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Two coupled phase oscillators

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Introduction to two coupled phase oscillators (Kuramoto model). Based on a lecture I gave to 4th year undergraduates at the University of Bristol.

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Two coupled phase oscillators

  1. 1. Synchronisation — Two oscillators Naoki Masuda Department of Engineering Mathematics naoki.masuda@bristol.ac.uk http://www.naokimasuda.net Modified from a lecture I gave in Bristol (EMATM001 Advanced Nonlinear Dynamics and Chaos) Naoki Masuda Synchronisation — Two oscillators 1 / 14
  2. 2. Synchronous oscillations (and synchronous movements in general) • Fireflies https://www.youtube.com/watch?v=0BOjTMkyfIA • Clapping https://www.youtube.com/watch?v=Go8jd8CSqzY • Candle frames https://www.youtube.com/watch?v=ndNBSgUd-vU • Dancing robots https://www.youtube.com/watch?v=SPlYYV4lC1g • Millennium Bridge https://www.youtube.com/watch?v=eAXVa__XWZ8 • Suprachiasmatic nucleus in the brain https://www.youtube.com/watch?v=dqZTrpgilzQ (4 day recording) • Students’ synchronized walking https://www.youtube.com/watch?v=E7cQtbMtODk • Metronomes https://www.youtube.com/watch?v=ZMApCadGSt0 Naoki Masuda Synchronisation — Two oscillators 2 / 14
  3. 3. Christian Huygens (1629–1695) • Dutch physicist, mathematician, astronomer and inventor, • Pendulum clock (1656) • ‘An odd sympathy’, an unexpected discovery he made at home (1665) Left figure: public domain; right figure: original drawing by Huygens Naoki Masuda Synchronisation — Two oscillators 3 / 14
  4. 4. Sync, but not oscillatory or dynamic in the end Deffuant model of collective opinion dynamics (Deffuant, et al., Advances in Complex Systems, 3, 87–98, 2000): Interact if and only if |xi (t) − xj (t)| < ϵ, { xi (t + 1) = xi (t) + κ [xj (t) − xi (t)] xj (t + 1) = xj (t) + κ [xi (t) − xj (t)] 0 20 40 60 80 100 time 0.0 0.2 0.4 0.6 0.8 1.0 agent'sopinion Dynamics of the Deffuant model. N = 100 agents, ϵ − 0.25, κ = 0.2. Naoki Masuda Synchronisation — Two oscillators 4 / 14
  5. 5. Two ways to synchronise Left: figure in the public domain. Right: clip from the video: https://www.youtube.com/watch?v=oJ2ZLr87lLY Q: Which of the two sync mechanisms is at work in the following examples? • Fireflies? • Clapping? • Millennium bridge? • Metronomes? • Candle frames? • Students’ sync walking? • Heart? • Circadian clock? • Dancing robots? Naoki Masuda Synchronisation — Two oscillators 5 / 14
  6. 6. Phase dynamics of two coupled phase oscillators { ˙ϕ1 = ω1 + κ sin(ϕ2 − ϕ1) ˙ϕ2 = ω2 + κ sin(ϕ1 − ϕ2) where ϕi (i = 1, 2) is the phase variable, ∈ [0, 2π), rotating, ωi is the angular velocity, and κ is the coupling strength. Q: 1 What happens if κ = 0? 2 Taylor expand the sin term and tell its role when ϕ1 and ϕ2 are not too far. 3 What do you expect as κ(> 0) increases? 4 What do you expect as κ goes negative large? 5 Synchronisation easier or harder as |ω2 − ω1| becomes larger? 6 Why sin? Naoki Masuda Synchronisation — Two oscillators 6 / 14
