On the critical parameter value for the higher sigma-exponent of linear differential systems

Abstract

where the least upper bound is taken over all possible sigma-perturbations Q corresponding to a given σ > 0 is called [1] the higher sigma-exponent (the Izobov exponent) of system (1). It is well known that the sets of characteristic exponents of the original and perturbed systems (1) and (2) coincide for all σ > 2M ; therefore, the relation ∇σ(A) = λn(A) is valid for such σ. It follows from the results in [2, 3] providing a complete description of the higher sigma-exponent treated as a function of the parameter σ that there exists a unique critical value σ0(A) ∈ [0, 2M ] such that ∇σ(A) = λn(A) for all σ > σ0(A) and ∇σ(A) > λn(A) for 0 < σ < σ0(A). By using the Lyapunov, Grobman, and Perron irregularity coefficients [4, pp. 67, 73; 5, p. 80] σL(A), σG(A), and σP(A), respectively, one can obtain more accurate upper bounds for σ0(A). In particular, it was shown in [6] that the inequality σ0(A) ≤ σL(A) holds, which, by [7], can be strengthened to the estimate σ0(A) ≤ σG(A). If n = 2, then, by [8], we have the even stronger inequality σ0(A) ≤ σP(A), which fails [9, 10] for systems of higher dimension. In the paper [9], the irregularity coefficient σλ(A) was constructed, which is also an upper bound for σ0(A) but, in addition, does not exceed σG(A) and coincides with σP(A) for two-dimensional systems. In the present paper, on the basis of the algorithm constructed in [1] for the computation of the higher sigma-exponent ∇σ(A), we obtain explicit formulas expressing the critical value σ0(A) in terms of the Cauchy matrix XA(t, s) of the unperturbed system (1). Let us introduce related notation. For each m ∈ N, by D0(m) we denote the set of all subsets d of the set {1, . . . ,m− 1} ⊂ N. If d ∈ D (m) is nonempty, then its elements di, i = 1, . . . , s, where s := |d| is the number of elements in d, are assumed to be numbered in ascending order, so that d = {d1, . . . , ds} and 0 < d1 < · · · < ds < m. The class of nonempty sets d ∈ D0(m) is denoted by D (m). We also set ‖d‖ := d1 + · · · + ds for d = ∅ and ‖d‖ := 0 for d = ∅. Furthermore, for convenience of computations, we assume that d0 = 0 and ds+1 = m for all d ∈ D (m). Under these assumptions, we define Ξ(m,d) := ∑s i=0 ln ‖XA (di+1, di)‖, where m ∈ N, d ∈ D (m), and s := |d|.