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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:
| m2  n1  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(2f 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  C1x2,k  x1, j , g1  0,0,...,g1,i ,...,0
g2,l  C1x1, 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 
xÝ2xÝ2y 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).