Saddle torus and mutual Saddle torus and mutual synchronization of periodic oscillatorssynchronization of periodic oscillators

N. Janson N. Janson 11,, A. Balanov A. Balanov 22,, V. Astakhov V. Astakhov 33, P.V.E. McClintock , P.V.E. McClintock 44

1

34

2

1

2

)(, 11 tf Φ

)(, 22 tf Φ

Frequency synchronization:f 1,2 – basic frequencies Φ 1,2 - phases

mn

ff

1

2 =

Phase synchronization:

n, m - integers

π<δδ<−Φ−Φ 2,|constnm| 12

mn

ff

1

2 ≠

Phase and frequency synchronization: definitionsPhase and frequency synchronization: definitions

torus

nonsynchronous

limit cycle

synchronous

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What happens in the phase space?What happens in the phase space?

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˙ x 1 = h1(x1,x2,...,xn)L˙ x j = h j(x1,x2,...,xn) + C sin(2πf f t)

L˙ x n = hn(x1,x2,...,xn)

Forced Periodic OscillatorsForced Periodic Oscillators

Forcing parameters:C – amplitudeff - forcing frequency

fff0,fr

Periodic oscillator

Uni-directional coupling, or forcing

Simplest form of forcing!

C

Synchronization Regions of Synchronization Regions of Forced Periodic OscillatorsForced Periodic Oscillators

1 21:21:4 ff:f0

CSynchronization regions

Weak coupling – phase lockingStrong coupling – suppression of natural dynamics

suppression

phase locking

frequency locking

Evolution of spectra:

11stst classical mechanism of synchronization classical mechanism of synchronization

Van der Pol oscillator under harmonic forcing

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˙ ̇ x − ε(1− x 2) ˙ x + ω 2x = C cosΩt

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ε=0.2;ω =1.0

frequency locking

11stst classical mechanism of synchronization classical mechanism of synchronization

Van der Pol oscillator under harmonic forcing

Stroboscopic section(in a period of forcing):€

˙ ̇ x − ε(1− x 2) ˙ x + ω 2x = C cosΩt

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ε=0.2;ω =1.0

22ndnd classical mechanism of synchronization classical mechanism of synchronizationsuppression of natural oscillations

Evolution of spectra:Van der Pol oscillator under harmonic forcing

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˙ ̇ x − ε(1− x 2) ˙ x + ω 2x = C cosΩt

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ε=0.2;ω =1.0

22ndnd classical mechanism of synchronization classical mechanism of synchronizationsuppression of natural oscillations

Van der Pol oscillator under harmonic forcingStroboscopic section

(in a period of forcing):€

˙ ̇ x − ε(1− x 2) ˙ x + ω 2x = C cosΩt

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ε=0.2;ω =1.0

Mutually Coupled Periodic OscillatorsMutually Coupled Periodic Oscillators

Mutual couplingf2

Periodic oscillator 2Periodic oscillator 2

f1

Periodic oscillator 1Periodic oscillator 1

General forms of coupling functions

Most popular coupling function: diffusive

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˙ x 1 = h1(x1) + g1(x1, x2)

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˙ x 2 = h2(x1) + g2(x1, x2)

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g1 = (g1,1,g1,2,...,g1,n )

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g2 = (g2,1,g2,2,...,g2,m )

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g1,i = g1,i x2,k − x1, j( )

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g2,l = g2,l x1, j − x2,k( )

Knowledge Knowledge vsvs Assumption. Question 1 Assumption. Question 1

What can be (and is often) assumed:What can be (and is often) assumed:

The structure of the phase space in mutually coupled periodicoscillators is similar to the one in forced oscillators.

Question 1: Is this true?

What was known previously: 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 vs Non-linear Coupling. Question 2.

Most earlier studies are devoted to linearly coupled systems.Question 2: Does nonlinear coupling produce substantially different effect to the one by linear coupling?

Linear Diffusive Symmetric Coupling in one Equation

Linearly diffusively coupled Van der Pol oscillatorsNon-linearly diffusively coupled Van der Pol oscillators€

g1,i = C1 x2,k − x1, j( ), g1 = 0,0,...,g1,i,...,0( )

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g2,l = C1 x1, j − x2,k( ), g2 = 0,0,...,g2,l ,...,0( )

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˙ x 1 = y1

˙ y 1 = (ε1 − x12)y1 − ω1

2x1 + C1(x2 − x1) + C2(x2 − x1)2

˙ x 2 = y2

˙ y 2 = (ε2 − x22)y2 − ω 2

2x2 + C1(x1 − x2) + C2(x1 − x2)2

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˙ x 1 = y1

˙ y 1 = (ε1 − x12)y1 − ω1

2x1 + C1(x2 − x1)˙ x 2 = y2

˙ y 2 = (ε2 − x22)y2 − ω 2

2x2 + C1(x1 − x2)

NoteNote

When two periodic oscillators are coupled mutually, the smallest dimension of phase space is 4.

Visualization of phase space structure is more difficultthan with forced systems.

One can reasonably well visualize only Poincare map.

System under StudySystem under StudyPossibly, the simplest case:

Two diffusively coupled Van der Pol oscillators

I.e. around 1:1 synchronization region.

