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Particle and kinetic descriptions of Kuramoto ensemble in the presence of adaptive couplings and noises : 변동성 결합력과 노이즈를 가지는 쿠라모토 모델의 입자, 운동 방정식에 관하여
| DC Field | Value | Language |
|---|---|---|
| dc.contributor.advisor | 하승열 | - |
| dc.contributor.author | 이재승 | - |
| dc.date.accessioned | 2018-11-12T00:58:58Z | - |
| dc.date.available | 2018-11-12T00:58:58Z | - |
| dc.date.issued | 2018-08 | - |
| dc.identifier.other | 000000151867 | - |
| dc.identifier.uri | https://hdl.handle.net/10371/143218 | - |
| dc.description | 학위논문 (박사)-- 서울대학교 대학원 : 자연과학대학 수리과학부, 2018. 8. 하승열. | - |
| dc.description.abstract | In this thesis, we investigate the particle and kinetic models of the Kuramoto-type model with adaptive coupling strengths and noise environment. In the classical Kuramoto model, the pairwise coupling strengths are assumed to be uniform and the oscillators are always supposed to be deterministic. For more realistic description of many-body systems in the real world, however, the need for improving the classical model arises. In this regard, we first introduce the Kuramoto-type models where the coupling strengths among the oscillators depend on the pairs and time. Then, we consider the behaviour of the Kuramoto oscillator ensemble when the number of oscillators goes to infinity($N \to \infty$) in the presence of phase lag and Gaussian noise. Finally, we study the swarming model on a unit sphere that can be regarded as a variation of the classical Kuramoto model. | - |
| dc.description.tableofcontents | Contents
Abstract i 1 Introduction 1 2 Preliminaries 10 2.1 Kuramoto model with adaptive couplings . . . . . . . . . . . . 10 2.2 Kuramoto-Sakaguchi-Fokker-Planck equation . . . . . . . . . . 12 2.2.1 Stability problems . . . . . . . . . . . . . . . . . . . . 14 2.2.2 Instability problems . . . . . . . . . . . . . . . . . . . . 15 2.3 From Lohe matrix model to particle swarm model . . . . . . . 16 3 Kuramoto oscillators with adaptive couplings: conservation law and fast learning 22 3.1 Synchronization of P-R oscillators . . . . . . . . . . . . . . . . 22 3.1.1 Dynamics of coupling strengths . . . . . . . . . . . . . 25 3.1.2 Synchronization estimates . . . . . . . . . . . . . . . . 28 3.2 Singular limit of the model A . . . . . . . . . . . . . . . . . . 32 3.2.1 Infinite learning rate limit ε → 0 . . . . . . . . . . . . 32 3.2.2 Synchronization of the limit system . . . . . . . . . . . 35 3.3 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . 42 3.3.1 The P-R model . . . . . . . . . . . . . . . . . . . . . . 42 3.3.2 Infinite learning rate limit . . . . . . . . . . . . . . . . 45 3.3.3 Evolution of order parameters . . . . . . . . . . . . . . 46 3.3.4 ACPS of the limit system . . . . . . . . . . . . . . . . 46 ii CONTENTS 4 Kuramoto-Sakaguchi-Fokker-Planck equation with frustra- tion I 49 4.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.1 Asymptotic stability . . . . . . . . . . . . . . . . . . . 49 4.1.2 Vanishing frustration limit . . . . . . . . . . . . . . . . 52 4.2 Stability of incoherent state . . . . . . . . . . . . . . . . . . . 53 4.2.1 A priori H 4 γ -estimates . . . . . . . . . . . . . . . . . . . 54 4.2.2 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . 61 4.3 Vanishing frustration limit . . . . . . . . . . . . . . . . . . . . 62 4.3.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . 62 4.3.2 Basic estimates . . . . . . . . . . . . . . . . . . . . . . 63 4.3.3 Proof of Theorem 4.1.2 . . . . . . . . . . . . . . . . . . 73 5 Kuramoto-Sakaguchi-Fokker-Planck equation with frustra- tion II 75 5.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2 Nonlinear instability of incoherent state I . . . . . . . . . . . . 78 5.2.1 Basic estimates . . . . . . . . . . . . . . . . . . . . . . 79 5.2.2 Unstable mode . . . . . . . . . . . . . . . . . . . . . . 81 5.2.3 A nonlinear problem . . . . . . . . . . . . . . . . . . . 83 5.2.4 Error estimates . . . . . . . . . . . . . . . . . . . . . . 88 5.2.5 Proof of Theorem 5.1.1 . . . . . . . . . . . . . . . . . . 92 5.3 Nonlinear instability of incoherent state II . . . . . . . . . . . 94 5.3.1 Preparatory estimates . . . . . . . . . . . . . . . . . . 95 5.3.2 Proof of Theorem 5.1.2 . . . . . . . . . . . . . . . . . . 100 6 Swarming model on a unit sphere 101 6.1 From particle swarm to kinetic swarm . . . . . . . . . . . . . . 102 6.2 A particle swarm model on the unit sphere . . . . . . . . . . . 106 6.2.1 Emergent dynamics . . . . . . . . . . . . . . . . . . . . 106 6.2.2 Uniform p -stability . . . . . . . . . . . . . . . . . . . . 108 6.3 The kinetic swarm model on the unit sphere . . . . . . . . . . 112 6.3.1 A measure-theoretic formulation . . . . . . . . . . . . . 112 6.3.2 Uniform mean-field limit . . . . . . . . . . . . . . . . . 113 6.4 A noisy particle swarm model and its kinetic limit . . . . . . . 117 iii CONTENTS 6.4.1 A stochastic mean-field limit . . . . . . . . . . . . . . . 117 6.4.2 Sobolev spaces on the unit sphere . . . . . . . . . . . . 123 6.4.3 A kinetic swarm model with diffusion . . . . . . . . . . 125 6.5 Nonlinear stability of the incoherent state . . . . . . . . . . . 136 6.5.1 Identical particles . . . . . . . . . . . . . . . . . . . . . 137 6.5.2 Nonidentical particles . . . . . . . . . . . . . . . . . . . 142 7 Conclusion and future works 147 Bibliography 149 Appendix A Proof of Proposition 3.2.1 160 Appendix B Implicit Runge Kutta method 162 Abstract (in Korean) 164 | - |
| dc.language.iso | en | - |
| dc.publisher | 서울대학교 대학원 | - |
| dc.subject.ddc | 510 | - |
| dc.title | Particle and kinetic descriptions of Kuramoto ensemble in the presence of adaptive couplings and noises | - |
| dc.title.alternative | 변동성 결합력과 노이즈를 가지는 쿠라모토 모델의 입자, 운동 방정식에 관하여 | - |
| dc.type | Thesis | - |
| dc.description.degree | Doctor | - |
| dc.contributor.affiliation | 자연과학대학 수리과학부 | - |
| dc.date.awarded | 2018-08 | - |
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