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

2. Chaos and Synchronization
 Lorenz Attractor
[Lorenz, J. Atmos. Sci. 20, 130 (1963).]
Butterfly Effect: Sensitive Dependence
on Initial Conditions
[Small Cause  Large Effect] z 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).

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

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).]

5. 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 a exceed a, a chaotic attractor
with a positive Lyapunov exponent s appears.  
( > 0  chaotic attractor,
 < 0  regular attractor) a a a a

6. Asymmetry Parameter  (0    1)
 Two Coupled 1D Maps
Coupling Function 
c: coupling parameter
Asymmetry Parameter  (0    1) 
 = 0: symmetric coupling
 exchange symmetry
  0: asymmetric coupling
( = 1: unidirectional coupling)
 Invariant Synchronization Line y = x
Synchronous Orbits on the Diagonal (, )
Asynchronous Orbits off the Diagonal () 

7. Transverse Stability of the Synchronous Chaotic Attractor
 Synchronous Chaotic Attractor (SCA)
on the Invariant Diagonal
SCA: Stable against the “Transverse Perturbation”  Chaos Synchronization
An Infinite Number of Unstable Periodic Orbits (UPOs) Embedded in the SCA
and Forming Its Skeleton
 Intimately Associated with the Transverse Stability of the SCA

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)}
 Chaos Synchronization
“Strength” of {PSs} > “Strength” of {PRs}
  < 0 (SCA: transversely stable)
 Complete Desynchronization
“Strength” of {PSs} < “Strength” of {PRs}
  > 0 (SCA: transversely unstable chaotic saddle)
Chaos Synchronization c
Blowout
Bifurcation
Blowout
Bifurcation

9. Loss of Chaos Synchronization
 Unidirectionally Coupled 1D Maps (=1)
c: Coupling Parameter
 State Diagram
• 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 Synchronous UPOs: Transversely Stable Periodic Saddles
 No Bursting (attracted to the diagonal without any bursting)
• Transition to Weak Synchronization (Gray and Dark Gray Regions, <0) via
a First Transverse Bifurcation of a Periodic Saddle
Some UPOs: Transversely Unstable  Local Bursting

10. Fate of Local Bursting for the Case of Weak Synchronization
(Riddling)
Weak
Synchronization
(Bubbling)
Strong
Synchronization c
Blowout
Bifurcation
First
Transverse
Bifurcation
First
Transverse
Bifurcation
Blowout
Bifurcation
• Weak Synchronization: Some UPOs: Transversely Unstable  Local Bursting
 Fate of Local Bursting Dependent on the Global Dynamics
 Attractor Bubbling (Gray Region)
Folded Back of a Locally Repelled Trajectory
 Transient Intermittent Bursting (<0)
 Basin Riddling (Dark Gray Region)
Basin of the SCA: Riddled with a Dense Set
of “Holes,” Leading to
Another Attractor
Attracted to Another Distant Attractor

11. Transcritical Transition to Basin Riddling
 Transcritical Bifurcation (Stability Exchange)
[S.-Y. Kim and W. Lim, Phys. Rev. E 63, (2001).
S.-Y. Kim and W. Lim, Prog. Theor. Phys. 105, 187 (2001).
S.-Y. Kim and W. Lim, Phys. Rev. E 64, (2001).]
Riddling
Strong synchronization
Bubbling c
Transcritical Bif.
Supercritical Period-Doubling Bif.
 Riddling Transition (Basin: riddled with a dense set of “holes”)
 an absorbing area
surrounding the SCA
(: repeller on the
basin boundary
: saddle on the
diagonal)
Contact between
the SCA and the
basin boundary
 Riddled Basin
After the transcritical bifurcation, the basin becomes “riddled” with a dense set of “holes” leading to divergent orbits.
As c decreases from ct,l, the basin riddling is intensified.

12. Characterization of the Riddled Basin
 Divergence Exponent
Divergence Probability P(d) ~ d
Measure of the Basin Riddling
[c  Blowout Bifurcation Point cb,l (=-2.963)
 ()  P(d): Increase] c
 Uncertainty Exponent
Uncertainty Probability P() ~ 
Two Initial Condition: Uncertain if their final states are different  Fine Scaled Riddling of the SCA
[c  cb,l  ()  P(): Increase] c

13. Effect of Parameter Mismatching on Weak Synchronization
[A. Jalnin and S.-Y. Kim, Phys. Rev. E 65, (2002). S.-Y. Kim, W. Lim, and Y. Kim, Prog. Theor. Phys. 107, 239 (2002).]
 Unidirectionally Coupled Nonidentical 1D Maps
 : Mismatching Parameter
 Effect of Parameter Mismatching (=0.001)
(1) Strong Synchronization (Small Mismatching  Small Effect)
Slightly Perturbed SCA ( Mismatching Strength)
All UPOs (embedded in the SCA): Transversely Stable
 No Parameter Sensitivity
(2) Weak Synchronization (Small Mismatching  Large Effect)
Local Transverse Repulsion of Periodic Repellers  Parameter Sensitivity
 Bubbling
 Riddling
SCA with a Riddled
Basin (Gray) 
Chaotic Transient (Black)
with a Finite Lifetime
SCA 
Bubbling Attractor
(exhibiting persistent
intermittent bursting)

14. Characterization of the Parameter Sensitivity of a
Weakly Stable SCA
 Characterization of Parameter Sensitivity
• Measured by Calculating a Derivative of the Transverse Variable un (=xn-yn) with respect to the
Mismatching Parameter along a Synchronous Trajectory
 Boundedness of SN
Intermittent Behavior
Looking only at the Maximum Values of |SN|:
Representative Value (by Taking the Minimum Value of in an Ensemble of Randomly Chosen
Initial Orbit Points)
Parameter Sensitivity Function:
• Strong Synchronization (SS)
N: Bounded  No Parameter Sensitivity
• Weak Synchronization (WS)
N ~ N: Unbounded  Parameter Sensitivity
: Parameter Sensitivity Exponent (PSE)
Used to Measure the Degree of Parameter Sensitivity
c=-0.7 (WS)
c=-1.5 (SS)

