1. Synchronization in Coupled Chaotic Oscillators
Sang-Yoon Kim
Department of Physics
Kangwon National University
Korea
Synchronization in Coupled Periodic Oscillators
Synchronous Pendulum Clocks
Synchronously Flashing Fireflies 1

2. Chaos and Synchronization
 Lorenz Attractor
[Lorenz, J. Atmos. Sci. 20, 130 (1963).] z
Butterfly Effect: Sensitive Dependence
on Initial Conditions
[Small Cause  Large Effect] y x
 Coupled Chaotic (Chemical) Oscillators
[H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983).]
• Other Pioneering Works
• A.S. Pikovsky, Z. Phys. B 50, 149 (1984).
• V.S. Afraimovich, N.N. Verichev, and
M.I. Rabinovich, Radiophys. Quantum
Electron. 29, 795 (1986).
• L.M. Pecora and T.L. Carrol, Phys. Rev.
Lett. 64, 821 (1990). 2

3. Secure Communication (Application)
[K.M. Cuomo and A.V. Oppenheim, Phys. Rev. Lett. 71, 65 (1993).]
(Secret Message)
Chaotic
System
Chaotic
System  + -
Transmitter
Receiver
Frequency (kHz)
Secret Message Spectrum
Chaotic Masking
Spectrum
 Encoding by Using Chaotic Masking
 Decoding by Using Chaos Synchronization 3

4. Several Types of Chaos Synchronization
Different degrees of correlation between the interacting subsystems
 Identical Subsystems
 Complete Synchronization
[H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983).]
 Nonidentical Subsystems
 • Generalized Synchronization
[N.F. Rulkov et.al., Phys. Rev. E 51, 980 (1995).]
• Phase Synchronization
[M. Rosenblum, A.S. Pikovsky, and J. Kurths, Phys. Rev. Lett 76, 1804 (1996).]
• Lag Synchronization
[M. Rosenblum, A.S. Pikovsky, and J. Kurths, Phys. Rev. Lett 78, 4193 (1997).] 4

5. Complete Synchronization in Coupled 1D Maps
 1D Map (Representative model exhibiting universal scaling behavior)
(x: seasonly breeding inset population)
Iterates: (trajectory)  Attractor
 Period-Doubling Transition to Chaos
An infinite sequence of period doubling
bifurcations ends at a finite accumulation
point
When exceed , a chaotic attractor
with positive Lyapunov exponent s appears.  
( > 0  chaotic attractor,
 < 0  regular attractor) 5

6. Asymmetry parameter  (0    1)  = 0: symmetric coupling
 Coupled 1D Maps
Coupling function 
C: coupling parameter
Asymmetry parameter  (0    1) 
 = 0: symmetric coupling
 exchange symmetry
 = 1: unidirectional coupling
Invariant synchronization line y = x
Synchronous orbits on the diagonal (, )
Asynchronous orbits off the diagonal ()  6

7. Transverse Stability of the Synchronous Chaotic Attractor
Synchronous Chaotic Attractor (SCA) on The Invariant Synchronization Line
SCA: Stable against the “Transverse Perturbation”  Chaos Synchronization
An infinite number of Unstable Periodic Orbits (UPOs) embedded in the SCA
and forming its skeleton
 Characterization of the Macroscopic Phenomena Associated with the Transverse
Stability of the SCA in terms of UPOs (Periodic Orbit Theory) 7

8. Competition between Periodic Saddles and Repellers
 Investigation of Transverse Stability of the SCA in Terms of UPOs
(: transverse Lyapunov exponent of the SCA)
{UPOs} = {Transversely Stable Periodic Saddles (PSs)}
+ {Transversely Unstable Periodic Repellers (PRs)}
“Strength” of {PSs} > “Strength” of {PRs}
 < 0 (SCA: transversely stable)
 Chaos Synchronization
“Strength” of {PSs} < “Strength” of {PRs}
 > 0 (SCA: transversely unstable chaotic saddle)
 Complete Desynchronization
Chaos Synchronization C
Blow-out
Bifurcation
Blow-out
Bifurcation 8

9. Chaotic Synchronization in Unidirectionally Coupled 1D Maps
c: Coupling Parameter, : Noise Strength
(i) (i=1,2): Uniform Random Variable with and
 Phase Diagram (=0)
• Appearance of a Synchronous Chaotic Attractor (SCA)
on the Invariant Diagonal when Passing a Critical Line
(heavy solid line).
a=1.82
• an Infinite Number of
UPOs inside the SCA.
a= …
• Strong Synchronization (Hatched Region, <0)
All Sync. UPOs: Transversely Stable Periodic Saddles  No Bursting
• Weak Synchronization (Gray and Dark Gray Regions, <0)
{Sync. UPOs} = {Periodic Saddles (PSs)} + {Periodic Repellers (PRs)}
{PSs}: Dominant  SCA: Transversely Stable (<0), {PRs}  Local Bursting 9

