Characterization for Rectifiable and Nonrectifiable Attractivity of Nonautonomous Systems of Linear Differential Equations. (16th June 2013) - Naito, Yūki; Pašić, Mervan
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 .