  7. 7. Analysis of a two-oscillator system ˙ϕ1 = ω1 + κ sin(ϕ2 − ϕ1) ˙ϕ2 = ω2 + κ sin(ϕ1 − ϕ2) • Let ψ ≡ ϕ2 − ϕ1 and ∆ω = ω2 − ω1. What dynamics does ψ obey? ˙ψ = ∆ω − 2κ sin ψ Worked example 11.1 Show that this system have a solution (i.e. ˙ψ = 0) when ∆ω 2κ ≤ 1 Is this condition intuitive? Naoki Masuda Synchronisation — Two oscillators 7 / 14
  8. 8. Analysis of a two-oscillator system Worked example 11.2 Analyse ˙ψ = ∆ω − 2κ sin ψ by drawing a bifurcation diagram in terms of κ. Which bifurcation happens where? Perfect synchrony (i.e. ψ = 0) happens? For small positive κ, what is happening? Naoki Masuda Synchronisation — Two oscillators 8 / 14
  9. 9. Analysis of a two-oscillator system Worked example 11.3 Do a linear stability analysis of phase-locked solutions (why are they so called?) of ˙ψ = ∆ω − 2κ sin ψ when κ > ∆ω/2. A: By setting ˙ψ = 0, we get sin ψ∗ = ∆ω/2κ. Set ψ = ψ∗ + ϵ, where ϵ is small, to obtain ˙ϵ = ∆ω − 2κ sin(ψ∗ + ϵ) = ∆ω − 2κ(sin ψ∗ + ϵ cos ψ∗ ) = −2κ cos ψ∗ · ϵ So the in-phase solution (0 < ψ∗ < π/2, assuming ω1 < ω2) is linearly stable, whereas the anti-phase solution (π/2 < ψ∗ < π) is linearly unstable. Naoki Masuda Synchronisation — Two oscillators 9 / 14
  10. 10. Analysis of a two-oscillator system Worked example 11.4 What is the oscillation frequency when the phase locking is happening? ˙ϕ1 = ω1 + κ sin(ϕ2 − ϕ1) ˙ϕ2 = ω2 + κ sin(ϕ1 − ϕ2) ˙ψ = ∆ω − 2κ sin ψ Is the solution intuitive? A: Under phase locking, sin ψ∗ = ∆ω/2κ. So, ˙ϕ1 = ω1 + κ sin ψ∗ = ω1 + ω2 2 Naoki Masuda Synchronisation — Two oscillators 10 / 14
  11. 11. Back to Huygens Oliveira & Melo, Scientific Reports, 5, 11548 (2015) https://doi.org/10.1038/srep11548 • Andronov clock model (1966) ¨θ + µ · sign( ˙θ) + ω2 θ = 0 • Plus kicking in a constant energy to compensate the loss of kinetic energy due to dry friction • µ(> 0): dry friction coefficient, at θ ≈ 0 in each cycle • ω: natural angular frequency of the pendulum This and the following figures are from the Oliveira & Melo paper, which has been published under CC BY license. Naoki Masuda Synchronisation — Two oscillators 11 / 14
  12. 12. Two clocks • Assumption: When one clock receives a kick, the impact propagates in the wall to instantaneously perturb the other clock slightly. • Sound travels fast. { ¨θ1 + µ1 · sign( ˙θ1) + ω2 1θ1 = −α1F(θ2), ¨θ2 + µ2 · sign( ˙θ2) + ω2 2θ2 = −α2F(θ1). plus kicking, with ω1 = ω + ϵ and ω2 = ω − ϵ Naoki Masuda Synchronisation — Two oscillators 12 / 14
  13. 13. Flavour of analysis • ϕn: The phase of clock 2 when the phase of clock 1 is 2nπ. • Derive the Poincar´e map: ϕn+1 = T(ϕn) • Can show that T has a stable fixed point near π. • What does this mean physically? • Consistent with Huygens’ observation. Simulations Red: ϵ = 1.5 × 10−4 rad/s Black: ϵ = 3 × 10−3 rad/s ω = 4.4879 rad/s Naoki Masuda Synchronisation — Two oscillators 13 / 14
  14. 14. Experiments In the bottom panel, the free clock freq of the two clocks are closer than in the top panel. Naoki Masuda Synchronisation — Two oscillators 14 / 14

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