Асимптотика решений системы сингулярно возмущенных дифференциальных уравнений

Abstract

Solutions of nonlinear singularly perturbed differential equations are studied in this work. The matrix functions have multiple eigen values, which change the stability conditions the region under consideration several times. For these eigen values, the stability conditions are satisfied, but the stability change points, initial and critical, coincide. This is not typical for ongoing work in this direction. It is not clear what the estimate is at this point. If we consider this point as the starting point, then to the right of it will be, to the left more, at the point itself . If this point is taken as the point of change of stability, then we obtain a similar estimate. If this point is a critical point, then we will not approach this point. Therefore, we choose the starting point so that it remains the area under consideration. To obtain the asymptotics of solutions, the method of successive approximations is used. The solution of the stated problem is considered in the complex plane. Eigen values are analytic functions, therefore, using level lines, both parts of analytic functions can be covered with lines of the region under consideration. When investigating the solution, integration paths were chosen and the corresponding estimate was obtained. As a result, the uniqueness and uniform convergence of solutions are proved.