Characterization for Rectifiable and Nonrectifiable Attractivity of Nonautonomous Systems of Linear Differential Equations. (16th June 2013)

• Record Type: Journal Article
• Authors: Naito, Yūki; Pašić, Mervan
• Other Names: Yoshida Norio Academic Editor.
• Is Part Of: International journal of differential equations. Volume 2013(2013)
• Publication Date: 2013-06-16
• Subjects: Differential equations -- Periodicals; Differential equations; Periodicals; 515.35
• DOI: 10.1155/2013/740980
• Languages: English
• ISSNs: 1687-9643

Abstract

Abstract : We study a new kind of asymptotic behaviour neart = 0 for the nonautonomous system of two linear differential equations:x ' ( t ) = A ( t ) x ( t ), t ∈ ( 0, t 0 ], where the matrix-valued functionA = A ( t ) has a kind of singularity att = 0 . It is called rectifiable (resp., nonrectifiable) attractivity of the zero solution, which means that∥ x ( t ) ∥ 2 → 0 ast → 0 and the length of the solution curve ofx is finite (resp., infinite) for everyx ≠ 0 . It is characterized in terms of certain asymptotic behaviour of the eigenvalues ofA ( t ) neart = 0 . Consequently, the main results are applied to a system of two linear differential equations with polynomial coefficients which are singular att = 0 .