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

Abstract

We study a new kind of asymptotic behaviour near $t=\mathrm{0}$ for the nonautonomous system of two linear differential equations: $\mathbf{x}\mathrm{\text{'}}(t)=A(t)\mathbf{x}(t)$, $t\in (\mathrm{0},t_{\mathrm{0}}]$, where the matrix-valued function $A=A(t)$ has a kind of singularity at $t=\mathrm{0}$. It is called rectifiable (resp., nonrectifiable) attractivity of the zero solution, which means that $\parallel \mathbf{x}(t){\parallel }_{\mathrm{2}}\to \mathrm{0}$ as $t\to \mathrm{0}$ and the length of the solution curve of $\mathbf{x}$ is finite (resp., infinite) for every $\mathbf{x}\ne \mathbf{0}$. It is characterized in terms of certain asymptotic behaviour of the eigenvalues of $A(t)$ near $t=\mathrm{0}$. Consequently, the main results are applied to a system of two linear differential equations with polynomial coefficients which are singular at $t=\mathrm{0}$.