Finitely and Infinitely Sheeted Solutions in Some Classes of Nonlinear ODEs

In this paper we examine an integrable and a non-integrable class of the first order nonlinear ordinary differential equations of the type $\dot{x}= x - x^n + \varepsilon g(t), x \in \mathbb{C}, n \in \mathbb{N}$. We exploit, using the analysis proposed in [1], the asymptotic formulas which give the location of the singularities in the complex plane and show that there is an essential difference regarding the formation and the density of the singularities between the cases $g(t)=1$ and $g(t)=t$. Our analytical results are combined with a numerical study of the solutions in the complex time plane.