10. Fate of Local Bursting for the Case of Weak Synchronization
Dependent on the Existence of an Absorbing Area, Controlling the Global Dynamics and Acting as a Bounded Trapping Area.
 Attractor Bubbling (in the presence of an absorbing area, Gray Region)
Folded Back of a Locally Repelled Trajectory
 Transient Intermittent Bursting (<0)
 Basin Riddling (in the absence of an absorbing area, Dark Gray Region)
Basin of the SCA: Riddled with a Dense Set
of “Holes,” Leading to
Another Attractor
Attracted to Another Distant Attractor 10

11. Noise Effect on the Chaos Synchronization
 Strong Synchronization [Small Noise  Small Effect]
All UPOs (embedded in the SCA): Transversely Stable  No Noise Sensitivity
Slightly Perturbed SCA ( Noise Strength)
Evolution of Transverse Variable un (=xn-yn) vs. n
a=1.82
c=-1.5
=0.0005
 Weak Synchronization [Small Noise  Large Effect]
Local Transverse Repulsion of Periodic Repellers  Noise Sensitivity
• Bubbling
Transient Intermittent Bursting (=0)  Persistent Intermittent Bursting
of the Bubbling Attractor Filling an Absorbing Area for Any Small Noise.
a=1.82
c=-0.7
=0.0005
a=1.82, c=-2.91
=0.0005
• Riddling
SCA (=0)with a Basin (Gray) Riddled with a Dense Set of “Holes,” Leading to Divergent Orbits (White)
 Chaotic Transient (Black) with a Finite Lifetime. 11

12. Quantitative Characterization of the Noise Sensitivity of the SCA
[S.-Y. Kim, W. Lim, A. Jalnine, and S.P. Kuznetsov, Phys. Rev. E 67, (2003); A. Jalnine and S.-Y. Kim,
Phys. Rev. E 65, (2002); S.-Y. Kim, W. Lim, and Y. Kim, Prog. Theor. Phys. 107, (2002).]
 Noise Sensitivity Induced by Local Transverse Repulsion
Local Transverse Repulsion of Periodic Repellers Embedded in the Weakly Stable SCA
A Typical Trajectory may Have Segments of Arbitrarily Long Length M with
Positive Local (M-Time) Transverse Lyapunov Exponents 
The Weakly-Stable SCA for =0 Becomes Sensitive with respect to Any Small Noise
Strength . M
 Characterization of Noise Sensitivity
• Measured by Calculating a Derivative of the Transverse Variable un (=xn-yn) with
respect to the Noise Strength along a Synchronous Trajectory
R N-k ( x k ) *
R M ( x k ) *
: Local (M-time) Transverse Stability Multiplier, M
= exp(M )
: Local (M-time) Transverse Lyapunov exponent
: Synchronous Orbit for =0, 12

13.  Boundedness of (by Looking only at the Maximum Values of )
Representative Value (by Taking the Minimum Value of in an Ensemble of Randomly
Chosen Initial Orbit Points)
Noise Sensitivity Function:
a=1.82
• Strong Synchronization (SS)
N: Bounded  No Noise Sensitivity
• Weak Synchronization (WS)
: Unbounded  Noise Sensitivity
: Noise Sensitivity Exponent (NSE)
The NSE  Measures the Degree of Noise Sensitivity of the Weakly Stable SCA. 13

14. NSEs for the Bubbling and Riddling Cases
 One-Band SCA for a=1.82 and Its Transverse Stability
Weak Synchronization
Strong Synchronization
Weak Synchronization c
 > 0
 = 0
 > 0
Blow-out Bif.
Riddling Transition
Bubbling Transition
Blow-out Bif.
 NSEs
• Monotonic Increase of  (denoted by open circles)
with respect to the Variance of c from the
Bubbling or Riddling Transition Point to the
Blow-Out Bifurcation Point
( the Strength of Local Transverse Repulsion
of the Embedded Periodic Repellers Increases.) 14

15. Parameter-Mismatching Effect on the Weak Synchronization
 Coupled Non-Identical 1D Maps
 : Mismatching Parameter
 Comparison of the Parameter-Sensitivity Case with the Noise-Sensitivity Case
• Parameter Sensitivity
R N-k ( x k ) *
[ fa(x) = -x2: Bounded Chaotic Signal ]
• Noise Sensitivity
R N-k ( x k ) *
[: Bounded Random Variable]
Growth of for Large N must be Determined by the Local Transverse Stability
Multipliers RM, as in the Noise-Sensitivity Case.
N ~ N  for both the Parameter-Mismatching and Noise Cases:
Parameter Sensitivity Exponent (PSE) (crosses) = NSE (open circles) 15

