Regularized asymptotic solutions to singularly perturbed problems in a critical case
The author constructs regularized asymptotic solutions to the singularly perturbed problem \(\varepsilon dy/dt= Ay+ f(y)+ \varepsilon \Phi(y, t)\), \(y(0, \varepsilon)= y^0\), \(t\in [0, T]\), where \(y\in \mathbb{R}^2\), \(f(0)= 0\), \(fy'(0)= 0\), \(\varepsilon> 0\) is a small parameter, and the \(2\times 2\) matrix \(A\) has the purely imaginary eigenvalues \(\lambda_{1, 2}= \pm i\omega\) \((\omega> 0)\). The construction is based on a normal form algorithm developed by \textit{Yu. A. Gubin} and the author in [Differ. Equations 20, 665-674 (1984); translation from Differ. Uravn. 20, 930-941 (1984; Zbl 0561.34039)].
scientific article; zbMATH DE number 794584