In the paper [ Differ. Uravn ., 2007, vol. 43, no. 2, pp. 191–202], we defined the noncoinciding irreducibility sets N 2 ( a, σ ) and N 3 ( a, σ ), σ ∈ (0, 2 a ], of all n -dimensional linear differential systems with piecewise continuous coefficient matrices A(t) bounded on the half-line [0,+∞) with norms || A ( t )|| ≤ a < +∞ for each of which there exists a linear differential system that cannot be reduced to it by Lyapunov transformations and whose coefficient matrix B ( t ) satisfies the condition || B ( t ) - A ( t )|| ≤ const × e − σt , t ≥ 0, or the more general condition that the Lyapunov exponent of the difference B ( t ) - A ( t ) does not exceed - σ , respectively. In the present paper, we study the properties of irreducibility sets treated as functions of the parameters σ and a .

The paper [2] defines the noncoinciding irreducibility sets N 2 ( a, σ ) and N 3 ( a, σ ), σ ∈ (0, 2 a ], of all n -dimensional linear differential systems with piecewise continuous coefficient matrices A ( t ) such that ‖ A ( t )‖ ≤ a < + ∞ for t ∈ [0,+ ∞ ) and there exists a linear differential system that is not Lyapunov reducible to the original system and has coefficient matrix B ( t ) satisfying [for the case of N 2 ( a, σ )] the condition B(t) - A(t)⩽ const × e^ - σ t ,t ⩾ 0, or [for the case of N 3 ( a, σ )] the more general condition that the Lyapunov exponent of the difference B ( t ) − A ( t ) does not exceed − σ . For these sets, which are related by the obvious inclusions N_i (a,σ _1 ) ⊇ N_i (a,σ _2 ),0 < σ _1 < σ _2 ⩽ 2a,i = 2,3, , we prove that (i) they strictly decrease with increasing parameter σ ∈ (0, 2 a ], N i ( a, σ 1 ) ⊃ N i ( a, σ 2 ) for σ 1 < σ 2 ; (ii) there is a strict inclusion N 2 ( a, σ ) ⊂ N 3 ( a, σ ) for all σ ∈ (0, 2 a ].

We establish that the reducibility exponent ( Differentsial’nye Uravneniya , 2007, vol. 43, no. 2, pp. 191–202) of each linear system ẋ = A(t)x, x ∈ℝ^n , t ⩾ 0 , with piecewise continuous bounded coefficient matrix A does not belong to the set of values of σ for which the perturbed system (1 A + Q ) with an arbitrary piecewise continuous perturbation Q satisfying the condition lim _t → + ∞ t^ - 1lnQ(t)⩽ - σ is reducible to the original system (1 A ) by some Lyapunov transformation.

A great number of papers (it seems impossible to compile the complete bibliography of them) are dedicated to the investigation of the classic notion of Lyapunov’s reducibility (see [1, p. 43]) of linear systems. Here we are interested in properties of the coefficient of reducibility r2(A) and the exponent of reducibility r3(A) of (1A) with respect to perturbations (2) and (31)–(32), respectively. Definition 1 (see [2]). The infimum of the set R2(A) (the set R3(A)) of all values of σ > 0 such that perturbed system (1A+Q) with any perturbation Q satisfying condition (2) (conditions (31)–(32)) is reducible to the initial system (1A) is called the coefficient of reducibility r2(A) (the exponent of reducibility r3(A)) of (1A). To further investigate the properties of r2(A) and r3(A), we will use the following definition which is equivalent to Definition 1.

The class of linear differential systems with coefficient matrices which are commutative with their integrals is considered. The results on asymptotic equivalence of these systems and their distribution among linear systems are given.

It is known (e.g., see [1, p. 274, Th. 21.1.1]) that any linear system is almost reducible by a Lyapunov transformation [2] to a diagonal system with real diagonal entries. Since a diagonal system is necessarily a system with a functionally commutative coefficient matrix, it follows that any linear system is almost reducible to a system with a functionally commutative coefficient matrix. On the other hand, it was proved in [3] that there exist linear systems that cannot be reduced by Lyapunov transformations to systems with functionally commutative coefficient matrices. Systems considered in that paper are proper systems, which, by virtue of the Basov-Bogdanov-Grobman criterion (e.g., see [4, p. 77]), can be reduced by generalized Lyapunov transformations [4, p. 78] to diagonal systems, which are necessarily systems with functionally commutative coefficient matrices. The aim of the present paper is to prove the existence of linear systems that cannot be reduced to systems with functionally commutative coefficient matrices by generalized Lyapunov transformations.

An asymptotic equivalence of linear differential systems and Lappo-Danilevski systems is investigated. It is proved that there exist linear systems which are not asymptotically equivalent to Lappo-Danilevski systems.

An asymptotical equivalence of linear differential equations is studied. It is shown that for any second order linear differential equation with bounded coefficients there exists the asymptotically equivalent equation with piecewise constant coefficients and a special sequence of points of discontinuity.