A bifurcation approach to the synchronization of coupled van der Pol oscillators

Transcription

1 A bifurcation approach to the synchronization of coupled van der Pol oscillators Departmento de Matemática Aplicada II & Instituto de Matemáticas de la Universidad de Sevilla (IMUS) Joint work with G. Paccosi and A. Figliola Universidad Nacional de General Sarmiento (Buenos Aires, Argentina) Puebla, February 2012

2 What is synchronization? Synchronization = Adjustment of rhythms of oscillating objects due to their weak interaction 1 1 Synchronization A universal concept in nonlinear sciences, A. Pikovsky, M. Rosenblum and J. Kurths

3 What is synchronization? Synchronization = Adjustment of rhythms of oscillating objects due to their weak interaction. Calendario Maya Tzolkin (divine) = 260 days. Haab (civil) = 365 days. La Rueda calendárica, Serie Inicial or Cuenta larga /5 = 18,980 days = 52 years. Three synchronized calendars (conmensurate) The length of the year for the mayas is 365, which is more accurate than the Gregorian calendar 365, st December 2012 is the end of a year cycle of the Cuenta Larga calendar that started in a.c.

4 Outline Message of the talk Simulation or Continuation IVP or BVP Motivation: broad band synchronization [1]. Structure of the synchronization tongues: isolas. The onset of 1: k synchronization; a BVP approach. The role of the tori. Conclusions [1] A. P. Kuznetsov, J. P. Roman. Physica D (2009) Properties of synchronization in the systems of non-identical coupled van der Pol-Duffing oscillators. Broadband synchronization.

5 Outline Message of the talk Simulation or Continuation IVP or BVP Motivation: broad band synchronization [1]. Structure of the synchronization tongues: isolas. The onset of 1: k synchronization; a BVP approach. The role of the tori. Conclusions [1] A. P. Kuznetsov, J. P. Roman. Physica D (2009) Properties of synchronization in the systems of non-identical coupled van der Pol-Duffing oscillators. Broadband synchronization.

6 Outline Message of the talk Simulation or Continuation IVP or BVP Motivation: broad band synchronization [1]. Structure of the synchronization tongues: isolas. The onset of 1: k synchronization; a BVP approach. The role of the tori. Conclusions [1] A. P. Kuznetsov, J. P. Roman. Physica D (2009) Properties of synchronization in the systems of non-identical coupled van der Pol-Duffing oscillators. Broadband synchronization.

7 Outline Message of the talk Simulation or Continuation IVP or BVP Motivation: broad band synchronization [1]. Structure of the synchronization tongues: isolas. The onset of 1: k synchronization; a BVP approach. The role of the tori. Conclusions [1] A. P. Kuznetsov, J. P. Roman. Physica D (2009) Properties of synchronization in the systems of non-identical coupled van der Pol-Duffing oscillators. Broadband synchronization.

8 The model: velocity coupled van der Pol oscillators Oscillator 1 Oscillator 2 X Ẋ λ 1 1 µ Y λ 2 Ẏ 1 + δ {ẍ ( λ1 x 2) ẋ + x = µ (ẏ ẋ) ÿ ( λ 2 y 2) ẏ + (1 + δ)y = µ (ẋ ẏ) The λ s control the Hopf bifurcation and are taken as λ 1 = 1 + λ and λ 2 = 1. δ is the frequency mismatch. µ parametrizes the dissipative coupling.

9 Measuring synchrony in time simulation [1] ynamic regime chart for the system (1) for = 1 25, = 1, = 0 and phase plane po

10 Measuring synchrony II: Poincaré-like sections [1] 1: 1 1: 3 1: 5

11 Bifurcation set: λ = 0 [1] Fig. 1. (a) Dynamic regime chart and (b) its enlarged fragment for the system (1) for = = 1, = 0.

12 Bifurcation set: λ = 0,25 [1] ynamic regime chart for the system (1) for = 1 25, = 1, = 0 and phase plane portraits in characteristic areas of the paramet

13 Bifurcation set: λ = 1 [1]

14 Bifurcation set: λ = 0,25 Zoom[1]

15 Questions Are the change of colours bifurcations? Can we compute the separations curves? What is the structure of the synchronization tongues? What happens in the white regions? Do sync-people speak the same language as the dinsys-people?

16 Bifurcation approach: λ = 0,25 Start from the trivial unique equilibrium. Scan either in the µ or δ direction with a fixed value of λ. Locate a bifurcation point and follow it in two parameters. Repeat the process with the periodic orbits and tori (if possible) B Hopf Bifurcation Oscillator death area!

