Estimating the Norm of Solution of Singularly Perturbed Quasilinear Problems for ODE Systems with Nonlinear Normal Matrices on the Semiaxis

Using 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 problem, unitary transformation, normal matrix, stability