15. Characterization of the Bubbling Attractor and the Chaotic Transient
 Parameter Sensitivity Exponents (PSEs)
• Monotonic Increase of  () as c is Changed toward the
Blowout Bifurcation Point
(Due to the Increase in the Strength of Local Transverse
Repulsion of the Embedded Periodic Repellers.)
Scaling for the Average Characteristic Time
a=1.82, c=-0.7, =0.001
1500
3000
0.8
0.0
-0.8 n un
Average Laminar Length (Interburst Interval) of
the Bubbling Attractor:  ~  -
 Average Lifetime of Chaotic Transient:  ~  -
Reciprocal Relation between the Scaling Exponent 
and the PSE 
 ~  -1/
  () =1/ ()

16. Effect of Noise on Weak Synchronization
[S.-Y. Kim, W. Lim, A. Jalnin, and S.-P. Kuznetsov, Phys. Rev. E 67, (2003).]
 Unidirectionally Coupled Noisy 1D Maps
 Characterization of the Noise Sensitivity (=0.0005)
Bubbling Attractor
Chaotic Transient
: Bounded Noise → Boundedness of SN: Determined by RM (same as in the parameter mismatching case)
Characterization of the Bubbling Attractor and
the Chaotic Transient
 ~  - ;  () =1/ ()
(: average time spending
near the diagonal)
[Noise Sensitivity Exponent() = PSE()]
Noise Effect = Parameter Mismatching Effect

17. Dynamical Consequence of Blowout Bifurcations
[S.-Y. Kim, W. Lim, E. Ott, and B. Hunt, Phys. Rev. E 68, (2003).]
 Two Coupled 1D maps
: parameter tuning the degree of asymmetry of the coupling
=0  Symmetric Coupling Case, =1  Unidirectional Coupling Case
Synchronization c
Blowout
Bifurcation
Blowout
Bifurcation
 Appearance of an Asynchronous Attractor via a Blowout Bifurcation of the SCA
Asynchronous Hyperchaotic
Attractor with 2>0 for =0
Asynchronous Chaotic
Attractor with 2<0 for =1

18. Mechanism for the Appearance of the Asynchronous Hyperchaotic and Chaotic Attractors
Decomposition of the 2nd Lyapunov Exponent 2
d = |v| [|(x-y)/2|]: Transverse Variable
d < d*: Laminar Component (Off State)
d > d*: Bursting Component (On State)
Intermittent
Asynchronous
Attractor Born
via a Blowout
Bifurcation
d (t) d*
Weighted 2nd Lyapunov Exponent for the
Laminar (Bursting) Component
Competition between the Laminar and Bursting Components
 : =0,
: =0.852
 : =1
Sign of 2 (= b2 - |l2|): Determined by the Competition between the Laminar and Bursting Components

19. Partial Synchronization in Three Coupled Chaotic Systems
[W. Lim and S.-Y. Kim, Phys. Rev. E 71, (2005).]
Three Coupled 1D Maps
Occurrence of the Partial Synchronization
 Fully Synchronized Attractor (FSA) for the Case of Strong Coupling
Breakdown of the Full Synchronization via a Blowout Bifurcation
Partial Synchronization (PS)
Complete Desynchronization
: Two-Cluster State for p=0

20. Transverse Stability of Two-Cluster States
 Reduced 2D Map Governing the Dynamics of a Two-Cluster State
Two-Cluster State:
 Intermittent Two-Cluster State Born via the Blowout Bifurcation of the FSA
Unidirectional Coupling Case
Symmetric Coupling Case
 Threshold Value p* (~ 0.146) s.t.
• 0p<p*  Two-Cluster State: Transversely Stable (<0)  Occurrence of the PS
• p*<p1/3  Two-Cluster State: Transversely Unstable (>0)
 Occurrence of the Complete Desynchronization

21. Mechanism for the Occurrence of the Partial Synchronization
Decomposition of the Transverse Lyapunov Exponent 
d = |V| [V(X-Y)/2]: Transverse Variable
d < d*: Laminar Component, d > d*: Bursting Component.
Intermittent
Two-Cluster
State Born
via a Blowout
Bifurcation
d (t)
Weighted Transverse Lyapunov Exponent for the
Laminar (Bursting) Component d*
Competition between the Laminar and Bursting Components
: p=0,
: p=0.146
: p=1/3
Sign of  (= b - |l|): Determined by the Competition between the Laminar and Bursting Components

22. Strong Synchronization
Summary
 Investigation of Loss of Chaos Synchronization in Two Coupled Chaotic Systems
Weak Synchronization
(Riddling)
(Bubbling)
Strong Synchronization
Blowout
Bifurcation
First Transverse
1. Transcritical Transition to Basin Riddling in Asymmetrically Coupled Chaotic Systems
2. Characterization of the Parameter Mismatching and Noise Effects on Weak Synchronization
in terms of the PSEs and NSEs
3. Investigation of Dynamical Origin for the Occurrence of Hyperchaos and Chaos via
Blowout Bifurcation through Competition between the Laminar and Bursting Components
 Investigation of Loss of Full Synchronization in Three Coupled Chaotic Systems
Partial Synchronization (Clustering)
Full Synchronization
Complete Desynchronization
1. Investigation of the Dynamical Mechanism for the Occurrence of the Partial
Synchronization through Competition between the Laminar and Bursting Components