Characterization for Rectifiable and Nonrectifiable Attractivity of Nonautonomous Systems of Linear Differential Equations

We study a new kind of asymptotic behaviour near for the nonautonomous system of two linear differential equations: , , where the matrix-valued function has a kind of singularity at . It is called rectifiable (resp., nonrectifiable) attractivity of the zero solution, which means that as and the length of the solution curve of is finite (resp., infinite) for every . It is characterized in terms of certain asymptotic behaviour of the eigenvalues of near . Consequently, the main results are applied to a system of two linear differential equations with polynomial coefficients which are singular at . 1. Introduction We are concerned with the two-dimensional nonautonomous linear differential systems: where , and , is a matrix-valued function such that , where denotes the space of all real matrices. A solution of linear system (1) is a function , such that . In appendix of the paper, we show, there exists a unique solution of linear system (1). By the uniqueness of solution of (1), it is clear that if , then for all . It is called the zero solution of linear system (1). The zero solution of linear system (1) is said to be attractive as if for every the corresponding solution of (1) satisfies as , where denotes the standard Euclidean norm on . Since we are dealing with the attractive zero solution of (1), from we conclude that . For a continuous function , let denote a curve in corresponding to determined by where . It is often said that is parametrized by . Let . The curve is said to be a Jordan curve if is injective (one-to-one) on . Hence, is nonself-intersecting; it has two ends and it is a compact connected set in . In appendix of the paper, see Theorem A.3; we give some simple sufficient conditions on the matrix-valued function such that the solution curve of every nontrivial solution of (1) is a Jordan curve in . The assumption that is a Jordan curve is important for the process of measuring the length of (the rectification of ) as follows. If is a Jordan curve, then the length of is denoted by length and it is defined as usual by (see [1–3]) where the supremum is taken over all finite dissections of the interval . Definition 1. The zero solution of linear system (1) is said to be rectifiable (resp., nonrectifiable) attractive as ; if for every such that the of corresponding solution of (1) is a Jordan curve in , one has as and length (resp., length ). Let , , be two complex conjugate eigenvalues of . The main purpose of the paper is to characterize attractivity, rectifiable and nonrectifiable attractivity, as of the zero solution of system (1) in the