1 3 2 4 Saddle torus and mutual synchronization of periodic oscillators N. Janson 1, A. Balanov 2, V. Astakhov 3, P.V.E. McClintock 4 Phase and frequency synchronization: definitions f1 , 1 (t) 1 2 f2 , 2 (t ) f 1,2 – basic frequencies f2 n f1 m 1,2 - phases Frequency synchronization: f2 n f1 m n, m - integers Phase synchronization: | m2 n1 const | , 2 What happens in the phase space? torus nonsynchronous limit cycle synchronous Forced Periodic Oscillators Periodic oscillator C f0,fr ff Uni-directional coupling, or forcing xÝ1 h1(x1,x2,...,xn ) xÝj h j (x1,x2,...,xn ) xÝn hn (x1,x2,...,xn ) C sin(2f f t) Simplest form of forcing! Forcing parameters: C – amplitude ff - forcing frequency Synchronization Regions of Forced Periodic Oscillators Weak coupling – phase locking Strong coupling – suppression of natural dynamics C Synchronization regions suppression phase locking 1:4 1:2 1 2 ff:f0 1st classical mechanism of synchronization frequency locking Ý (1 x 2 ) xÝ 2 x C cos t xÝ 0.2; 1.0 Van der Pol oscillator under harmonic forcing Evolution of spectra: 1st classical mechanism of synchronization frequency locking Ý (1 x 2 ) xÝ 2 x C cos t xÝ 0.2; 1.0 Van der Pol oscillator under harmonic forcing Stroboscopic section (in a period of forcing): 2nd classical mechanism of synchronization suppression of natural oscillations Ý (1 x 2 ) xÝ 2 x C cos t xÝ 0.2; 1.0 Van der Pol oscillator under harmonic forcing Evolution of spectra: 2nd classical mechanism of synchronization suppression of natural oscillations Ý (1 x 2 ) xÝ 2 x C cos t xÝ 0.2; 1.0 Van der Pol oscillator under harmonic forcing Stroboscopic section (in a period of forcing): Mutually Coupled Periodic Oscillators Periodic oscillator 1 Periodic oscillator 2 f2 f1 Mutual coupling xÝ1 h1 (x1 ) g1 (x1, x2 ) xÝ2 h2 (x1 ) g2 (x1, x2 ) General forms of coupling functions g1 (g1,1,g1,2,...,g1,n ) g2 (g2,1,g2,2,...,g2,m ) Most popular coupling function: diffusive g1,i g1,i x2,k x1, j g2,l g2,l x1, j x2,k Knowledge vs Assumption. Question 1 What can be (and is often) assumed: The structure of the phase space in mutually coupled periodic oscillators is similar to the one in forced oscillators. Question 1: Is this true? What was known previously: Multistability inside synchronization region is common in mutually coupled periodic oscillators*. * Aronson 1986; Astakhov et al 1990; J. Rasmussen et al 1996; D. Postnov et al 1999; S. Park et al 1999; Vadivasova et al 2000; E. Mosekilde et al 2003. Linear vs Non-linear Coupling. Question 2. Linear Diffusive Symmetric Coupling in one Equation g1,i C1x2,k x1, j , g1 0,0,...,g1,i ,...,0 g2,l C1x1, j x2,k , g2 0,0,...,g2,l ,...,0 Non-linearly Linearly diffusively diffusively coupled coupled VanVan der der Pol Pol oscillators oscillators xÝ1xÝ 1 y1y1 2 2 2 yÝ1yÝ (( x12x)y C1C(x x1x)1 ) C2 (x 2 x1) 2 1 1 1 1 )y 1 1 1x 11x 1 1 (x 2 2 xÝ2xÝ2y 2y 2 2 2 2 yÝ2yÝ2(( x 22x)y C1C(x x 2x)2 ) C2 (x1 x 2 ) 2 2 2 2 )y 2 2 2 x22x 2 1 (x 1 1 Most earlier studies are devoted to linearly coupled systems. Question 2: Does nonlinear coupling produce substantially different effect to the one by linear coupling? Note When two periodic oscillators are coupled mutually, the smallest dimension of phase space is 4. Visualization of phase space structure is more difficult than with forced systems. One can reasonably well visualize only Poincare map. System under Study Possibly, the simplest case: Two diffusively coupled Van der Pol oscillators xÝ1 y1 yÝ1 (1 x12 )y1 12 x1 C1 (x 2 x1 ) C2 (x 2 x1) 2 xÝ2 y 2 yÝ2 (2 x 22 )y 2 22 x 2 C1 (x1 x 2 ) C2 (x1 x 2 ) 2 1 2 0.5 p , 1 2 1 1 0.8 p 1.2 Nonlinearities of partial oscillators Close to partial basic frequencies Detuning between oscilators I.e. around 1:1 synchronization region. Established Facts about the System (1) 1. The dimension of phase space is 4. 