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

Controlling chaos via wavelet transform was proposed by Wei [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 λ2 (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 λ2 (α,0) and λ2 (α,1) for any wavelet dimension jN and block dimension nN. Our results complete and strengthen the work of Shieh [J. Math. Phys. 47, 082701.1-082701.10 (2006)] and Juang and Li [J. Math. Phys. 47, 072704.1-072704.16 (2006)].