| Summary: | 碩士 === 國立清華大學 === 物理系 === 102 === 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.
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