Exponential dichotomy on the real line: SVD and QR methods

Abstract

In this work we show when and how techniques based on the\ud singular value decomposition (SVD) and the QR decomposition of\ud a fundamental matrix solution can be used to infer if a system\ud enjoys—or not—exponential dichotomy on the whole real line

Detecting exponential dichotomy on the real line: SVD and QR algorithms

“…This last topic we addressed in [17]. In line with this previous work of ours, in the present paper we base our numerical algorithms on the SVD and QR factorizations of a fundamental matrix solution.…”

“…As a consequence, to verify if (1) has ED in R, we will check that (see [17]) (ED-1) It has ED in R + and R − . [That is, 0 / ∈ ± ED .]…”

“…To verify if (1) possesses exponential dichotomy (ED for short) in R, in this paper we implement, and test, algorithms which follow the theoretical development we laid down in [17]. In that work, we gave theoretical justification for two different techniques to detect whether or not (1) admits ED in R. These techniques are based on the singular value decomposition (SVD) and on the QR factorization of a fundamental matrix solution of (1), respectively.…”

“…In that work, we gave theoretical justification for two different techniques to detect whether or not (1) admits ED in R. These techniques are based on the singular value decomposition (SVD) and on the QR factorization of a fundamental matrix solution of (1), respectively. Although in [17] we considered the case of real valued coefficient matrix (in (1), A ∈ R n×n ), the extension to the complex case (which is the case considered in this paper) is straightforward and appropriate comments will be made here below as needed.…”

“…The fundamental result on which we developed our analysis in [17] is the following one of Coppel (see [8]): "A given linear system possesses ED in R if and only if it possesses ED in the two half-lines, and the forward stable subspace and the backward stable subspace 1 are complementary". So, there are two different tasks:…”