Synchronization: Bringing Order to Chaos

Transcription

1 Synchronization: Bringing Order to Chaos A. Pikovsky Institut for Physics and Astronomy, University of Potsdam, Germany Florence, May 14, /68

2 Historical introduction Christiaan Huygens ( ) first observed a synchronization of two pendulum clocks 2/68

3 He described:... It is quite worths noting that when we suspended two clocks so constructed from two hooks imbedded in the same wooden beam, the motions of each pendulum in opposite swings were so much in agreement that they never receded the least bit from each other and the sound of each was always heard simultaneously. Further, if this agreement was disturbed by some interference, it reestablished itself in a short time. For a long time I was amazed at this unexpected result, but after a careful examination finally found that the cause of this is due to the motion of the beam, even though this is hardly perceptible. 3/68

4 Lord Rayleigh described synchronization in acoustical systems: When two organ-pipes of the same pitch stand side by side, complications ensue which not unfrequently give trouble in practice. In extreme cases the pipes may almost reduce one another to silence. Even when the mutual influence is more moderate, it may still go so far as to cause the pipes to speak in absolute unison, in spite of inevitable small differences. 4/68

5 Modern Times W. H. Eccles and J. H. Vincent applied for a British Patent confirming their discovery of the synchronization property of a triode generator Edward Appleton and Balthasar van der Pol extended the experiments of Eccles and Vincent and made the first step in the theoretical study of this effect ( ) 5/68

6 Biological observations Jean-Jacques Dortous de Mairan reported in 1729 on his experiments with the haricot bean and found a circadian rhythm (24-hours-rhythm): motion of leaves continues even without variations of the illuminance Engelbert Kaempfer wrote after his voyage to Siam in 1680: The glowworms... represent another shew, which settle on some Trees, like a fiery cloud, with this surprising circumstance, that a whole swarm of these insects, having taken possession of one Tree, and spread themselves over its branches, sometimes hide their Light all at once, and a moment after make it appear again with the utmost regularity and exactness.... 6/68

7 Basic effect A pendulum clock generates a (nearly) periodic motion characterized by the period T and the frequency ω = 2π T α period T α time 7/68

8 α1,2 period T 1 Two such clocks have different periods frequencies α α2 period T 2 time Being coupled, they adjust their rhythms and have the same frequency ω 1 < Ω < ω 2. There are different possibilities: in-phase and out-of-phase α1,2 α1,2 time time in phase out of phase 8/68

9 Synchronization occurs within a whole region of parameters Ω synchronization region ω ω frequency mismatch (difference of natural frequencies) Ω difference of observed frequencies 9/68

10 Self-sustained oscillations Synchronization is possible for self-sustained oscillators only Self-sustained oscillators - generate periodic oscillations - without periodic forces - are active/dissipative nonlinear systems - are described by autonomous ODEs - are represented by a limit cycle on the phase plane (plane of all variables) 10/68

11 y x x time PHASE is the variable proportional to the fraction of the period amplitude measures deviations from the cycle - amplitude (form) of oscillations is fixed and stable - PHASE of oscillations is free A amplitude φ 2π 0 phase 11/68

12 Examples: amplifier with a feedback loop clocks: pendulum, electronic,... Speaker Amplifier input output Microphone /68

13 metronom lasers, elsctronic generators, whistle, Josephson junction, spin-torque oscillators M Damping Spin torque z Precession H 0 y x Fixed layer Spacer Free layer 13/68

14 The concept can be extended to non-physical systems! Ecosystems, predator-pray, rhythmic (e.g. circadian) processes in cells and organizms relaxation integrate-and-fire oscillators t 1 t 2 threshold level accumulation e.g., firing neuron threshold level "firing" water outflow water level T T time time cell potential time 14/68

15 Autonomous oscillator - amplitude (form) of oscillations is fixed and stable - PHASE of oscillations is free A amplitude φ 2π 0 phase A θ θ = ω 0 (Lyapunov Exp. 0) Ȧ = γ(a A 0 ) (LE γ) 15/68

