Nonlinear Dynamics: Synchronisation

Transcription

1 Nonlinear Dynamics: Synchronisation Bristol Centre for Complexity Sciences Ian Ross BRIDGE, School of Geographical Sciences, University of Bristol October 19, / 16

2 I: Introduction 2 / 16

3 I: Fireflies FIREFLY MOVIE HERE From The Trials of Life, copyright BBC (1990). 3 / 16

4 I: Applause APPLAUSE SOUND CLIP HERE From Clappers: A History of Applause, copyright BBC (2003). 4 / 16

5 I: Other examples Circadian rhythms Wing/respiration frequency of birds in flight Gait/breathing when running Muscular contractions in mammalian intestine Belousov-Zhabotinskii reaction 5 / 16

6 II: A basic example fireflies 6 / 16

7 II: Vector fields on the circle The state space Most basic abstraction of an oscillation Ignore amplitude of oscillations and concentrate on phase State space is the circle, S 1 States are parameterised by the angle θ mod 2π 7 / 16

8 II: Vector fields on the circle The setup We have a continuous-time dynamical system in the form of a one-dimensional ODE: dθ dt = f (θ). We need f (θ + 2π) = f (θ) for single-valuedness and smoothness. ON OFF 7 / 16

9 II: Vector fields on the circle A simple example dθ dt = sin θ Equilibria at f (θ ) = sin θ = 0, i.e. at θ = 0 and θ = π. The stability of the equilibria can be determined as: df df dθ > 0 θ=0 dθ < 0 θ=π unstable stable θ * = π θ * = 0 7 / 16

10 II: Vector fields on the circle Uniform oscillator dθ dt = ω with ω a constant. This has solution θ(t) = θ(0) + ωt and the period of the oscillation is T = 2π ω. 7 / 16

11 II: Vector fields on the circle Non-uniform oscillator dθ dt = ω a sin θ For a = 0, this is just the uniform oscillator. Putting a 0 introduces a non-uniformity into the flow. dθ/dt a < ω SLOW θ FAST 7 / 16

12 II: Vector fields on the circle Non-uniform oscillator dθ dt = ω a sin θ For a = 0, this is just the uniform oscillator. Putting a 0 introduces a non-uniformity into the flow. dθ/dt a = ω θ 7 / 16

13 II: Vector fields on the circle Non-uniform oscillator dθ dt = ω a sin θ For a = 0, this is just the uniform oscillator. Putting a 0 introduces a non-uniformity into the flow. dθ/dt a > ω θ * 2 θ * 1 θ * 1 θ * 2 θ 7 / 16

14 II: Vector fields on the circle Non-uniform oscillator dθ dt = ω a sin θ For a = 0, this is just the uniform oscillator. Putting a 0 introduces a non-uniformity into the flow. T T = 2π ω 2 a 2 2π/ω ω a 7 / 16

15 II: Firefly phenomonology Experimental results 770 ms 750 ms Interval (ms) Time (s) Ermentrout & Rinzel (1984), Am. J. Physiol. 246, R / 16

16 II: Firefly phenomonology Experimental results Forcing period: 770 ms Probability Phase (φ/2π) Ermentrout & Rinzel (1984), Am. J. Physiol. 246, R / 16

17 II: Firefly phenomonology Experimental results Forcing period: 750 ms Probability Phase (φ/2π) Ermentrout & Rinzel (1984), Am. J. Physiol. 246, R / 16

18 II: Firefly phenomonology Interpretation Fireflies flash at a natural interval of about 0.9 s. There is a weak coupling between different fireflies. Entrainment is possible when f drive f natural, where f drive is the driving frequency and f natural the natural frequency. For a given f = f drive f natural, we can find the frequency difference in the coupled system, F for f < δ crit, F = 0. 8 / 16

19 II: Firefly model A simple model 1 Call the firefly phase θ(t), with a flash occurring when θ = 0. The unforced system has dθ/dt = ω. Call the phase of the stimulus Θ, with dθ/dt = Ω and with a flash occurring when Θ = 0. If the stimulus is ahead of the natural cycle, the firefly speeds up. If the stimulus is behind the natural cycle, the firefly slows down. Model this as dθ = ω + A sin(θ θ) dt with A > 0, implying that if Θ is ahead of θ, i.e. 0 < Θ θ < π, the firefly speeds up (dθ/dt > ω). 9 / 16

20 II: Firefly model A simple model 2 Consider the phase shift between the stimulus and the firefly, φ = Θ θ. Then dφ dt = dθ dt dθ = Ω ω A sin φ. dt Putting τ = At, µ = (Ω ω)/a, this gives a non-uniform oscillator as we examined before: dφ = µ sin φ. dτ The parameter µ measures the frequency difference as compared to the restoring strength A for small µ, we expect entrainment. 9 / 16

21 II: Firefly model Results dφ/dτ φ µ = 0 φ = 0 lim t φ = 0 Here, the firefly eventually synchronises to the stimulus exactly, with zero phase difference. 9 / 16

22 II: Firefly model Results dφ/dτ φ 0 < µ < 1 φ > 0 lim t φ = φ = non-zero constant Here, there is also entrainment, but with a non-zero phase different the firefly lags behind the stimulus. 9 / 16

23 II: Firefly model Results dφ/dτ φ µ > 1 No φ lim t φ does not exist Here, there is no entrainment, but a continuous drift in the phase difference between the stimulus and the firefly. 9 / 16

24 II: Firefly model Predictions 1. Entrainment will occur for a range of driving frequencies given by ω A Ω ω + A, with continuous phase drift outside this range. 2. For any Ω where entrainment is possible, the phase shift between the stimulus and the firefly will be given by sin φ = Ω ω A. Both of these predictions could be tested experimentally. Real fireflies are a bit more complicated than this. Our model is good for some species (e.g. Pteroptyx cribellata), but others are able to alter their natural frequency ω and this requires a more complex model (e.g. Pteroptyx malaccae). 9 / 16

25 III: Some theory 10 / 16

26 III: Self-sustained oscillations Periodic solutions In our firefly example, we abstracted the periodic solution to a simple phase oscillator this can be done in the general case as well. Synchronisation relies on the existence of self-sustained oscillations. 11 / 16

27 III: Self-sustained oscillations Stable periodic orbits are exponentially attracting Periodic solutions of dynamical systems can be stable or unstable, just as for equilibrium points. A stable periodic orbit (or limit cycle) exponentially attracts nearby trajectories. 11 / 16

28 III: Persistence of phase perturbations Perturbations Perturb by ( ρ, φ) Exponential attraction: ρ 0 Phase perturbation, φ, persists Easy to modify phase synchronisation Zero Lyapunov exponent along the orbit 12 / 16

29 IV: Generalisations Mutual synchronisation of two oscillators Collective synchronisation of globally coupled oscillators Fractional resonances Synchronisation of chaotic systems Extended media 13 / 16

30 V: Noise and synchronisation 14 / 16

31 V: Stochastic resonance example system STOCHASTIC RESONANCE MOVIE HERE 15 / 16

32 V: Stochastic resonance example system Timestep 15 / 16

33 V: Stochastic resonance Dansgaard-Oeschger events Results with purely stochastic forcing A. Ganopolski & S. Rahmstorf (2002), Phys. Rev. Lett. 88(3), art. no / 16

34 V: Stochastic resonance Dansgaard-Oeschger events Results with combined stochastic and periodic forcing A. Ganopolski & S. Rahmstorf (2002), Phys. Rev. Lett. 88(3), art. no / 16