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Effects of the underlying topologies on Coupled Oscillators

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### Effects of the underlying topologies on Coupled Oscillators

Soon-Hyung Yook

Outlook

- Effects on the Underlying Topology on Dynamical Systems on Complex Networks
- Historical view
- Synchronization of Coupled Oscillators
- coupled chaotic oscillators
- Mean field interaction
- finding the topological properties causing synchronization on complex networks
- Modular structure

Historical remarks

In a letter to his father 1665 he described his observations of 2 pendulums fastened at the same beam:

Über lange Zeit war ich verblüfft über das unerwartete Resultat, aber nach vorsichtiger Untersuchung fand ich dieUrsache in der Bewegung des Stabes, obwohl sie kaum sichtbar ist.Die Ursache ist, dass die Pendel durch ihr Gewicht die Uhren etwas in Bewegung versetzen.Diese Bewegung überträgt sich auf den Stab und hat den Effekt, dass die Pendel letztlich entgegengesetzt schwingen, und dadurch kommt die Bewegung letztlich zum Stillstand. Aber diese Ursache kommt nur zum Zuge, wenn die entgegenge setzten Bewegungen der Uhren genau gleich und gleichförmig sind…..

C.Huygens 1629-1695

Bennett et al.(Atlanta) redo the experiment: Huygens was lucky………..

Corron et al. PRE 66 (2002)

Synchronization of Coupled Oscillators- Synchronization:
- To share the common time
- Collective Dynamics
- Examples in the nature:
- Josephson junction array [Daniels et al. PRE 67]
- living organisms such as biological rhythm of the heart or lightening of fire-flies [Mosekilde et al. World Scientific 2002]
- electrical circuit [Corron et al. PRE 66]

Phase Synchronization

- In many cases, the most interesting physical quantity is the phase.
- Winfree model
- Kuramoto model:
- Synchronization transition

Coupled Nonlinear Oscillators

- Coupled Oscillator: Zanette&Mikhailov (ZM) PRE 57 (1998)
- Each oscillator can be described by m-dimensional dynamic variable
- Without a coupling:
- The evolution of dynamic variable of i-th oscillator with coupling:
- Evolution of deviation: e’=1

Rössler Oscillator

- Phys. Lett. Rössler 1976
- Lorenz equation

Coupled Rössler Oscillator

- Special Case of ZM description
- Distance:

Order parameter

- What is the phase transition?
- Transition between two equilibrium phases of matter whose signature is a singularity or discontinuity in some observable quantities. some quantities change abruptly
- First order phase transition: discontinuity in a first derivative of a free energy discontinuity in a entropy
- Second order phase transition: discontinuity in a second derivative of a free energy
- What is the order parameter?
- A parameter which distinguishes the ordered and disordered phases
- Ex. Liquid-gas: density ferromagnetic: magnetization
- superconductor: electron pair amplitude

Zanette&Mikhailov (ZM) PRE 57 (1998)

Order Parameters & Synchronization Transition- Two different order parameters
- The number of pairs of oscillators at zero distance
- The number of oscillators which has at least one oscillator at d=0.

Small-World Networks

Starting with regular lattice

Add shortcuts with probability f

Scale-Free Networks

Probability to add m link between new and old node

Underlying Geometryk=2

k=1

k=4

Interpolation between 1<m<2 on SF net.

- With probability q

a new node has m=2

- With probability 1-q

a new node has m=1

k=2

k=2

k=1

k=1

k=4

k=4

meffective=1+q

n(d) for 1<m<2

- normalized number of oscillators at distance d

Different Behavior Between m=1 and m>1 on SF Networks

- m=1 tree structure
- m>1 networks have loops by definition
- difference between the average shortest path-length <l >
- different average connectivity <k>

Add shortcuts between nodes in the outermost shell

Distance distribution

Bethe Lattice with Closed LoopsBethe Lattice with shortcuts

Comparison to the SW and SF networks

Change of the Shortest Path Length DistributionConnect the nodes in the outermost shell to the central node with probability p1

Distance Distribution

Bethe Lattice with Long Range LoopsPhase of coupled limit-cycle oscillator

Kuramoto model

Effects of modular structure on synchronization

Def. : order parameter for phase

Modular structureOrder parameter for different m

m: number of modules

Especially, for weak coupling (small e):

maximum around m=10

Order parameterLaplacian matrix G

Ratio of eigenvalues

lmax: largest eigenvalue

l1: nonzero smallest eigenvalue

[L. M. Pecora and T. L. Carroll PRL 80, K. S. Fink et al. PRE 61]

Master stability function gives the stability condition

Laplacian Matrixc: constant: depends on the Lyapunov exponent

Summary & Conclusions

- We studied the effect of the underlying topology on the synchronization-transition of coupled chaotic oscillators.
- We observed there is no synchronization on the BA scale-free networks with m=1. However, for BA model with m>1 and SW networks, we found that the coupled chaotic oscillators show a complete synchronization depending on the coupling strength e.
- By comparing with the results on the Bethe lattice, we found that not only for the loops but also for the shortcuts which make the shortest path length distribution be bounded around the average shortest path length also necessary for global synchronization.
- The loops should remove dead-ends.
- We also studied the effect of the modular structure on the coupled limit-cycle oscillator.

Different Behavior Between m=1 and m>1 on SF Networks

- m=1 tree structure
- m>1 networks have loops by definition
- difference between the average shortest path-length <l >
- different average connectivity <k>

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