1 Partial Synchronization in Coupled Chaotic Systems  Cooperative Behaviors in Three Coupled Maps 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  Three-Cluster State Three Coupled Logistic Maps (Representative Model) (Each 1D map is coupled to all the other ones with equal strength.) 1st Cluster (N 1 ) 2nd Cluster (N 2 ) 3rd Cluster (N 3 )

3 p i (=N i /N): “coupling weight factor” corresponding to the fraction of the total population in the ith cluster  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

4 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

5 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

6 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 (0  p  1/3)

7 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*)

8 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 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)

9 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

10 Effect of Parameter Mismatching on Partial Synchronization  Three Unidirectionally Coupled Nonidentical Logistic Maps  Effect of Parameter Mismatching  Partially Synchronized Attractor on the  23 (  {(x (1), x (2), x (3) ) | x (2) =x (3) }) Plane in the Ideal Case without Mismatching (  2 =  3 = 0)  Attractor Bubbling (Persistent Intermittent Bursting from the  23 Plane) a=1.95, c=0.42,  2 =0.001,  3 =0 a=1.95, c=0.42 a=1.95, c=0.42,  2 =0,  3 =0.001 p 2 =p 3 =p, p 1 =1-2p (0  p  1/3)  Reduced 2D Map Governing the Dynamics of a Two-Cluster State mismatching parameters

11 Distribution of Local Transverse Lyapunov Exponents  Probability Distribution P of Local M-time Transverse Lyapunov Exponents  Fraction of Positive Local Lyapunov Exponents Significant Positive Tail which does not Vanish Even for Large M  Parameter Sensitivity A Typical Trajectory Has Segments of Arbitrarily Long M with Positive Local Transverse Lyapunov Exponents (due to the local transverse repulsion of unstable orbits embedded in the partially synchronized attractor)  Parameter Sensitivity of the Partially Synchronized Attractor Power-Law Decay a=1.95, c=0.42 a=1.95 c=0.42 a=1.95

12 Characterization of the Parameter Sensitivity of a Partially Synchronized Attractor  Characterization of Parameter Sensitivity Measured by Calculating a Derivative of the Transverse Variable Denoting the Deviation from the  23 Plane with Respect to  2 along a Partially Synchronous Trajectory  Representative Value (by Taking the Minimum Value of  N (X 0,Y 0 ) in an Ensemble of Randomly Chosen Initial Orbit Points) Parameter Sensitivity Function:  N ~ N  : Unbounded  Parameter Sensitivity  : Parameter Sensitivity Exponent (PSE) Used to Measure the Degree of Parameter Sensitivity  Boundedness of S N Looking only at the Maximum Values of |S N |: Intermittent Behavior a=1.95 c=0.42 a=1.95 c=0.42

13 Characterization of the Bubbling Attractor  Parameter Sensitivity Exponents (PSEs) of the Partially Synchronized Attractor on the  23 Plane  Scaling for the Average Characteristic Time   (  ) =1/  (  )  ~   1/   Average Laminar Length (i.e., average time spending near the  23 plane) of the Bubbling Attractor:  ~  -   Reciprocal Relation between the Scaling Exponent  and the PSE   c 1 * (=0.4398) > c > 0.372: Increase of   > c > (decreasing part of  ): Decrease of   > c > c 2 * (=0.3376): Increase of  Increase of   More Sensitive with Respect to the Parameter Mismatching a=1.95  Partially Synchronized Attractor  Bubbling Attractor (in the Presence of Parameter Mismatching) 

14 Effect of Noise on the Partially Synchronized Attractor  Characterization of the Noise Sensitivity of the Partially Synchronized Attractor (  2 =0.0005,  1 =  3 =0)  Three Unidirectionally Coupled Noisy 1D Maps  2 : Bounded Noise → Boundedness of S N : Determined by R M (same as in the parameter mismatching case) Noise Sensitivity Exponent(  ) = PSE(  )  Noise Effect = Parameter Mismatching Effect  Characterization of the Bubbling Attractor  ~  -  ;  (  ) =1/  (  ) Bubbling Attractor for a=1.95 and c=0.42 (  : average time spending near the diagonal)

15 Partial Synchronization in Three Coupled Pendula  Three Coupled Pendula  Transverse Stability of Two-Cluster States on the  23 Plane Born via a Blowout Bifurcation of the FSA Threshold Value p * (~0.17) s.t. 0p<p *0p<p * p * <p  1/3  Two-Cluster State: Transversely Stable  Occurrence of the PS    Two-Cluster State: Transversely Unstable  Occurrence of the Complete Desynchronization (  : p=0,  : p=0.17,  : p=1/3)  =1,  =0.5, A=0.85 d * =10 -4

16  Unidirectional Coupling Case (p=0) Two-Cluster State: Transversely Stable  Occurrence of the PS  Symmetric Coupling Case (p=1/3) Two-Cluster State: Transversely Unstable  Occurrence of the Complete Desynchronization  =1,  =0.5, A=0.85, c= ~0.648, 2 ~  0.013, 3 ~  0.013, 4 ~  3.790, 5 ~  4.388, 6 ~   =1,  =0.5, A=0.85, c= ~0.626, 2 ~0.015, 3 ~0.013, 4 ~  3.794, 5 ~  4.390, 6 ~  4.415

17 Effect of Parameter Mismatching on Partial Synchronization in Three Coupled Pendula  Three Unidirectionally Coupled Nonidentical Pendula  Effect of Parameter Mismatching  Attractor Bubbling (Persistent Intermittent Bursting from the  23 Plane)  =1,  =0.5, A=0.85, c=0.6,  2 =0.001, and  3 =0

18 Characterization of the Parameter Sensitivity of a Partially Synchronized Attractor  Parameter Sensitivity of a Partially Synchronized Attractor Characterized by Differentiating the Transverse Variable Denoting the Deviation from the  23 Subspace with Respect to  2 at a Discrete Time t=n. Parameter Sensitivity Function: (  : Parameter Sensitivity Exponent)  : Used to Measure the Degree of Parameter Sensitivity  Characterization of the Bubbling Attractor   (  ) =1/  (  )  Average Laminar Length (Interburst Interval) of the Bubbling Attractor:  ~  -   Reciprocal Relation between the Scaling Exponent  and the PSE   ~   1/  A=0.85

19  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  Similar Results: Found in High-Dimensional Invertible Period-Doubling Systems such as Coupled Parametrically Forced Pendula  Occurrence of the PS  Occurrence of the Complete Desynchronization  Effect of the Parameter Mismatching and Noise on the Partial Synchronization  Characterized in terms of the PSE and NSE  Reciprocal Relation between the Scaling Exponent  for the Average Laminar Length and the PSE(NSE)  (  =1/  )