1. Effect of the Coupling Range on the Occurrence of Partial Synchronization
Woochang Lim and Sang-Yoon Kim
Department of Physics
Kangwon National University
 Four Coupled 1D Maps

2. Transverse Stability of the Fully Synchronized Attractor (FSA)
=0.7
w=0
• Longitudinal Lyapunov Exponent of the FSA
• Transverse Lyapunov Exponent of the FSA

3. • Blowout Bifurcation of the FSA
For >*, ,1<0   FSA on the Main Diagonal
Occurrence of the Blowout Bifurcation for =1* 
• FSA: Transversely Unstable (,1>0) for <1*
• Appearance of a New Asynchronous Attractor

4. Type of Asynchronous Attractors Born via a Blowout Bifurcation
Appearance of an Intermittent Two-Cluster State on the Invariant 1 ({(x1, x2, x3, x4) | x1=x2, x3=x4}) and 2 ({(x1, x2, x3, x4) | x1=x4, x2=x3}) Plane through a Blowout Bifurcation of the FSA
 Local Coupling Case (w=0)
 Global Coupling Case (w=1)
Two-Cluster State: Transversely Unstable
Occurrence of the Complete
Desynchronization
Two-Cluster State: Transversely Stable
 Occurrence of the PS
 Symmetric Coupling Case (p=1/3)

5. Two-Cluster States on the 1 Plane
 Reduced 2D Map Governing the Dynamics of a Two-Cluster State
Two-Cluster State:
For numerical accuracy, we introduce new coordinates:
Local Coupling Case (w=0)
Global Coupling Case (w=1)

6. Transverse Stability of Two-Cluster States
 Transverse Lyapunov Exponent of the Two-Cluster State
We obtain four Lyapunov exponents through the Gram-Schmidt Reorthonomalization procedure, and the first and second Lyapunov exponents corresponds to the longitudinal Lyapunov exponents (| |,1 and | |,2), while the third and fourth Lyapunov exponents corresponds to the transverse Lyapunov exponents (,1 and ,2).
Threshold Value w* ( ~ 0.66) s.t.
• 0  w <w*  Two-Cluster State: Transversely Stable (,1<0)  Occurrence of the PS
• w*< w  1  Two-Cluster State: Transversely Unstable (,1>0)
 Occurrence of the Complete Desynchronization

7. Mechanism for the Occurrence of the Partial Synchronization
 Intermittent Two-Cluster State Born via a Blowout Bifurcation
d = |V|: Transverse Bursting Variable
d*: Threshold Value s.t.
d < d*: Laminar Component (Off State),
d > d*: Bursting Component (On State).
d (t)
We numerically follow a trajectory segment with large length L (=108), and calculate its largest transverse Lyapunov exponent.
Decomposition of the Transverse Lyapunov Exponent  of the Two-Cluster State
: Weighted Largest Transverse Lyapunov Exponent for the Laminar
(Bursting) Component
Fraction of the Time Spent in the i Component
(Li: Time Spent in the i Component)
: Transverse Lyapunov Exponent of the i Component

8. Competition between the Laminar and Bursting Components
Local Coupling Case (w=0)
Global Coupling Case (w=1)
Sign of , : Determined via the Competition of the Laminar and
Bursting Components
Threshold Value w* ( ~ 0.66) s.t.
• 0  w < w*   Two-Cluster State: Transversely Stable (,1<0)
 Occurrence of the PS
• w*< w  1   Two-Cluster State: Transversely Unstable (,1>0)
 Occurrence of the Complete Desynchronization

9. Summary
Effect of the Coupling Range on the Occurrence of the Partial Synchronization
in Coupled 1D Maps
Sign of the Largest Transverse Lyapunov Exponent of the Two-Cluster
State Born via a Blowout Bifurcation of the FSA:
Determined via the Competition of the Laminar and Bursting Components
• (,1<0)  Two-Cluster State: Transversely Stable
 Occurrence of the PS
• (,1>0)  Two-Cluster State: Transversely Unstable
 Occurrence of the Complete Desynchronization