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Some problems arising from the dynamics of the Kuramoto oscillators : 쿠라모토 진동자들의 동역학에서 일어나는 문제들에 대한 고찰

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dc.contributor.advisor하승열-
dc.contributor.author김용덕-
dc.date.accessioned2017-07-14T00:40:37Z-
dc.date.available2017-07-14T00:40:37Z-
dc.date.issued2014-02-
dc.identifier.other000000016728-
dc.identifier.urihttps://hdl.handle.net/10371/121274-
dc.description학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2014. 2. 하승열.-
dc.description.abstractIn this thesis, we study several problems on the ensemble of Kuramoto oscillators. We present the nonlinear stability of the phase-locked states using a robust $\ell_1$-metric as a Lyapunov functional. We show that the phase-locked states are congruent each other in the sense that one phase-locked state is the simply translation of the other and phase-shift is the difference of averaged initial phases. We also show the contraction property for measure valued solutions of the kinetic Kuramoto model. We next consider the effect of interaction frustration on the complete synchronization of Kuramoto oscillators. In general, interaction frustration hinders the formation of complete frequency synchronization. For more quantitative estimates, we considered three Kuramoto-type models. Our first model is for an ensemble of Kuramoto oscillators with uniform interaction frustration. Our second model is, as a special case of the first model, a mixture of two identical Kuramoto oscillator groups with distinct natural frequencies. Our third model is like the Kuramoto model for identical oscillators on the bipartite graph. Finally, we investigate the intricate interplay between the inertia and frustration in an ensemble of Kuramoto oscillators. We cannot apply the explicit macro-micro decomposition to reduce the dynamics of initial phases to that of fluctuations. However, we can still derive second-order differential inequalities for the phase or frequency diameters so that the second-order Gronwall inequality method still works well. Moreover, both the analytical and numerical studies demonstrate this fact.-
dc.description.tableofcontentsAbstract
1 Introduction
2 Preliminaries
2.1 The Kuramoto model
2.2 The kinetic Kuramoto equation
3 Nonlinear stability
3.1Orbital stability of phase-locked states
3.2Stability estimate of the kinetic Kuramoto equation
3.2.1 Alternative formulation of the KKE
3.2.2 Strict contractivity in the Wasserstein distance
4 Kuramoto type models with frustration
4.1Kuramoto model with frustration
4.2 Synchronization estimate for Model A
4.2.1 Existence of a trapping region
4.2.2 Entrance to the exponential stability regime
4.2.3 Relaxation estimate
4.3Synchronization estimates for Model B
4.3.1Existence of a trapping region
4.3.2From mixed stage to segregated stage
4.3.3Formation of the two-point cluster configuration
4.4Synchronization estimates for Model C
4.4.1 Existence of a trapping region
4.4.2 Relaxation estimate
4.5 Numerical examples
4.5.1 Model A
4.5.2 Model B
4.5.3 Model C
5Kuramoto model with inertia and frustration
5.1The Kuramoto model with inertia
5.2Synchronization estimate: identical oscillators
5.2.1 Notations
5.2.2 Complete synchronization
5.3Synchronization estimate: non-identical oscillators
5.3.1 A small inertia regime
5.3.2 A large inertia regime
5.4Numerical simulations
5.4.1 Identical oscillators
5.4.2 Nonidentical oscillators
6 Conclusion and future works
6.1 Conclusion
6.2 Future works
Abstract (in Korean)
Acknowledgement (in Korean)
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dc.formatapplication/pdf-
dc.format.extent2371470 bytes-
dc.format.mediumapplication/pdf-
dc.language.isoko-
dc.publisher서울대학교 대학원-
dc.subjectKuramoto model-
dc.subjectorbital stability-
dc.subjectcontraction-
dc.subjectfrustration-
dc.subjectsynchronization-
dc.subject.ddc510-
dc.titleSome problems arising from the dynamics of the Kuramoto oscillators-
dc.title.alternative쿠라모토 진동자들의 동역학에서 일어나는 문제들에 대한 고찰-
dc.typeThesis-
dc.contributor.AlternativeAuthorYongduck Kim-
dc.description.degreeDoctor-
dc.citation.pages111-
dc.contributor.affiliation자연과학대학 수리과학부-
dc.date.awarded2014-02-
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