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We consider two generalizations $$A{\cal E}_n$$ and $$W{\cal E}_n$$ of the class $${\cal E}_n$$ of linear $$n$$-dimensional exponentially dichotomous systems on the half-line. The definition of these classes differs from the classical definition of exponential dichotomy in terms of estimates for the norms of solutions in that we allow these estimates to hold starting from some time depending on the solution (the class $$A{\cal E}_n)$$ or the constant factors in the estimates to depend on the solution (the class $$W{\cal E}_n)$$. We give an equivalent definition of the class $$W{\cal E}_n$$ in the language of Bohl exponents and prove that the proper inclusions $${\cal E}_n\subset A{\cal E}_n \subset W{\cal E}_n$$ hold for $$n\geq 2$$.

We consider families of n-dimensional (n ≥ 2) linear differential systems on the time half-line with parameter belonging to a metric space. We obtain a complete description of the Lyapunov irregularity coefficient as a function of the parameter for families whose dependence on the parameter is continuous in the sense of uniform convergence on the time half-line. As a corollary, we completely describe the parametric dependence of the Lyapunov irregularity coefficient of a regular linear system with a linear parametric perturbation decaying at infinity uniformly with respect to the parameter.

For parametric families of n-dimensional linear differential systems on the time semiaxis with parameter varying in a metric space, we consider two functions of the parameter defined as the dimension of the subspace of solutions that have the characteristic exponent that, respectively, is less than or does not exceed a given real number. A complete description is derived both for the functions themselves and for the vector function composed of them, for the families of systems continuous in one of the two topologies: uniform or compact-open. In addition, the Lebesgue sets and the sets of points of upper and lower semicontinuity are described for the indicated functions.

We consider families of n-dimensional (n ≥ 2) linear differential systems on the time semiaxis with parameter varying in a metric space. For such families continuously depending on the parameter in the sense of uniform convergence on the time semiaxis, we completely describe the spectra of their Lyapunov exponents as vector functions of the parameter.

The theorem that claims that the spectra (ranges) of upper and lower Sergeev frequencies of zeros, signs, and roots of a linear differential equation of order > 2 with continuous coefficients belong to the class of Suslin sets on the nonnegative half-line of the extended numerical line is inverted for the spectra of upper frequencies of third-order equations under the assumption that the spectra contain zero. In addition, we construct examples of third-order equations with continuous coefficients whose Lebesgue sets of the upper Sergeev frequency of signs belong to the exact first Borel class, and the Lebesgue sets of upper Sergeev frequencies of zeros and roots belong to the exact second Borel class.

The kinematic similarity set for a pair of two piecewise continuous bounded matrices (i.e., matrix-valued functions) A(·) and B(·) defined on the time semi-axis is defined as the set of all real μ for which the matrices μA(·) and μB(·) are kinematically similar. It is shown that a subset of the real axis is a kinematic similarity set for a pair of matrices if it contains the origin and is an Fσ-set.

We obtain formulas for the computation of a certain asymptotic characteristic of piecewise continuous functions defined on the half-line and use it to state the well-known necessary and sufficient condition on a linear homogeneous differential system for the corresponding inhomogeneous system with arbitrary inhomogeneity whose characteristic exponent is nonpositive to have a solution with nonpositive characteristic exponent. We give a new form of this necessary and sufficient condition similar to the Perron-Maizel condition for the exponential dichotomy of the system.

We consider linear piecewise differentiable transformations of linear differential systems. We show that the set of such transformations preserving the Lyapunov exponents of each system with finite Lyapunov exponents coincides with the group of generalized Lyapunov transformations. In addition, we show that the set of transformations bringing each linear system with bounded coefficients on the half-line to a system of the same class and preserving its Lyapunov exponents is a group containing the group of Lyapunov transformations as a proper subgroup, and we describe that group. We obtain descriptions of other groups and semigroups of linear piecewise differentiable transformations considered in the theory of Lyapunov exponents and the theory of stability.

We consider families of linear differential systems depending on a real parameter that occurs only as a factor multiplying the matrix of the system. The stability (respectively, asymptotic stability) set of such a family is defined as the set of parameter values for which the corresponding systems in the family are stable (respectively, asymptotically stable). We show that a pair (P,Q) of sets on the real line is a pair consisting of the stability set P and the asymptotic stability set Q of some family if and only if the following conditions are satisfied: 0 ∈ P; if Q ≠ ∅, then P and Q are Fσ- and Fσδ-sets, respectively, lie on one of the closed rays issuing from zero, and satisfy Q ⊂ P\{0}; if Q = ∅, then P is an Fσ-set on the real line. In addition, for any pair (P,Q) of sets with these properties, the coefficient matrix of a family with multiplying parameter whose stability and asymptotic stability sets coincide with P and Q, respectively, can be chosen to be infinitely differentiable and uniformly bounded on the time half-line. The same problem of complete description of pairs consisting of stability and asymptotic stability sets is also solved for general one-parameter families of linear differential systems whose solutions continuously depend on a parameter.

Proper linear differential systems (whose coefficients are not necessarily bounded on the half-line) are defined as systems for which there exists a generalized Lyapunov transformation reducing them to a diagonal system with constant coefficients (Basov). We prove that Lyapunov’s original definition of a proper system and the Perron and Vinograd criteria hold for the class of proper systems as well as for the class of proper systems with uniformly bounded coefficients. We show that the Lyapunov properness criterion for a triangular system fails for systems with unbounded coefficients; namely, we construct an improper system with the following properties: the Lyapunov exponents of all nonzero solutions of that system are finite and exact, and for an arbitrary reduction of this system by a generalized Lyapunov transformation to triangular form, its diagonal coefficients have finite exact mean values, whose set with regard of multiplicities is independent of the choice of the transformation. In addition, we show that the main property of proper systems with uniformly bounded coefficients (preservation of conditional exponential stability as well as the dimension of the exponentially stable manifold and the exponent of the asymptotic behavior of solutions under perturbations of higher-order smallness) holds for proper systems with unbounded coefficients as well.

We consider families of linear differential systems depending on a real parameter that occurs only as a factor multiplying the matrix of the system. The asymptotic stability set of such a family is defined as the set of all parameter values for which the corresponding systems in the family are asymptotically stable. We prove that a set on the real axis is the asymptotic stability set of such a family if and only if it is an Fσδ-set lying entirely on an open ray with origin at zero. In addition, for any set of this kind, the coefficient matrix of a family whose asymptotic stability set coincides with this set can be chosen to be infinitely differentiable and uniformly bounded on the time half-line.

We consider families of linear differential systems continuously depending on a real parameter with continuous (or piecewise continuous) coefficients on the half-line. The improperness set of such a family is defined as the set of all parameter values for which the corresponding systems in the family are Lyapunov improper. We show that a subset of the real axis is the improperness set of some family if and only if it is a Gδσ-set. The result remains valid for families in which the matrices of the systems are bounded on the half-line. Almost the same result holds for families in which the parameter occurs only as a factor multiplying the system matrix: their improperness sets are the Gδσ-sets not containing zero. For families of the last kind with bounded coefficient matrix, we show that their improperness set is an arbitrary open subset of the real line.