A Frequency-weighted Kuramoto Model

1. A Frequency-weighted Kuramoto Model Lecturer: 王瀚清 Adaptive Network and Control Lab Electronic Engineering Department of Fudan University

2. Outlines • Background • Our Model • Simulations • Analysis • Conclusion

3. Outlines • Background • Our Model • Simulations • Analysis • Conclusion

4. The Original Kuramoto Model • A classical and useful tool to analyze networks of coupled oscillators • Drawbacks: Idealized assumptions and constraints • All-to-all, Equal-weighted • The distribution of natural frequencies should be unimodal and symmetric • Extension • Practically, the couplings among the oscillators should be influenced by their own charateristics, i.e. Power grid • Unimodal distributions are not universal, especially in human dynamics.

5. Outlines • Background • Our Model • Simulations • Analysis • Conclusion

6. Description and Definition Governing equations: The order parameter With this definition, Eq.1 becomes When , Eq.2 becomes Frequency-distribution

7. Outlines • Background • Our Model • Simulations • Analysis • Conclusion

8. Instructions • In most of the simulations: we set N=1000, , . • The left figures shows the results when . • In the bottom figure, we illustrate how will the final value of varies with the coupling strength . • The oscillatory part in the r-K figure indicates that the final value of will oscillate instead of converging to a steady value, as shown in the top figure. • We averaged the results when the final value of is converged.

9. The Cases of Odd Beta: • A threshold of the coupling strength exists. • Below the threshold, the final value will oscillate. • Exceeding the threshold, the final value will converge to a steady value. However with the coupling strength increasing, decreases.

10. A Microscopic View • The oscillators will spontaneously split into two clusters of different synchronization when . And these two clusters locate roughly at the opposite sides on the unit-circle. • With coupling strength increasing, more oscillators will be locked to the two clusters, and run at a common frequency

11. The Cases of Even Beta: • These also exists a threshold . • Below the threshold, the final values oscillate significantly. • Above the threshold ,the final values will come to a steady value.

12. A Microscopic View • The oscillators would spontaneously split into two clusters even when .In this condition, however, the two clusters running in opposite directions at the same frequency. • When , the two locked clusters move closer to each other, and will finally stop near 0, with a tiny difference between their average phases.

13. Others Distributions • Uniform Distribution • Gaussian Distribution

14. Outlines • Background • Our Model • Simulations • Analysis • Conclusion

15. Two Questions • What kind of oscillators compose the two clusters we observed previously？ • Why significant synchronization is only possible for even beta, and why increasing coupling strength will decrease the order parameter when beta is odd？

16. Even Beta: illustrating with the case • Suppose there is a oscillator with its natural frequency , and its phase is . Meanwhile, there is another oscillator , whose natural frequency and phase satisfy . Then we proved that this condition will always be satisfied. • We assume that the locked positive cluster and locked negative cluster run at frequencies respectively, then we can decide which oscillator can be locked. • Consider the case where the coupling strength is large enough to lock all oscillators, then we can get the solution of and finally get the following:

17. Odd beta: illustrating with the case • The symmetry of network lose when beta is odd, which can be also proved. • We suppose that the locked clusters, positive and negative run in the same direction with frequency . With these, we derived which oscillator can be locked and find some can’t be locked no matter how large the coupling strength is. • We get for the locked oscillators, and only consider their contribution to the order parameter, because the unsynchronized oscillator’s influence on order parameter is relatively small.

18. Chimera States when beta is odd

19. Chimera States • The so-called Chimera States were first found by Kuramoto in a reaction and diffusion system, where identical nodes behave quite differently. And Strogatz named this phenomena as ‘Chimera States’ • Recently, the Chimera States in a heterogeneous network have also been studied. However, Our model is different from the previous work in the following aspects: • In previous work , the Kuramoto oscillator networks with observed Chimera States are usually phase-delayed. • In previous work , oscillators were deliberately divided into several groups, and the coupling strengths in each group are different.

20. Outlines • Background • Our Model • Simulations • Analysis • Conclusion

21. Conclusion • Proposed a frequency-weighted Kuramoto model • Investigated its dynamics by numeric simulations • Analyzed the observations with mathematical method. Thank you for listening! Q&A