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Asymptotic estimates of the norm of the integral matrix of a two-dimensional system of differential equations in the neighborhood of a singular point of the system. (Russian) Zbl 0569.34049

Let Y(\(\tau)\) be the solution matrix of the second order matrix differential equation \(dY/d\tau =Y(A\tau^{-2}+B\tau^{\mu -2})\) where \(A=-\left( \begin{matrix} \xi_ 1&0\\ 1&\xi_ 2\end{matrix}\right)\), \(B=- \left( \begin{matrix} \eta_ 1&\nabla\\ 0&\eta_ 2\end{matrix}\right)\) \(\xi_ i\), \(\eta_ i\) and \(\nabla\) being complex numbers, \(\mu \in (0,+\infty)\) and \((\xi_ 2-\xi_ 1)(\eta_ 2-\eta_ 1)+\nabla \neq 0,\) \(\nabla \neq 0\). For \(\tau\) \(\to \infty\) the following asymptotic behaviour is proved \(| Y(\tau)| \leq a\cdot e^{-c}\), \(\mu\in (0,1)\), \(| Y(\tau)| =O(\tau^ k)\), \(\eta =0\), \(| Y(\tau)| =O(\exp (c\tau^{\mu -1})),\) \(\mu \in (1,+\infty)\) where a,\(\kappa\),c, are numbers that can be computed by explicit formulae using the coefficients \(\xi_ i\), \(\eta_ i\), \(\nabla\).
Reviewer: V.Răsvan

MSC:

34E05 Asymptotic expansions of solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34A30 Linear ordinary differential equations and systems
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