16 Forced oscillator With small periodic external force (e.g. εsinωt): only the phase θ is affected dθ dt = ω 0 +εg(θ,ψ) dψ dt = ω ψ = ωt is the phase of the external force, G(, ) is 2π-periodic If ω 0 ω then ϕ = θ(t) ψ(t) is slow perform averaging by keeping only slow terms (e.g. sin(θ ψ)) dϕ = ω +εsinϕ dt Parameters in the Adler equation (1946): ω = ω 0 ω detuning ε forcing strength 16/68

17 Solutions of the Adler equation dϕ dt = ω+εsinϕ Fixed point for ω < ε: Frequency entrainment Ω = θ = ω Phase locking ϕ = θ ψ = const Periodic orbit for ω > ε: an asynchronous quasiperiodic motion ω < ε ω > ε ϕ ϕ 17/68

18 Phase dynamics as a motion of an overdamped particle in an inclined potential dϕ dt = du(ϕ) dϕ U(ϕ) = ω ϕ+εcosϕ ω < ε ω > ε 18/68

19 Synchronization region Arnold tongue ε Ω ω_0 ω synchronization region ω Unusual situation: synchronization occurs for very small force ε 0, but cannot be obtain with a simple perturbation method: the perturbation theory is singular due to a degeneracy (vanishing Lyapunov exponent) 19/68

20 More generally: synchronization of higher order is possible, whith a relation Ω ω = m n ε 2:1 1:1 2:3 1:2 1:3 ω 0 /2 ω 0 3ω 0 /2 2ω 0 3ω 0 ω Ω/ω 2 1 ω /2 0 ω 0 3ω 0 /2 2ω 0 3ω 0 ω 20/68

21 The simplest ways to observe synchronization: Lissajous figure Ω/ω = 1/1 quasiperiodicity Ω/ω = 1/2 x force 21/68

22 Stroboscopic observation: Plot phase at each period of forcing quasiperiodicity synchrony 22/68

23 Example: Periodically driven Josephson junction Synchronization regions Shapiro steps, frequency voltage V = 2e ϕ [Lab. Nat. de métrologie et d essais] 23/68

24 Example: Radio-controlled clocks Atomic clocks in the PTB, Braunschweig Radio-controlled clocks 24/68

25 Example: circadian rhythm hours Light - dark Constant conditions asleep awake Jet-lag is the result of the phase difference shift one needs to resynchronize 25/68

26 One can control re-synchronization (eg for shift-work in space) [E. Klerman, Brigham and Women s Hospital, Boston] 26/68

27 Mutual synchronization Two non-coupled self-sustained oscillators: dθ 1 dt Two weakly coupled oscillators: = ω 1 dθ 2 dt = ω 2 dθ 1 dt = ω 1 +εg 1 (θ 1,θ 2 ) dθ 2 dt = ω 2 +εg 2 (θ 1,θ 2 ) For ω 1 ω 2 the phase difference ϕ = θ 1 θ 2 is slow averaging leads to the Adler equation dϕ dt = ω +εsinϕ Parameters: ω = ω 1 ω 2 detuning ε coupling strength 27/68

28 Interaction of two periodic oscillators may be attractive ore repulsive: one observes in phase or out of phase synchronization, correspondingly /68

29 Example: classical experiments by Appleton x y Beat frequency oscilloscope Condenser readings 29/68

30 Example: Metronoms Attracting coupling: synchrony in phase Repulsive coupling: synchrony out of phase 30/68

31 Effect of noise The Langevin dynamics of the phase = the Langevin dynamics of an overdamped particle in an inclined ( ω) potential ρ = mean flow = smooth function of parameters ε > ω ε < ω Ω without noise with noise ω 31/68

32 Large noise: no synchronization Small noise: rare irregular phase slips and long phase-locked intervals 10 Phase difference ϕ no force small noise large noise time π 0 ϕ π 32/68

