Asymptotic Solutions of Resonant Nonlinear Singularly Perturbed Problems in the Case of Intersecting Eigenvalues of the Limit Operator

Abstract

Lomov’s regularization method is generalized to resonant, weakly nonlinear, singularly perturbed systems in the case of intersecting roots of the characteristic equation of the limit operator. For constructing asymptotic solutions, the regularization of the original problem by using normal forms developed by the authors is performed. In the absence of resonance, the regularizing normal form is linear, whereas in the presence of resonances, it is nonlinear. In this paper, the resonant case of a weakly nonlinear problem is considered. By using an algorithm of normal forms, we construct an asymptotic solution of any order (with respect to a parameter) and justify this algorithm.