16. Characterization of the Bubbling Attractor and Chaotic Transient
 Average Laminar Length (Interburst Interval) of the Bubbling Attractor
a=1.82
c=-0.7
=0.0005
(threshold value)  Laminar State
(threshold value)  Bursting State
 ~  -
(: open circles)
 Average Lifetime of Chaotic Transient
(threshold value)  Regarded as have Escaped
 ~  -
 Reciprocal Relation between the Scaling Exponent  and the NSE 
Characteristic Time  at Which the Magnitude
of the Deviation from the Diagonal
Becomes the Threshold Value u* is Given by
 ~  -1/
  (open circles) =1/ (crosses) 16

17. Parametric Noise Effect on the Weak Synchronization
 : Noise Strength
(i) (i=1,2): uniform random variable with and
 Noise Sensitivity Exponents
• Parametric Noise
R N-k ( x k ) *
[ fa(x,a)(=-x2): Bounded Chaotic Variable, : Bounded Random Variable]
• Additive Noise
R N-k ( x k ) *
NSEs for Both Cases of the Additive and Parametric Noise Become the Same. 17

18. Noise Effect on Weak Synchronization for the Mutually-Coupled Case
=0: Symmetric Coupling, 0<1: Asymmetric Coupling
 Characterization of the Noise Sensitivity
R N-k ( x k ) *
(2-)c
• By a Scale Change c  c/(2-), the Result for the NSE in the Unidirectionally Coupled
Case (=1) may be Converted into that for the Mutually Coupled Case (0<1).
[In the Same Way, the PSE for the Mutually Coupled Case can also be Obtained.] 18

19. Strong Synchronization
Summary
The NSE (PSE)  Measures the Degree of the Sensitivity of the SCA with respect to the
Noise (Parameter Mismatching);
(cf. The Transverse Lyapunov Exponent  Measures the Degree of the Sensitivity to
the Initial Orbit Points.)
• Noise Effect = Parameter-Mismatching Effect ( NSE=PSE)
 Characterization of the Bubbling Attractor and Chaotic Transient
Power-Law Scaling of the Average Laminar Length and Average Lifetime: ~- (-)
• Reciprocal Relation between the Scaling Exponent  and the NSE (PSE) 
=1/
N ~ N 
Weak Synchronization
Strong Synchronization
Weak Synchronization c
< 0
 > 0
< 0
 = 0
< 0
 > 0
Blow-out
Bif.
First
Transverse Bif.
First
Transverse Bif.
Blow-out
Bif.  19

20. Analogy between the Characteristic Quantities of the Weakly
Stable SCA and the Strange Nonchaotic Attractor (SNA)
Common Property: A Typical Trajectory on Both the Weakly Stable SCA and the SNA
may Have Segments of Arbitrarily Long Length M with Positive
Local (M-Time) Lyapunov Exponents
 Quasiperiodically Forced System
The Phase Sensitivity Exponent Measures the Degree of Strangeness of the SNA.
Smooth Torus
SNA
No Phase Sensitivity
Phase Sensitivity
 Chaos Synchronization
The Noise (Parameter) Sensitivity Exponent Measures the Degree of the Noise (Parameter) Sensitivity of the Weakly Stable SCA.
Strong Synchronization
Weak Synchronization
No Noise (Parameter) Sensitivity
Noise (Parameter) Sensitivity 20

21. Quasiperiodically Forced Systems
: Irrational No. 21

22. Phase Sensitivity Exponent to Characterize Strangeness of an Attractor
• Smooth Torus
(a=3.38, = )
N: Bounded
 No Phase Sensitivity
• SNA
(a=3.38, = )
N ~ N: Unbounded
(( ): PSE)
Phase Sensitivity
 Strange Geometry ~ _
 Phase Sensitivity with Respect to the Phase of Quasiperiodic Forcing:
Measured by Calculating a Derivative x/ along a Trajectory and Finding its Maximum Value:
Phase Sensitivity Function:
(Taking the minimum value of N(x0,0) with
respect to an ensemble of randomly chosen
initial conditions) 22

23. Distribution of Local Transverse Lyapunov Exponents
 Probability Distribution PM of Local M-time Transverse Lyapunov Exponents
The Variance of from Its Average Value Decreases Inversely with M:
c = -0.7
D=0.054
 Fraction of Positive Local Lyapunov Exponents
• A Typical Trajectory Has Segments of Arbitrarily
Long M with Positive Local Lyapunov Exponents.
 The Weakly-Stable SCA Has a Noise or
Parameter Sensitivity. 23