Аңдатпа
This paper considers impulsive systems with singularities. The main novelty of this study is that the impulses (impulsive functions) and the initial value are singular. The asymptotic convergence of the solution to a singularly perturbed initial problem with an infinitely large initial value, as ε→0, to the solution to a corresponding modified degenerate initial problem is proved. It is established that the solution to the initial problem at point t=0 has an initial jump phenomenon, and the value of this initial jump is determined. The theoretical results are supported by illustrative examples with simulations. Singularly perturbed problems are characterized by the presence of a small parameter multiplying the highest derivatives in the differential equations. This leads to rapid changes in the solution near the boundary or at certain points inside the domain. In our problem, symmetry is violated due to the emergence of a boundary layer at the initial point and at the moments of discontinuity. As a result, the problem as a whole is asymmetric. Such asymmetry in the behavior of the solution is a main feature of singularly perturbed problems, setting them apart from regularly perturbed problems in which the solutions usually exhibit smoother changes.