Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)8801Research ArticleEstimating the Norm of Solution of Singularly Perturbed Quasilinear Problems for ODE Systems with Nonlinear Normal Matrices on the SemiaxisKonyaevY ADepartment of Mathematics-WorknehA ZDepartment of Mathematics-Peoples’ Friendship University of Russia15042013451008092016Copyright © 2013,2013Using the method of unitary transformation, the singularly perturbed quasi-linear systems of ordinary diﬀerential equations with nonlinear normal matrices on the semiaxis were studied, which in some cases can lead to the existence of countable number of additional boundary layers. For such system, most problems arise in the study of the stability of their solution especially in critical cases where the spectrum deﬁned by the matrix lies (or touches) the imaginary axis. The proposed method allows us to study the traditional Lyapunov functions. We have shown suﬃcient conditions for stability (and asymptotic stability) and given the evaluation of the norm of the solution for such problems, which clariﬁes or supplements previously known results. In addition in the paper we have included some non-trivial examples of nonlinear singularly perturbed problems for quasi-linear systems of ordinary diﬀerential equations with nonlinear normal matrices.singularly perturbed initial value problemunitary transformationnormal matrixstabilityсингулярно возмущённые задачиметод унитарных преобразованийнормальная матрицаустойчивость