1 Dynamical Routes to Clusters and Scaling in Globally Coupled Chaotic Systems Sang-Yoon Kim Department of Physics Kangwon National University Globally Coupled Systems (Each element is coupled to all the other ones with equal strength) logical Examples rtbeats, Circadian Rhythms, Brain Rhythms, Flashing of Fireflies biological Examples phson Junction Array, Multimode Laser, Electrochemical Oscillator Incoherent State (i: index for the element, : Ensemble Average) each element’s motion: independent) (collective motion) Coherent State Stationary Snapshots Nonstationary Snapshots t = n t = n+1 Synchronized Flashing of Fireflies
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11 Dynamical Routes to Clusters and Scaling in Globally Coupled Chaotic Systems Sang-Yoon Kim Department of Physics Kangwon National University Globally.
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11
Dynamical Routes to Clusters and Scaling in Globally Coupled Chaotic Systems
Sang-Yoon Kim
Department of Physics
Kangwon National University
Globally Coupled Systems (Each element is coupled to all the other ones with equal strength)
Biological Examples Heartbeats, Circadian Rhythms, Brain Rhythms, Flashing of Fireflies
Fully Synchronized Chaotic Attractor on the Invariant Diagonal
Complete Chaos Synchronization
Transverse Lyapunov Exponent of the FSA
1D Reduced Map Governing the Dynamicsof the Fully Synchronized Attractor (FSA):
M
tt
MXf
M 1
|)('|ln1
lim|1|ln
21 1)( ttt aXXfX
: Transverse Lyapunov exponent associated with perturbation transverse to the diagonal
For strong coupling, < 0 Complete Synchronization
For < *(~ 0.2901), > 0 FSA: Transversely Unstable Transition to Clustering State
a=0.15
a=0.15
1313
)].([ cluster 2nd :)()(
),( cluster 1st :)()()(
121
121
1
1
NNNYtxtx
NXtxtxtx
tNN
tN
Two-Cluster States on an Invariant 2D Plane
Two-Cluster States
Transverse Lyapunov Exponents
,1 (,2): Transverse Lyapunov exponent associated with perturbation
breaking the synchrony of the 1st (2nd) cluster,1<0 and ,2<0 Two-cluster state: Transversely Stable Attractor in the original N-D state space
2D Reduced Map Governing the Dynamicsof the Two-Cluster State:
)].()([)1()(
)],()([)(
1
1
tttt
tttt
YfXfpYfY
XfYfpXfX
M
tt
M
M
tt
MYf
MXf
M 12,
11, |)('|ln
1lim|1|ln,|)('|ln
1lim|1|ln
p (=N2/N): Asymmetry Parameter (fraction of the total population of elements in the 2nd cluster)
0 (Unidirectional coupling) < p 1/2 (Symmetric coupling)
a=0.15=0.05
1414
Classification of Periodic Orbits in Terms of the Period and Phase Shift (q,s)
Scaling Associated with Periodic Orbits for the Two-Cluster Case
Scaling near the Zero-Coupling Critical Point (a*, 0) for p=1/2
q different orbits with period q distinguished by the phase shift s (=1,…,q-1) in the two uncoupled (=0) logistic maps
(Synchronous) In-phase orbit on the diagonal (s = 0) (Asynchronous) Anti-phase (180o out-of-phase) orbit with time shift of half a period (s = q/2)(Asynchronous) Non-antiphase orbits (Other s) Two orbits with phase shifts s and q- s: Conjugate-phase orbits (under the exchange X Y for p=1/2)
0,,4 * aaq
Stability Diagrams of the Conjugate-Phase Periodic Orbits
[1. S.P.Kuznetsov, Radiophysics and Quantum Electronics 28, 681 (1985). 2. S.-Y. Kim and H. Kook, Phys. Rev. E 48, 785 (1993). 3. S.-Y. Kim and H. Kook, Phys. Lett. A 178, 258 (1993).]
Renormalization Results: Scaling Factor for the Coupling Parameter =2 [i.e., ’(=2)]
1515
Dynamical Routes to Two-Cluster States (p=1/2)
Dynamical Route to Two-Cluster State for a=0.15 (Two Stages)
[ for the inertial coupling case; 2 for the dissipative coupling case]
[ Renormalization Results: 1. S.P.Kuznetsov, Radiophysics and Quantum Electronics 28, 681 (1985). 2. S.-Y. Kim and H. Kook, Phys. Rev. E 48, 785 (1993). 3. S.-Y. Kim and H. Kook, Phys. Lett. A 178, 258 (1993).]
[Linear Mean Field ‘Inertial Coupling’ (each element: maintaining the memory of its previous states)]
:))((1
cf.[1
N
jjt txf
NM Nonlinear Mean Field Dissipative Coupling
(Tendency of equalizing the states of the elements)]
1919
Successive Appearance of Similar Cluster States of Higher Ordersfor the Linear Coupling Case (P=1/2)
(1) 0th-Order Cluster State
(2) 1st-Order Renormalized State
(3) 2nd-Order Renormalized State
(No Complete Chaos Synchronization near the Zero Coupling Critical Point)
2/15.0 a
/6.0,/15.0 a
6.0,15.0 a 15.0a
/15.0a
2020
Asymmetric Effect on the Dynamical Routes to Clusters
p (asymmetry parameter): smaller
Conjugate-Phase Two-ClusterStates: Dominant
Appearance of Similar Cluster States2/15.0 a/15.0a
15.0a 35.0,15.0 pa
(Similar to the Dissipative Coupling Case)
2121
Dynamical Routes to Clusters and Scaling in Globally Coupled Oscillators
(Purpose: to examine the universality for the results obtained in globally coupled maps)