Nonlinearities of partial oscillators

Close to partial basic frequencies

Detuning between oscilators

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˙ x 1 = y1

˙ y 1 = (ε1 − x12)y1 − ω1

2x1 + C1(x2 − x1) + C2(x2 − x1)2

˙ x 2 = y2

˙ y 2 = (ε2 − x22)y2 − ω 2

2x2 + C1(x1 − x2) + C2(x1 − x2)2

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ε1 = ε2 = 0.5ω 2 = pω1, ω1 =10.8 < p <1.2

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 DiagramBifurcation Diagramfor Linear Coupling (Cfor Linear Coupling (C22=0)=0)

suppression

locking

Solid: Saddle-Node; Dashed: Torus Birth/Death

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 3Question 3In 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 What separates the basins of attraction of two cycles in suppression region?of two cycles in suppression region?

BDs for Nonlinear Coupling (different CBDs for Nonlinear Coupling (different C11))

suppression suppression

suppression suppression

locking

locking

locking

locking

Solid: Saddle-Node: Dashed: Torus Birth/Death

3-Parameter Bifurcation Diagram3-Parameter Bifurcation Diagram

p

C2

C1

p-Cp-C22 Bifurcation Diagram for C Bifurcation Diagram for C11 =0.1 =0.1Black solid: Saddle-Node: Dashed: Torus Birth/DeathGreen solid: line where stable torus vanishes

2 stable cycles

A stable cycle +A stable ergodic torus

Question 4Question 4

When the stable torus vanishes it -1. remains of finite size2. is not distorted3. 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?Why does the torus disappear?

??

SpeculationSpeculation

The assumption that the phase space structure is the sameas 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 FactsHypothesis to Fit the Facts

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.

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.

Suggestion: Saddle and twice saddle cycles could lie on the same saddle torus.

Inside locking region where two stable cycles coexist -

Hypothesis ContinuedHypothesis ContinuedConsequence: A stable and a saddle torus should lie on thesame manifold and intersect at two saddle cycles.

HypothesizedPoincare mapof all objectsin the phasespace

stable torus saddle torus

Stable cycleSaddle cycleTwice saddle cycle

Cutting the cylinder

and squeezing its

boundaries to a point

3. Fact: trajectoriesgo to the stable cyclesalong spirals in Poincare section.

St

The Hypothesis would Answer Question 3The Hypothesis would Answer Question 3

Answer:

The stable manifold of the saddle torus (blue one).

Question 3:

What separates the basins of What separates the basins of attraction of two cycles in attraction of two cycles in

suppression region? suppression region?

S1

S2

suppression

The Hypothesis would Answer Question 4The Hypothesis would Answer Question 4

Answer:

The stable torus merges with saddle torus and vanishes viaSN bifurcation.

Question 4:

At a non-linear coupling, the At a non-linear coupling, the just born torus disappears just born torus disappears while staying of finite size, while staying of finite size, smooth, with finite quasiperiod.smooth, with finite quasiperiod.Why does the torus disappear?Why does the torus disappear?

S2

S1

How to Verify the Hypothesis?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?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“plane” Thus: The “plane” can be calculated via calculation of basinsof attraction of the two stable cycles in direct time.

locking

How to Verify the Hypothesis?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 cyclesshould lie on it!

This is a test for the saddle torus validity.

How Saddle Torus was FoundHow Saddle Torus was Found

“Sphere”

Borderline betweenbasins of stable cycles (“plane”)

Stabletorus

Stable cycle

Saddle cycle

Twice saddle cycle

Saddletorus

Phase Space StructurePhase Space Structure

Numericallyestimated

1Hypothesized

locking

Phase Space StructurePhase Space Structure

1

2 3

Numericallyestimated

Hypothesized

S1

S2

T2

T*

Surface of the “Sphere” (Poincare section)Surface of the “Sphere” (Poincare section)

If systems are slightly non-identical, If systems are slightly non-identical, how will their BD change?how will their BD change?

p

C2

C1=0.1

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ε1 = ε2 = 0.5

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ε1 = 0.5, ε2 = 0.51

What if we Couple Other What if we Couple Other Oscillators?Oscillators?

FitzHugh-Nagumo FitzHugh-Nagumo mutually coupled oscillatorsmutually coupled oscillators

far from non-local bifurcationsfar from non-local bifurcations

Van der PolVan der Polmutually coupled oscillatorsmutually coupled oscillators

ConclusionsConclusions

1.1. The structure of the phase space in two mutually coupled The structure of the phase space in two mutually coupled 2D periodic oscillators, each being far from non-local 2D periodic oscillators, each being far from non-local bifurcations, is more complicated than in a forced bifurcations, is more complicated than in a forced oscillator. It involves a saddle torusoscillator. It involves a saddle torus. .

2.2. Nonlinearity in coupling function influences the structure of Nonlinearity in coupling function influences the structure of synchronization region and bifurcations. E.g. it can induce synchronization region and bifurcations. E.g. it can induce saddle-node bifurcation of tori.saddle-node bifurcation of tori.

3.3. An approach is proposed to vizualize a saddle torus.An approach is proposed to vizualize a saddle torus.

p-Cp-C11 Bifurcation Diagrams (SN lines) Bifurcation Diagrams (SN lines)

ε1= ε2 = 0.501=1

p

C2

p-Cp-C22 Bifurcation Diagrams Bifurcation Diagrams

ε1= ε2 = 0.501=1

p

C2

Some Basic PublicationsSome 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).