The rotation method and singular numbers of three-dimensional systems (Q2703854)

Here, the author continues the investigations begun in [Lower estimate for the minimal exponent of a three-dimensional system. (Russian, English) Differ. Equations 35, No. 10, 1407-1417 (1999); translation from Differ. Uravn. 35, No. 10, 1387-1387 (1999; Zbl 0981.34033)]. In that paper, he introduced the singular numbers \(\delta_i(X)\), \(i= 1,2,3\), of the Cauchy operator \(X\) of a third-order linear system \(\dot x= A(t)x\) and studied them to give an estimate (from below) on the minimal exponent of this system. In terms of the singular numbers for a given linear system with Cauchy operator \(X\), he introduced the idea of upper \(\varepsilon\)-separability, lower \(\varepsilon\)-separability, general \(\varepsilon\)-separability and general \(\varepsilon\)-inseparability on an arbitrary interval \([s,t]\).NEWLINENEWLINENEWLINEIn the present paper, he shows how, using Millionshchikov's rotation method for a linear third-order system that possesses general \(\varepsilon\)-separability on the \(k\)th interval \([(k- 1)T,kT]\) to construct a fundamental system of solutions with growth close to the given singular numbers of the Cauchy operator.NEWLINENEWLINENEWLINEIn developing the concept of \(\varepsilon\)-separability on an interval, the author introduces the notions of upper closeness, lower closeness and general closeness on an interval. Such systems are studied. In particlar, it is proved that if \([(k-1)T, kT]\), is an interval of upper closeness for the system \(\dot x= A(t)x\), then for sufficiently large \(T\) there exists a system \(\dot y= B(t)y\) for which there is general \(\varepsilon\)-separability on the \(k\)th interval, the norms of the matrices \(A(t)\) and \(B(t)\) are close on this interval, and the singular numbers of the Cauchy operators \(X\) and \(Y\) of these systems are also close. The author establishes a connection between the singular subspaces of these systems \(A\) and \(B\).