Asymptotic behavior of solutions to a Cauchy problem with a turning point in the case of change of stability (Q2663215)

The author considers the singularly perturbed planar Cauchy problem \[ \varepsilon x'(t,\varepsilon)=A(t)x(t,\varepsilon)+\varepsilon^{\alpha}f(t), \ t\in(t_0,+\infty), \ 0<\varepsilon\ll 1, \] \[ x(t_0,\varepsilon)=x^0(\varepsilon), \ ||x^0(\varepsilon)||=O(\varepsilon^{\beta}), \] where \(A(t)\) is an analytic \(2\times 2\) matrix with complex conjugate eigenvalues \(\lambda_{1,2}=(a-t)^{2n-1}(t\pm ik))\), \(n\in \mathbb{N},\) \(0 < a\in\mathbb{R},\) \(a=k\tan(\pi n/(2n+1)),\) \(0 < k\in\mathbb{R},\) \(\alpha>1-1/(2n)\) and \(\beta\geq\alpha.\) Using successive approximations and the Tikhonov theorem the author proves the convergence of the solutions of the Cauchy problem above to \(0\) for \(\varepsilon\to 0^+\) on the interval \([-a/2n,+\infty)\).