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Characterization for rectifiable and nonrectifiable attractivity of nonautonomous systems of linear differential equations. (English) Zbl 1294.34011

Summary: We study a new kind of asymptotic behaviour near \(t=0\) for the nonautonomous system of two linear differential equations: \[ x'(t)= A(t)x(t),\;t\in(0, t_0], \] where the matrix-valued function \(A= A(t)\) has a kind of singularity at \(t=0\). It is called rectifiable \(j\) (resp., nonrectifiable) attractivity of the zero solution, which means that \(\| x(t)\|_2\to 0\) as \(t\to 0\) and the length of the solution curve of \(x\) is finite (resp., infinite) for every \(x\neq 0\). 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 \(t=0\).

MSC:

34A30 Linear ordinary differential equations and systems
37C60 Nonautonomous smooth dynamical systems
34D05 Asymptotic properties of solutions to ordinary differential equations
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