1 Mechanism for the Partial Synchronization in Coupled Logistic Maps  Cooperative Behaviors in Many Coupled Maps Woochang Lim and Sang-Yoon Kim Department of Physics Kangwon National University  Fully Synchronized Attractor for the Case of Strong Coupling Breakdown of the Full Synchronization via a Blowout Bifurcation Partial Synchronization (PS)Complete Desynchronization : Clustering

2  N Globally Coupled 1D Maps  Reduced Map Governing the Dynamics of a Three-Cluster State p i (=N i /N): “coupling weight factor” corresponding to the fraction of the total population in the ith cluster  Three-Cluster State Coupled Logistic Maps (Representative Model)  Reduced 3D Map  Globally Coupled Maps with Different Coupling Weight Investigation of the PS along a path connecting the symmetric and unidirectional coupling cases: p 2 =p 3 =p, p 1 =1-2p (0  p  1/3) p 1 =p 2 =p 3 =1/3  Symmetric Coupling Case  No Occurrence of the PS p 1 =1 and p 2 =p 3 =0  Unidirectional Coupling Case  Occurrence of the PS

3 Transverse Stability of the Fully Synchronized Attractor (FSA) Longitudinal Lyapunov Exponent of the FSA Transverse Lyapunov Exponent of the FSA For c>c * (=0.4398),  <0   FSA on the Main Diagonal Occurrence of the Blowout Bifurcation for c=c *  FSA: Transversely Unstable (  >0) for c<c * Appearance of a New Asynchronous Attractor Transverse Lyapunov exponent a=1.95 a=1.95, c=0.5

4 Type of Asynchronous Attractors Born via a Blowout Bifurcation  Unidirectional Coupling Case (p=0) Two-Cluster State: Transversely Stable   Partially Synchronized Attractor on the  23 Plane  Occurrence of the PS  Symmetric Coupling Case (p=1/3) Appearance of an Intermittent Two-Cluster State on the Invariant  23 Plane (  {(X 1, X 2, X 3 ) | X 2 =X 3 }) through a Blowout Bifurcation of the FSA Two-Cluster State: Transversely Unstable   Completely Desynchronized (Hyperchaotic) Attractor Filling a 3D Subspace (containing the main diagonal)  Occurrence of the Complete Desynchronization

5 Two-Cluster States on the  23 Plane  Reduced 2D Map Governing the Dynamics of a Two-Cluster State For numerical accuracy, we introduce new coordinates: Two-Cluster State: Unidirectional Coupling Case Symmetric Coupling Case

6 Threshold Value p * ( 0.146) s.t. 0  p<p *  Two-Cluster State: Transversely Stable (  <0)  Occurrence of the PS p * 0)  Occurrence of the Complete Desynchronization Transverse Stability of Two-Cluster States  Transverse Lyapunov Exponent of the Two-Cluster State (c  cc*)(c  cc*)

7 Mechanism for the Occurrence of the Partial Synchronization  Intermittent Two-Cluster State Born via a Blowout Bifurcation  Decomposition of the Transverse Lyapunov Exponent  of the Two-Cluster State Fraction of the Time Spent in the i Component (L i : Time Spent in the i Component) Transverse Lyapunov Exponent of the i Component (primed summation is performed in each i component) : Weighted Transverse Lyapunov Exponent for the Laminar (Bursting) Component d = |V|: Transverse Bursting Variable d * : Threshold Value s.t. d < d * : Laminar Component (Off State), d > d * : Bursting Component (On State). We numerically follow a trajectory segment with large length L (=10 8 ), and calculate its transverse Lyapunov exponent: d (t)

8 Threshold Value p * ( 0.146) s.t. 0p<p *0p<p * p * <p  1/3  Two-Cluster State: Transversely Stable  Occurrence of the PS Sign of  : Determined via the Competition of the Laminar and Bursting Components    Two-Cluster State: Transversely Unstable  Occurrence of the Complete Desynchronization (  : p=0,  : p=0.146,  : p=1/3)  Competition between the Laminar and Bursting Components  Laminar Component  Bursting Component a=1.95, d * =10 -4

9  Mechanism for the Occurrence of the Partial Synchronization in Coupled 1D Maps Sign of the 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 Summary  Two-Cluster State: Transversely Stable  Occurrence of the PS  Two-Cluster State: Transversely Unstable  Occurrence of the Complete Desynchronization