2. There is 1 fixed point at the origin for all parameter values. 3. There are 6 cycles involved for parameters at least around synchronization region 1:1. Namely, • 2 stable cycles S1 and S2 • 2 saddle cycles (with one-dimensional unstable manifold) S1* and S2* • 2 twice saddle cycles (with two-dimensional unstable manifolds) U1 and U2 Bifurcation Diagram for Linear Coupling (C2=0) Solid: Saddle-Node; Dashed: Torus Birth/Death suppression locking Established Facts about the System (2) 4. In locking region a pair of stable and a pair of saddle cycles lie on the same stable resonant torus. 5. Any saddle cycle can merge with any stable cycle (because they lie on the same torus). 6. Any saddle cycle can merge with any twice saddle cycle. 7. In suppression region there are no saddle or twice saddle cycles. Question 3 In locking region – 1. Two stable cycles coexist. 2. Their basins of attractions are supposedly separated by manifolds of saddle cycles (like in case of a forced oscillator) In suppression region 1. Two stable cycles coexist. 2. There are no saddle or twice saddle cycles (they have died via SN bifurcation). 3. Therefore, there are no separating manifolds of saddle cycles. Question 3: What separates the basins of attraction of two cycles in suppression region? BDs for Nonlinear Coupling (different C1) Solid: Saddle-Node: Dashed: Torus Birth/Death suppression suppression locking locking suppression locking suppression locking 3-Parameter Bifurcation Diagram C2 p C1 p-C2 Bifurcation Diagram for C1 =0.1 Black solid: Saddle-Node: Dashed: Torus Birth/Death Green solid: line where stable torus vanishes A stable cycle + A stable ergodic torus 2 stable cycles Question 4 ?? When the stable torus vanishes it 1. remains of finite size 2. is not distorted 3. its quasiperiod does not tend to infinity Thus: There are no signs of the torus merging with some saddle cycle, i.e. no homoclinic-like bifurcation. 3. For a double check, such a stray cycle was sought for and not found. Question 4: Why does the torus disappear? Speculation The assumption that the phase space structure is the same as for a forced periodic oscillator does not explain the phenomena observed. We need to reveal the true phase space structure that explains all observed phenomena. We hypothesize the following structure Hypothesis to Fit the Facts Inside locking region where two stable cycles coexist 1. Fact: All stable and saddle cycles lie on the same stable torus. 2. Fact: Any saddle cycle can merge with any twice saddle cycle. Suggestion: Saddle and twice saddle cycles could lie on the same saddle torus. To imagine a saddle torus, draw analogy with a saddle cycle in 3D. • A saddle cycle is a closed curve in the original 3D space. A saddle torus is a closed curve in 3D Poincare section of a 4D space. • A saddle cycle is an intersection of two 2D manifolds in 3D space. A saddle torus is an intersection of two 2D manifolds in 3D Poincare section. Hypothesis Continued Consequence: A stable and a saddle torus should lie on the same manifold and intersect at two saddle cycles. 