The Wavelet Transform Method for the Eigenvalues of an Asymmetrical Sparse Coupled Matrix and its Synchronous Applications

• Main Authors: Chun-Kai Hsu, 許竣凱
• Other Authors: Chin-Lung Li
• Format: Others
• Language: zh-TW
• Published: 2011
• Online Access: http://ndltd.ncl.edu.tw/handle/73763588223852880290

碩士 === 國立新竹教育大學 === 人資處數學教育碩士班 === 99 === Networks of coupled chaotic oscillators model many systems of interest in physics, electrical engineering, biology, laser systems, etc. In particular, complete chaotic synchronization, all oscillators acquiring identical chaotic behavior, has received much attention analytically. In 2007, global stability of synchronization in networks is studied by Juang et al. [Chaos, 17, 033111.11]. Their results apply to quite general connection topology. 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. Therefore, the greater can greatly increase the applicable ranges of the coupling strengths. In 2002, Wei et al.[ Phys. Rev. Lett. 89, 284103.4] proposed a wavelet transform method to alter the connection topology. The wavelet transform method was reported that by modifying a tiny fraction of the wavelet subspace of a coupling matrix, the transverse stability of the synchronous manifold M of a coupled chaotic system could be dramatically enhanced. In other words, the wavelet transform method can greatly increase the applicable ranges of the coupling strengths and the number of oscillators for synchronization of networks of coupled chaotic systems. In this thesis, three kinds of circulant connection topologies are studied. First, the eigenvalues formulas for these coupling matrices are analytical found. Second, we sort the second largest eigenvalue by the eigenvalues formulas. Finally, we discuss how the wavelet transform method affects the synchronous phenomena in coupled chaotic systems for the coupling matrix with circulant connection topology.