## 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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### References:

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