**Alexander Eliseev †,‡ and Tatjana Ratnikova \*,‡**

Research Group on Macro-Structural Modeling of the Russian Economy, National Research University "MPEI", 111250 Moscow, Russia; predikat@bk.ru


Received: 17 September 2019; Accepted: 30 October 2019; Published: 1 November 2019

**Abstract:** By Lomov's S.A. regularization method, we constructed an asymptotic solution of the singularly perturbed Cauchy problem in a two-dimensional case in the case of violation of stability conditions of the limit-operator spectrum. In particular, the problem with a "simple" turning point was considered, i.e., one eigenvalue vanishes for *t* = 0 and has the form *t <sup>m</sup>*/*na*(*t*) (limit operator is discretely irreversible). The regularization method allows us to construct an asymptotic solution that is uniform over the entire segment [0, *T*], and under additional conditions on the parameters of the singularly perturbed problem and its right-hand side, the exact solution.

**Keywords:** singularly perturbed Cauchy problem; regularized asymptotic solution; rational "simple" turning point
