本篇論文的研究目的，是為了瞭解有限個數的震盪子，在一維空間中以降冪耦合(power-law coupling)所產生的同步化現象。我們使用連續邊界條件的Kuramoto 模型，並讓系統中的震盪子有柯西分布(Cauchy distribution)的自然頻率。我們發現，當震盪子之間以短距離的降冪耦合，震盪子將個別與局部的平均場耦合；反之，當震盪子之間的耦合是長距離(空間中所有震盪子等價地耦合)，每一個震盪子將和整個系統的平均場耦合。為了定量分析短距離的耦合與長距離的耦合對同步化現象的影響，我們定義了一個測量參數ξ，是任兩個相鄰震盪子的局部平均場之差。這個測量參數ξ包含了兩個資訊：有限的震盪子個數和降冪指數，其中降冪指數表示耦合距離的長短。測量參數ξ也代表了平均而言，震盪子的局部平均場(local mean field)與系統的平均場(mean field)之差。當測量參數ξ趨近於零，這個同步化系統實質上是長距離耦合，我們能預測相位角的同調性(phase coherence)。當測量參數ξ大於零，我們發現相位角同調性(phase coherence)會隨時間做週期性地振盪。造成振盪的原因，是因為震盪子出現局部的同步化而非全體同步化，使得系統中的震盪子分裂成較小的同步群體。我們研究判斷條件(occurrence condition)使得同步化系統出現振盪的相位角同調性，發現這個條件與震盪子自然頻率的排列方式有關。我們提出一個系統性的方法，能夠預測振盪的相位角同調性的發生，以及預測震盪子分裂成小群體的細部位置。

The focus of our research is to understand the synchronization phenomena among
a finite size network of oscillators with power-law coupling in an one dimensional
chain. Renormalized Kuramoto model with periodic boundary conditions and
Cauchy distribution of natural frequency are used in this thesis. In locally coupled
system the oscillator couples to its local mean field rather than the mean field in
globally coupling case. In order to analyze synchronization phenomenon of locally
coupled oscillators, we develop a measurement tool, ξ, as the difference of local
order parameter between any two nearest neighbours. The variation contains the
information of system size N and the decay rate of coupling α, and its value shows
how the local mean field deviates from the mean field. As variation decreases to
zero, the phase coherence is independent of the local couplings and system size,
which means that the locally coupled system at low variation is a globally coupled
system. In addition, the phase coherence becomes oscillatory with time as the
variation ξ becomes larger. We find that the occurrence of the oscillatory state is
due to the splitting of oscillators into multiple groups; oscillators in the same group
remain synchronized. The formation of oscillatory state is shown to be sensitive
to initial spatial arrangements of natural frequency of oscillators. We propose a
systematic approach to predict the occurrence condition of oscillatory states.

Contents i
List of Tables ii
List of Figures v
1 Introduction 1
2 Model 6
2.1 Kuramoto model with global interaction . . . . . . . . . . . . . . . 6
2.2 Renormalized Kuramoto model . . . . . . . . . . . . . . . . . . . . 10
2.3 Local mean field and Variation . . . . . . . . . . . . . . . . . . . . 14
2.4 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . 16
3 Result - Synchronization of oscillators with power-law interaction
in one dimension 21
3.1 Plateau Formation and Oscillatory State . . . . . . . . . . . . . . . 21
3.2 Dependence of Dynamical Behaviour on Variation . . . . . . . . . . 32
4 Conclusion and Future Work 41
Reference 46
Appendix 49
A. Coherence of integration form in all-to-all coupling . . . . . . . . . . . 49
B. Coherence of integration form in local field . . . . . . . . . . . . . . . 51

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