33 Phase of a chaotic oscillator 10 Rössler attractor: ẋ = y z ẏ = x +0.15y ż = 0.4+z(x 8.5) y 0 10 x phase should correspond to the zero Lyapunov exponent! 33/68

34 naive definition of the phase: θ = arctan(y/x) More advanced: From the condition of maximally uniform rotation [J. Schwabedal et al, PRE (2012)] y y +z x x 34/68

35 For the topologically simple attractors all definitions are good Lorenz attractor: ẋ = 10(y x) ẏ = 28x y xz ż = 8/3z +xy z (x 2 +y 2 ) 1/2 35/68

36 Phase dynamics in a chaotic oscillator A model phase equation: dθ dt = ω 0 +F(A) (first return time to the surface of section depends on the coordinate on the surface) A: chaotic phase diffusion broad spectrum (θ(t) θ(0) ω 0 t) 2 D p t D p measures coherence of chaos (θ ω0t)/2π time 36/68

37 dθ dt = ω 0 +F(A) F(A) is like effective noise Synchronization of chaotic oscillators synchronization of noisy periodic oscillators phase synchronization can be observed while the amplitudes remain chaotic 37/68

38 Synchronization of a chaotic oscillator by external force If the phase is well-defined Ω = dθ dt is easy to calculate (e.g. Ω = 2πlim t N t /t, N is a number of maxima) Forced Rössler oscillator: ẋ = y z +E cos(ωt) ẏ = x +ay Ω ω ż = 0.4+z(x 8.5) ω E phase is locked, amplitude is chaotic 38/68

39 Stroboscopic observation Autonomous chaotic oscillator: phases are distributed from 0 to 2π. Under periodic forcing: if the phase is locked, then W(θ,t) has a sharp peak near θ = ωt +const. 15 synchronized asynchronous 5 y x x 39/68

40 Phase synchronization of chaotic gas discharge by periodic pacing [Tacos et al, Phys. Rev. Lett. 85, 2929 (2000)] Experimental setup: FIG. 2. Schematic representation of our experimental setup. 40/68

41 Phase plane projections in non-synchronized and synchronized cases 0.01 a) 0.01 b) I (n+12) 0 I (n+12) I (n) I (n) 41/68

42 Synchronization region: Pacer Amplitude (V) Frequency (Hz) 42/68

43 Electrochemical chaotic oscillator [Kiss and Hudson, Phys. Rev. E 64, (2001)] 43/68

44 The synchronized oscillator remains chaotic: 44/68

45 Frequency difference as a function of driving frequency for different amplitudes of forcing: Ω synchronization region ω 45/68

46 Synchronization region: ε ω_0 ω 46/68

47 Unified description of regular, noisy, and chaotic oscillators autonomous oscillators forced oscillators periodic noisy chaotic 47/68

48 Many coupled oscillators Typical setups: Lattices Networks Global coupling 48/68

49 Lattice of phase oscillators dϕ n dt = ω n +εq(ϕ n+1 ϕ n )+εq(ϕ n 1 ϕ n ) 2 2 a) 0 2 b) 0 2 Small ε: no synchrony Intermediate ε: clusters Large ε: full synchrony (one cluster) Ω k c) site k 49/68

50 Continuous phase profile In a large system frequencies can be entrained but the phases widely distributed For a continuous phase profile in the case of noisy/chaotic oscillators one obtains a KPZ equation ϕ t = ω(x)+ε 2 ϕ+β( ϕ) 2 +ξ(x,t) Roughening means lack of phase synchrony on large scales 50/68

51 Dissipative vs conservative coupling dϕ n = ω +q(ϕ n+1 ϕ n )+q(ϕ n 1 ϕ n ) dt For the phase difference v n = ϕ n+1 ϕ n we get dv n dt = q( v n)+q(v n+1 ) q( v n 1 ) q(v n ) Odd and even parts of the coupling function: q(v) = εsinv +αcosv: dv n dt = α d cosv +ε d sinv where d and d are discrete nabla and Laplacian operators 51/68