17 One param. bifurcation diagrams: λ = 0,25 δ = 10

18 Two param. bifurcation diagrams: λ = 0,25 µ B Hopf Bifurcation Oscillator death area! A Hopf Bifurcation Hopf Hopf Bifurcation Oscillator death area 1:1 1:3 1:

19 One param. bifurcation diagrams: λ = 0,25 δ = 9,5

20 One param. bifurcation diagrams: λ = 0,25 δ = 9,45

21 One param. bifurcation diagrams: λ = 0,25 δ = 9,45

22 Looking for the 1:3 sync.: degraded top λ = 0, x µ= = y y x 10 3

23 Looking for the 1:3 sync.: degraded top λ = 0, µ= = y y

24 Looking for the 1:3 sync.: degraded top λ = 0, µ= = y y

25 Looking for the 1:3 sync.: degraded top λ = 0, µ= = y y

26 Looking for the 1:3 sync.: degraded top λ = 0,25 The upper border of the degraded top is not a bifurcation. It is a geometrical feature (tangency). Can we trace in two parameters the locus of such events? Yes, by setting up an appropriate BVP dx dτ = Tx 1 dy dτ = Ty 1 dx 1 dτ = T [(λ 1 x 2 )x 1 x µ(x 1 y 1 )] dx 1 dτ = T [(1 y 2 )y 1 (1 + δ)y µ(y 1 x 1 )] with periodicity boundary conditions: x(0) = x(1), ẋ(0) = x 1 (0) = ẋ(1) = x 1 (1) y(0) = y(1), ẏ(0) = y 1 (0) = ẏ(1) = y 1 (1) phase condition: y (0) = y1(0) = 0 tangency condition: y1(0) = 0

27 Looking for the 1:3 sync.: degraded top λ = 0,25 The upper border of the degraded top is not a bifurcation. It is a geometrical feature (tangency). Can we trace in two parameters the locus of such events? Yes, by setting up an appropriate BVP dx dτ = Tx 1 dy dτ = Ty 1 dx 1 dτ = T [(λ 1 x 2 )x 1 x µ(x 1 y 1 )] dx 1 dτ = T [(1 y 2 )y 1 (1 + δ)y µ(y 1 x 1 )] with periodicity boundary conditions: x(0) = x(1), ẋ(0) = x 1 (0) = ẋ(1) = x 1 (1) y(0) = y(1), ẏ(0) = y 1 (0) = ẏ(1) = y 1 (1) phase condition: y (0) = y1(0) = 0 tangency condition: y1(0) = 0

28 Looking for the 1:3 sync.: degraded top λ = 0,25 The upper border of the degraded top is not a bifurcation. It is a geometrical feature (tangency). Can we trace in two parameters the locus of such events? Yes, by setting up an appropriate BVP dx dτ = Tx 1 dy dτ = Ty 1 dx 1 dτ = T [(λ 1 x 2 )x 1 x µ(x 1 y 1 )] dx 1 dτ = T [(1 y 2 )y 1 (1 + δ)y µ(y 1 x 1 )] with periodicity boundary conditions: x(0) = x(1), ẋ(0) = x 1 (0) = ẋ(1) = x 1 (1) y(0) = y(1), ẏ(0) = y 1 (0) = ẏ(1) = y 1 (1) phase condition: y (0) = y1(0) = 0 tangency condition: y1(0) = 0

29 Looking for the 1:3 sync.: degraded top λ = 0,25 The upper border of the degraded top is not a bifurcation. It is a geometrical feature (tangency). Can we trace in two parameters the locus of such events? Yes, by setting up an appropriate BVP dx dτ = Tx 1 dy dτ = Ty 1 dx 1 dτ = T [(λ 1 x 2 )x 1 x µ(x 1 y 1 )] dx 1 dτ = T [(1 y 2 )y 1 (1 + δ)y µ(y 1 x 1 )] with periodicity boundary conditions: x(0) = x(1), ẋ(0) = x 1 (0) = ẋ(1) = x 1 (1) y(0) = y(1), ẏ(0) = y 1 (0) = ẏ(1) = y 1 (1) phase condition: y (0) = y1(0) = 0 tangency condition: y1(0) = 0

30 Looking for the 1:3 sync.: tangencies λ = 0, A synchronization s boundary 1:3 synchronization s boundary 1:5 µ

31 The degraded top: λ = 0, µ

32 The complete picture: λ = 0, B 1:1 1:3 1:5 oscillator death area µ

33 The complete picture: λ = 0, A 1:1 1:3 1:5 oscillator death area µ

34 The complete picture: λ = C 1:1 1:3 1:5 oscillator death area!

35 Conclusions Message of the talk Simulation or Continuation IVP or BVP

36 Conclusions Message of the talk Simulation AND Continuation IVP AND BVP The interaction of the tori with the resonance tongues seems to be related to the isolas formation. The resonances along the tori bifurcation curve gives rises to periodic curves in the white region. The geometric mechanism and some of the results may be applicable to other systems.