Eigenvalue problems and their application to the wavelet method of chaotic control

Controlling chaos via wavelet transform was recently proposed by Wei, Zhan, and Lai [Phys. Rev. Lett. 89, 284103 (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 ₁(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= l (respectively, 0) gives the nearest neighbor coupling with periodic (respectively, Neumann) boundary conditions. The first, rigorous work to understand the eigenvalues of C(α, 1) was provided by Shieh et al. [J. Math. Phys. (to be published)]. The purpose of this paper is twofold. First, we apply a different approach to obtain the explicit formulas for the eigenvalues of C(αa, 1) and C(α,0). This, in turn, yields some new information concerning λ₁(α, 1). Second, we shed some light on the question whether the wavelet method works for general coupling schemes. In particular, we show that the wavelet method is also good for the nearest neighbor coupling with Neumann boundary conditions.