1. Effects of the underlying topologies on Coupled Oscillators Soon-Hyung Yook

2. 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

3. 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………..

4. 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] Corron et al. PRE 66 (2002)

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

6. 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:  ’=1

7. Rössler Oscillator Phys. Lett. Rössler 1976 Lorenz equation

8. Coupled Rössler Oscillator Special Case of ZM description Distance:

9. 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

10. 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. Zanette&Mikhailov (ZM) PRE 57 (1998)

11. Underlying Geometry Small-World Networks Starting with regular lattice Add shortcuts with probability  Scale-Free Networks Probability to add m link between new and old node k=4 k=2 k=1

12. Synchronization Transition on SW Networks Distance Distribution Two order parameters

13. Distance Distribution on SF Networks m=1 m=2

14. Interpolation between 1<m<2 on SF net. With probability q  a new node has m=2 k=4 k=2 k=1 With probability 1-q  a new node has m=1 k=4 k=2 k=1  m effective =1+q

15. n(d) for 1<m<2 normalized number of oscillators at distance d

16. Two Order Parameters on SF Networks r s

17. Two Order parameters on SF networks (cont.) r s

18. 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 different average connectivity

19. Bethe Lattice with Loops Construct small loops by adding a new links Distance distribution

20. Bethe Lattice with Closed Loops Add shortcuts between nodes in the outermost shell Distance distribution

21. Change of the Shortest Path Length Distribution Bethe Lattice with shortcuts Comparison to the SW and SF networks

22. Bethe Lattice with Long Range Loops Connect the nodes in the outermost shell to the central node with probability p 1 Distance Distribution

23. Modular structure Phase of coupled limit-cycle oscillator  Kuramoto model Effects of modular structure on synchronization Def. : order parameter for phase

24. Order parameter Order parameter for different m m: number of modules Especially, for weak coupling (small  ):  maximum around m=10

25. Laplacian Matrix Laplacian matrix G Ratio of eigenvalues max : largest eigenvalue 1 : 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 c: constant: depends on the Lyapunov exponent

26. 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 . 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.

28. Average cluster size & number of cluster

29. Size Dependent Behavior of Two Order Parameters on SF Networks:m=2 r s

30. 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 different average connectivity