Asymptotics of the Solution of the Cauchy Problem with an Unstable Spectrum and Prolonging Loss of Stability

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.