| Summary: | 博士 === 國立交通大學 === 應用數學系所 === 95 === The purpose of this thesis is two-fold. First, global synchronization in lattices of coupled chaotic systems is studied. Second, how wavelet transforms affect the synchronization of the corresponding systems is theoretically addressed. Based on the concept of matrix measures, global stability of synchronization in networks is obtained. Our results apply to quite general connectivity topology. Moreover, by merely checking the structure of the vector field of the single oscillator, we shall be able to determine if the system is globally synchronized. In addition, a rigorous lower bound on the coupling strength for global synchronization of all oscillators is also obtained. The lower bound on the coupling strength for synchronization is proportional to the inverse of the magnitude of the second largest eigenvalue λ2 of the coupling matrix. However, for a typical connectivity topology such as the diffusively coupled matrix, λ2 moves closer to the origin, as the number of nodes increases. Consequently, a larger coupling strength is required to realize synchronization. In [48], Wei et al, proposed a wavelet transform to alter the connectivity topology. In doing so, λ2=λ2 (α) becomes a quantity depending on wavelet parameter α. It is found there that a critical wavelet parameter αc can be chosen to move λ2 (αc) away from the origin regardless the number of nodes. This in turn greatly reduces the size of the critical coupling strength. Such phenomena are analytically verified when the coupling matrix is diffusively coupled with periodic and Neumann boundary conditions.
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