)

Coupled Stuart- Landau Oscillators:

z1 = (1 + iω1 − |z1(t)|2)z1(t) + K [z2(t)− z1(t)]

z2 = (1 + iω2 − |z2(t)|2)z2(t) + K [z1(t)− z2(t)]


In polar coordinates

r1 = r1(1− K − r21 ) + Kr2 cos(θ2 − θ1)

r2 = r2(1− K − r22 ) + Kr1 cos(θ1 − θ2)

θ1 = ω1 + Kr1r2

sin(θ2 − θ1)

θ2 = ω2 + Kr1r2

sin(θ1 − θ2)

Weak coupling approximation:Separation of variables:Short time: Relaxation to limit cycleLong time : Phases interact =⇒ r1 r2 ≈ const


SL Oscillator: Strong coupling limit:

SL Oscillator: Strong coupling limit =⇒ Amplitude Effects

Linear Stability Analysis

|M − λI | =∣∣∣(1 + iω1 − 2|z1|2 − K − λ) −K

−K (1 + iω2 − 2|z2|2 − K − λ)

∣∣∣ = 0

(a) |z01 | = |z0

2 | = 0

λ2 − 2(a + i ω)λ+ (b1 + ib2) + c = 0

Here a = 1− K , b = a2 − ω2 + ∆2

4 , b2 = 2aω, c = −K 2


SL Oscillator: Strong coupling limit:

With λ = α + iβ, α = 0=⇒ Critical Conditions:K = 1K = 1

2 (1 + ∆2

4 ), (K ,∆) = (1, 2)


Globally Coupled array of SL oscillator:

zj = zj(1− |zj |2 + iωj) +K

N

N∑i=1

(zi − zj)

Order parameter: Re iψ =∑N

i=1 ziEvolution of order parameter/ Synchronization/Desynchronization.


Phase Oscillators:

Phase Oscillator DynamicsN=2

θ1 = ω1 + K sin(θ2 − θ1)

θ2 = ω2 + K sin(θ1 − θ2)

Identical Oscillatorsω1 = ω2, and define φ = θ2 − θ1

=⇒ φ = −2K sinφ.Equilibrium points: φ = 0 =⇒ θ1 = θ2: symmetric state.φ = π =⇒ θ1 = θ2 + π: antisymmetric state.Phase-locking: Synchrony

: Symmetry breaking


Phase Oscillators:

Examples: Animal gaits

3 -oscillators

4 -oscillators


Phase Oscillators:

Non-identical Oscillators:

θ1 = ω1 + K sin(θ2 − θ1)

θ2 = ω2 + K sin(θ1 − θ2)

=⇒ φ = ∆− 2K sinφ∆ = |ω1 − ω2|, φ = θ2 − θ1

Phase locking only if ∆ ≤ 2K

Then θ = θ1+θ22 = ω1+ω2

2 : Common frquencyFrequency entrainment


N- Coupled Oscillators- Kuromoto Model:

θi = ωi + KN

∑Nj=1 sin(θj − θi ), i = 1, 2, .....N

Consider frequency distribution as a unimodal functiong(ω) = g(−ω).

Global Coupling =⇒ Mean field approximation.

Define the complex order parameter

re iψ =1

N

N∑j=1

e iθj

=⇒ r(t): A measure of phase coherenceψ(t): Average phase



r =√

1− KcK for Lorenzian distribution g(ω) = r

π(γ2+ω2)

Second order phase transition:

θi = ωi + Kr sin(ψ − θi), i = 1, 2, 3, .....,N



r = 1: Synchrony0 < r < 1- Partial synchronizationr = 0 - Desynchronization(phase drift)



Kuromoto: For r = constant, the threshold condition forsynchrony is K ≥ Kc , Kc = 2

πg(0)


Synchronization of fireflies

Croaking of frogs.


Delay coupling

dφi (t)

dt= ω0 + K

∑j

sin [φj(t − τ)− φi (t)]

Nonlocal coupling

dφ(t)

dt= ω −

∫ π

−πG (x − x

′) sin

[φ(x , t)− φ(x

′, t) + α]dx

′]

G (y) =k

2exp(−K |y |)

Chimera states: Simultaneous existence of coherent andincoherent states.


References:

Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence,Springer-Verlag, Berlin, (1984).

S. H. Strogatz, From Kuramoto to Crawford: exploring theonset of synchronization in populations of coupled oscillators,Physica D 143, 1-20 (2000)

A. Sen, D. Ramana Reddy, G.L. Johnston & G.C. Sethia,Amplitude death, Synchrony, and Chimera in Delay CoupledLimit Cycle Oscillators, in ”Complex Delay Systems” Ed. F.Atay, Springer (2010).

M. Lakshmanan & D. V. Senthilkumar , Nonlinear Dynamicsof Time Delay Systems,Springer-Verlag, (2011).


Dynamics of Limit Cycle Oscillators/ Phase Oscillators

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