52 purely conservative coupling, traveling waves (compactons) dvn = d cos v = cos vn+1 cos vn 1 dt v π/ time space 52 / 68

53 Global coupling of oscillators Each-to-each coupling Mean field coupling N X Σ 53/68

54 Physical systems with global coupling Josephson junction arrays Generators (eg spin-torque oscillators, electrochemical oscillators,...) with a common load Multimode lasers 54/68

55 Example: Metronoms on a common plattform 55/68

56 Example: blinking fireflies Also electronic fireflies: 56/68

57 Genetic fireflies [Pridle et al, Nature (2011)] a Coupling Reporter c H 2 O 2 ndh sfgfp LuxR-AHL AHL luxi aiia 5,000 cells per biopixel 2.5 million total cells b Dissolved AHL Media H 2 O 2 vapour Oscillator d Fluorescence (arbitrary units) Biopixel number ,000 Time (min) Biopixel GFP Mean ,000 Time (min) 57/68

58 A macroscopic example: Millenium Bridge 58/68

59 Experiment with Millenium Bridge 59/68

60 Kuramoto model: coupled phase oscillators Generalize the Adler equation to an ensemble with all-to-all coupling φ i = ω i +ε 1 N sin(φ j φ i ) N Can be written as a mean-field coupling φ i = ω i +ε( X sinφ i +Y cosφ i ) X +iy = M = 1 N j=1 j e iφ j The natural frequencies are distributed around some mean frequency ω 0 ω 0 g(ω) ω 60/68

61 Synchronisation transition M small ε: no synchronization, phases are distributed uniformly, mean field = 0 large ε: synchronization, distribution of phases is nonuniform, mean field 0 61/68

62 Theory of transition Like the mean-field theory of ferromagnetic transition: a self-consistent equation for the mean field M = 2π 0 π/2 n(φ)e iφ dφ = Mε g(mεsinφ)cosφ e iφ dφ π/2 M ε c ε 62/68

63 Globally coupled chaotic oscillators Each chaotic oscillator is like a noisy phase oscillator A regular mean field appears at the critical coupling ε cr. Example: Rössler oscillators with Gaussian distribution of frequencies 10 (a) (b) 10 y 0 0 Y ẋ i = ω i y i z i +εx, ẏ i = ω i x i +ay i, 10 (c) (d) 10 ż i = 0.4+z i (x i 8.5), y Y x X 20 63/68

64 Experiment Experimental example: synchronization transition in ensemble of 64 chaotic electrochemical oscillators Kiss, Zhai, and Hudson, Science, Finite size of the ensemble yields fluctuations of the mean field 1 N rms(x) ε 64/68

65 Synchronisation transition at zero temperature [M. Rosenblum, A. P., PRL (2007) ] Identical oscillators = zero temerature Attraction: synchronization Repulsion: desynchronization Small each-to-each coupling coupling via linear mean field linear unit Y N Strong each-to-each coupling coupling via nonlinear mean field X Σ nonlinear unit Y N X Σ 65/68

66 Loss of synchrony with increase of coupling Attraction for small coupling Repulsion at large coupling A state on the border, where the mean field is finite but the oscillators are not locked (quasiperiodicity!) establishes 1.1 order parameter Rωosc, Ωmf coupling ε 66/68

67 Experiment [Temirbayev et al, PRE, 2013] Linear coupling Nonlinear coupling R 1 (a) (d) (b) (e) Amin [V] (c) (f) fmf, fi [khz] ε ε 67/68

68 Other topics Synchronization by common noise [Braun et al, EPL (2012)] Control of synchrony (eg for suppression of pathological neural synchrony at Parkinson [Montaseri et al, Chaos (2013)]) Synchrony in multifrequency populations (eg resonances ω 1 +ω 2 = ω 3, [Komarov an A.P., PRL (2013)]) Inverse problems: infer coupling function from the signals (eg ECG + respiration signals yield coupling Heart-Respiration, [Kralemann et al, Nat. Com. (2013)]) 68/68