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

Abstract EN

We study a new kind of asymptotic behaviour near t=0 for the nonautonomous system of two linear differential equations: x'(t)=A(t)x(t), t∈(0,t0], where the matrix-valued function A=A(t) has a kind of singularity at t=0.

It is called rectifiable (resp., nonrectifiable) attractivity of the zero solution, which means that ∥x(t)∥2→0 as t→0 and the length of the solution curve of x is finite (resp., infinite) for every x≠0.

It is characterized in terms of certain asymptotic behaviour of the eigenvalues of A(t) near t=0.

Consequently, the main results are applied to a system of two linear differential equations with polynomial coefficients which are singular at t=0.