tailieunhanh - Báo cáo "On stability of Lyapunov exponents "

In this paper we consider the upper (lower) - stability of Lyapunov exponents of linear differential equations in Rn . Sufficient conditions for the upper - stability of maximal exponent of linear systems under linear perturbations are given. The obtained results are extended to the system with nonlinear perturbations. | VNU Journal of Science Mathematics - Physics 24 2008 73-80 On stability of Lyapunov exponents Nguyen Sinh Bay1 Tran Thi Anh Hoa2 1 Department of Mathematics Vietnam University of Commerces Ho Tung Mau Hanoi Vietnam 2456 Minh Khai Hanoi Vietnam Received 21 March 2008 received in revised form 9 April 2008 Abstract. In this paper we consider the upper lower - stability of Lyapunov exponents of linear differential equations in Rn. Sufficient conditions for the upper - stability of maximal exponent of linear systems under linear perturbations are given. The obtained results are extended to the system with nonlinear perturbations. Keywork Lyapunov exponents upper lower - stability maximal exponent. 1. Introduction Let us consider a linear system of differential equations x A t x t t0 0. 1 where A t is a real n X n - matrix function continuous and bounded on to to . It is well known that the above assumption guarantees the boundesness of the Lyapunov exponents of system 1 . Denote by X1 X2 . Xn X1 X2 . Xn the Lyapunov exponents of system 1 . Definition 1. The maximal exponent Xn of system 1 is said to be upper - stable if for any given e 0 there exists ỗ ỗ e 0 such that for any continuous on t0 n X n - matrix B t satisfying B t II ỗ the maximal exponent hn of perturbed system x A t B f x 2 satisfies the inequality Vn. Xn Ễ. 3 If B t II ỗ implies h1 X1 e we say that the minimal exponent X1 of system 1 is lower -stable. In general the maximal minimal exponent of system 1 is not always upper lower - stable 1 . However if system 1 is redusible in the Lyapunov sense then its maximal minimal exponent is upper lower - stable. In particular if system 1 is periodic then it has this property 2 3 . A problem arises In what conditions the maximal minimal exponent of nonreducible systems is upper lower - stable The aim of this paper is to show a class of nonreducible systems having this property. Corresponding author. E-mail nsbay@ 73 74 . Bay . Hoa VNU Journal of .