Perturbed block circulant matrices and their application to the wavelet method of chaotic control

• Published in: Journal of mathematical physics 2006-12, Vol.47 (12), p.1
• Main Authors: Juang, Jonq, Li, Chin-Lung, Chang, Jing-Wei
• Format: Article
• Language: English
• Subjects: Eigenvalues; Exact sciences and technology; Mathematical methods in physics; Mathematical models; Mathematics; Matrix; Physics; Sciences and techniques of general use; Studies; Wavelet transforms
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.2400828

Controlling chaos via wavelet transform was proposed by Wei et al. [Phys. Rev. Lett. 89, 284103–1 284103–4 (2002)]. It was reported there that by modifying a tiny fraction of the wavelet subspace of a coupling matrix, the transverse stability of the synchronous manifold of a coupled chaotic system could be dramatically enhanced. The stability of chaotic synchronization is actually controlled by the second largest eigenvalue λ 2 ( α , β ) of the (wavelet) transformed coupling matrix C ( α , β ) for each α and β . Here β is a mixed boundary constant and α is a scalar factor. In particular, β = 1 (0) gives the nearest neighbor coupling with periodic (Neumann) boundary conditions. In this paper, we obtain two main results. First, the reduced eigenvalue problem for C ( α , 0 ) is completely solved. Some partial results for the reduced eigenvalue problem of C ( α , β ) are also obtained. Second, we are then able to understand behavior of λ 2 ( α , 0 ) and λ 2 ( α , 1 ) for any wavelet dimension j ∊ N and block dimension n ∊ N . Our results complete and strengthen the work of Shieh et al. [J. Math. Phys. 47, 082701–1 082701–10 (2006)] and Juang and Li [J. Math. Phys. 47, 072704–1 072704–16 (2006)].