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, .....
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators
Limit Cycle Oscillators
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=⇒.
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators
Limit Cycle Oscillators
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”
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators
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
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators
Stuart- Landau Oscillator
=⇒ Hopf bifurcation
Solution r(t) =√a
(1+e(−at+c))12
θ(t) = ωt + δ
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators
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)]
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators
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
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators
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
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators
SL Oscillator: Strong coupling limit:
With λ = α + iβ, α = 0=⇒ Critical Conditions:K = 1K = 1
2 (1 + ∆2
4 ), (K ,∆) = (1, 2)
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators
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.
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators
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
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators
Phase Oscillators:
Examples: Animal gaits
3 -oscillators
4 -oscillators
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase 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
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators
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
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators
N- Coupled Oscillators- Kuromoto Model:
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
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators
N- Coupled Oscillators- Kuromoto Model:
r = 1: Synchrony0 < r < 1- Partial synchronizationr = 0 - Desynchronization(phase drift)
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators
N- Coupled Oscillators- Kuromoto Model:
Kuromoto: For r = constant, the threshold condition forsynchrony is K ≥ Kc , Kc = 2
πg(0)
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators
Synchronization of fireflies
Croaking of frogs.
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators
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.
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators
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).
Prof. M. Lakshmanan Dynamics of Limit Cycle Oscillators/ Phase Oscillators