3. Fact: trajectories go to the stable cycles along spirals in Poincare section. Hypothesized Poincare map of all objects in the phase space stable torus saddle torus St Stable cycle Saddle cycle Twice saddle cycle The Hypothesis would Answer Question 3 Question 3: What separates the basins of attraction of two cycles in suppression region? Answer: The stable manifold of the saddle torus (blue one). suppression S1 S2 The Hypothesis would Answer Question 4 Question 4: At a non-linear coupling, the just born torus disappears while staying of finite size, smooth, with finite quasiperiod. Why does the torus disappear? Answer: The stable torus merges with saddle torus and vanishes via SN bifurcation. S2 S1 How to Verify the Hypothesis? The hypothesis looks nice, but needs verification. Problem: There seem to be no well established numerical methods for plotting a saddle torus Solution: If the hypothesis is correct, what follows from it is correct, too. We can check if the consequences are true. What follows from the hypothesis (1): • The section of a Saddle torus is intersection of two 2D surfaces. • One of them is the “sphere”. • “Sphere” is an attracting object. • Inside the “sphere” there should be a repelling object – and there is one! An unstable fixed point at the origin – a repeller!. How to Verify the Hypothesis? Reverse the time: • repeller becomes stable fixed point • the “sphere” will be the borderline between the basins of attraction of the fixed point and infinity Thus: The “sphere” can be calculated via calculation of basins of attraction in reversed time. What follows from the hypothesis (2): •In locking region two stable cycles coexist •Their basins of attraction are separated by the locking “plane” Thus: The “plane” can be calculated via calculation of basins of attraction of the two stable cycles in direct time. How to Verify the Hypothesis? The saddle torus can be found as intersection of a “sphere” and the “plane”. Note: This method does not require the knowledge of where the saddle or the twice saddle cycles are located. If the saddle torus is found, and if it fits the hypothesis, all 4 cycles should lie on it! This is a test for the saddle torus validity. How Saddle Torus was Found Borderline between basins of stable cycles (“plane”) Stable torus Saddle torus “Sphere” Stable cycle Saddle cycle Twice saddle cycle Phase Space Structure Hypothesized Numerically estimated locking 1 Phase Space Structure S2 Hypothesized T2 1 T* S1 Numerically estimated 2 3 Surface of the “Sphere” (Poincare section) If systems are slightly non-identical, how will their BD change? 1 2 0.5 C1=0.1 C2 p 1 0.5, 2 0.51 What if we Couple Other Oscillators? Van der Pol mutually coupled oscillators FitzHugh-Nagumo mutually coupled oscillators far from non-local bifurcations Conclusions 1. The structure of the phase space in two mutually coupled 2D periodic oscillators, each being far from non-local bifurcations, is more complicated than in a forced oscillator. It involves a saddle torus. 2. Nonlinearity in coupling function influences the structure of synchronization region and bifurcations. E.g. it can induce saddle-node bifurcation of tori. 3. An approach is proposed to vizualize a saddle torus. p-C1 Bifurcation Diagrams (SN lines) C2 p 1= 2 = 0.5 01=1 p-C2 Bifurcation Diagrams C2 p 1= 2 = 0.5 01=1 Some Basic Publications 1. V.I. Arnold “Geometrical Methods in the Theory of Ordinary Differential Equations” (New York, Springer, 1993). 2. P.S. Landa “Self-Oscillations in Systems with Finite Numbers of Degrees of Freedom” (Moscow, Nauka, 1980, in Russian). 3. A. Pikovsky, M. Rosenblum, J. Kurths “Synchronization: a Universal Concept in Nonlinear Science” (Cambridge University Press, 2001). 4. E. Mosekilde, Yu. Maistrenko, D. Postnov “Chaotic Synchronization. Application to Living Systems” (World Scientific, Singapore, Series A, v 42, 2002). 5. S. Boccaletti, J. Kurths, G. Osipov, D.L. Valladares, C.S. Zhou Phys. Rep. 366, 1 (2002). 6. V.S. Anishchenko, V.V. Astakhov, A.B. Neiman, T.E. Vadivasova, L. Schimansky-Geier “Nonlinear Dynamics of Chaotic and Stochastic Systems” (Springer, Berlin, Heidelberg 2002). 7. J. Guckenheimer, P. Holmes “Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields” (Springer-Verlag, New York, 1986).