Abstract

The article is devoted to construct a complete asymptotic expansion of the solution to the Cauchy problem for a linear analytical system of singularly perturbed ordinary differential equations of the first order. The peculiarities of the Cauchy problem are that a small parameter is present in front of the derivative, and the stability conditions are violated in the region under consideration. By modifying the method of boundary functions, a formal asymptotic expansion of the solution to the Cauchy problem is constructed. The remainder term of the expansion is estimated by the idea of L.S. Pontryagin entering the complex plane.

中文翻译：

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具有不稳定谱和长期失稳的柯西问题解的渐近性

 抽象的

本文致力于构造一阶奇摄动常微分方程线性解析系统柯西问题解的完整渐近展开式。柯西问题的特点是导数前面存在一个小参数，并且在所考虑的区域中违反了稳定性条件。通过修改边界函数的方法，构造了柯西问题解的形式渐近展开。展开式的余项是通过 L.S. 的思想来估计的。庞特里亚金进入复平面。