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

Abstract

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 lambda(2)(alpha,beta) of the (wavelet) transformed coupling matrix C(alpha,beta) for each alpha and beta. Here beta is a mixed boundary constant and alpha is a scalar factor. In particular, beta=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(alpha,0) is completely solved. Some partial results for the reduced eigenvalue problem of C(alpha,beta) are also obtained. Second, we are then able to understand behavior of lambda(2)(alpha,0) and lambda(2)(alpha,1) for any wavelet dimension j is an element of N and block dimension n is an element of N. 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)]. (c) 2006 American Institute of Physics.