ON EXPONENTIALLY SEPARATED DIFFERENTIAL SYSTEMS

Exponentially separated linear homogeneous system of ordinary differential equations with continuous limited coeffi cients in critical cases of Lyapunov exponents is considered. The generalized exponentially separated linear system of differential equations with regard to a monotonically increasing function is defined. It is established that if a linear homogeneous system of differential equations is generalized exponentially separated, Lyapunov’s generalized exponents are stable in a class of small perturbations. The work of Perron [12], see also [11, p. 193, theorem 9] was a source of definition of an exponential separation. Then these systems were studied in B.F.Bylov’s [6], R.E. Vinograd’s [7], V.M. Millionshchikov’s [9], [10], Lillo’s [8] works. The definition of an exponential separation has connection with the definition of an exponential dichotomy D.V. Anosov [5]. Some information on the theory of Lyapunov’s generalized exponents is contained in Aldibekov T.M.’s works [1,2,3,4]. In the paper [3] the definition of an exponential separation is spread to linear systems with unlimited coeffi cients. In the present work using Lyapunov’s generalized exponents, the subclass of linear systems with continuous limited coeffi cients is investigated, where the definition of an exponential separated loses its meaning. The class of linear homogeneous systems of differential equations is considered ẋ = A(t)x (0.1) where t ∈ I ≡ [t0,+∞), (t0 > 1), x ∈ R, A(t) is a continuous matrix of the dimension n× n, satisfying to the inequality ‖A(t)‖ ≤ Kψ(t), (K > 0) (0.2) ψ(t) is a continuous, positive, distinct from a constant, such a fixed function that the function

The work of Perron [12], see also [11, p. 193, theorem 9] was a source of de…nition of an exponential separation. Then these systems were studied in B.F.Bylov's [6], R.E. Vinograd's [7], V.M. Millionshchikov's [9], [10], Lillo's [8] works. The de…nition of an exponential separation has connection with the de…nition of an exponential dichotomy D.V. Anosov [5]. Some information on the theory of Lyapunov's generalized exponents is contained in Aldibekov T.M.'s works [1,2,3,4]. In the paper [3] the de…nition of an exponential separation is spread to linear systems with unlimited coe¢ cients. In the present work using Lyapunov's generalized exponents, the subclass of linear systems with continuous limited coe¢ cients is investigated, where the de…nition of an exponential separated loses its meaning.
The class of linear homogeneous systems of di¤erential equations is considered where t 2 I [t 0 ; +1); (t 0 > 1); x 2 R n ; A(t) is a continuous matrix of the dimension n n; satisfying to the inequality (t) is a continuous, positive, distinct from a constant, such a …xed function that the function 2000 Mathematics Subject Classi…cation. Primary 05C38, 15A15; Secondary 05A15, 15A18. c 2 0 1 4 A n ka ra U n ive rsity 101 and satis…es to the conditions: Note that Lyapunov's exponents of the linear system (0.1) accept zero values, i.e. a so-called critical case takes place.
The generalized exponent of a nonzero solution x(t) of the linear system (0.1) is determined by the formula The generalized exponents of the fundamental system of solutions, in which the sum of the generalized exponents of solutions is the smallest compared with other fundamental systems of solutions, are called Lyapunov's generalized exponents of the linear system (0.1).
As a rule, Lyapunov's generalized exponents of the linear system (0.1) are designated as follows n (A) n 1 (A) : : : De…nition 0.1. The linear system (0.1) satisfying to the condition (0.2) is called generalized exponentially separated, if it has solutions x 1 (t); : : : ; x n (t); such that for all t s t 0 the inequalities are ful…lled De…nition 0.2. If the linearly perturbed system where a continuous matrix of perturbation P (t); t t 0 satis…es to the conditions has Lyapunov's generalized exponents, which coincide with generalized Lyapunov's exponents of the linear system (0.1), we can say that the linear system (0.1) satisfying to the condition (0.2) has Lyapunov's stable generalized exponents.
Proof. Note the perturbed system in the coordinate form It is known that from lim t!1 kP (t)k (t) = 0 it follows that for any i 2 f1; : : : ; ng; k 2 f1; : : : ; ng the equality takes place Following the work [3, pp. 65-78] it is easily established that the linearly perturbed system (0.6) satisfying to the condition (0.7) has n linearly independent solutions x k = fx 1k ; x 2k ; : : : ; x nk g; k = 1; 2; : : : ; n; satisfying to equalities a) lim From b) it follows that for any " > 0 there exists such T 2 I; that for any t > T; k = 1; : : : ; n; inequalities take place Integrating, we receive Therefore, inequalities take place From a) it follows that the k-th coordinate of the solution x k is the leader, which implies that the equalities take place Here 1 (A d ); : : : ; n (A d ) are Lyapunov's generalized exponents of the system (0.4), besides they are di¤erent. Therefore, the fundamental system of solutions x 1 ; x 2 ; : : : ; x n organizes a normal base of the linearly perturbed system (0.5). Therefore, by the de…nition ln jx k (t)j = k (A d + P ); k 2 f1; : : : ; ng; are Lyapunov's generalized exponents of the system (0.5) and the equalities take place i (A d + P ) = i (A d ); i = 1; : : : ; n; Therefore, the linear system (0.4) has Lyapunov's stable generalized exponents. Theorem 1 is proved.
Proof. By the de…nition the linear system (0.1) has solutions x 1 (t); : : : ; x n (t); for which at all t s t 0 inequalities are ful…lled with some constants > 0; B 1: Hence, it follows that the solutions x 1 (t); : : : ; x n (t) have various generalized indices, therefore from the property of the generalized indices it follows that they organize a fundamental system of solutions of the linear system (0.1). Let and we present a fundamental matrix of solutions in an aspect where (t) = [' 1 ; : : : ; ' n ]; j det j > 0; D = diag(kx 1 k; kx 2 j; : : : ; kx n k) The equalities take place where a i (t); i = 1; : : : ; n; continuous functions at t t 0 : Hence integrating, for any t s t 0 we receive Let us carry out transformation in the system (0.1) taking as a matrix of transformation the matrix (t) x = y (0.12) Then we receive the linear system Note that in the transformation (0.12), matrixes (t); 1 (t) are continuous limited, and the matrix _ (t) is continuous and k _ (t)k K (t): Therefore, (0.12) is the generalized Lyapunov's transformation.
To the fundamental matrix X linear to the system (0.1) there corresponds the fundamental matrix Y of the linear system (0.13) and from the equality Substituting (0.9) in (0.14) we have Therefore, owing to uniqueness the equality takes place Thus, the generalized exponentially separated linear system (0.1) satisfying to the condition (0.2) by application of Lyapunov's generalized transformation is reduced to a diagonal system satisfying to the condition (0.11). As Lyapunov's generalized transformation keeps stability, from theorem 1 it follows that Lyapunov's generalized exponents of the linear system (0.1) are stable.
Theorem 2 is proved.
Example 0.5. Let us consider the linear system 8 > < > : Note that the linear system from diagonal coe¢ cients of this system has Lyapunov's generalized exponents 1 (q) = 1 2 ; 2 (q) = 1; where q(t) = p t and the diagonal system is generalized exponentially separated, and the approval of theorem 2 is ful…lled. Therefore, this system has the same Lyapunov's generalized exponents. This implies that the system is stable according to Lyapunov.