On periodic solutions to autonomous systems (Q1333626)

The authors consider a quasilinear system of autonomous ordinary differential equations, (1) \(\dot x= Ax+ \varepsilon f(x,\varepsilon)\), where \(\varepsilon\) is a small parameter. They assume that the unperturbed system \(\dot x= Ax\) has a periodic solution with period \(p^*\). In four theorems they give sufficient conditions for the perturbed system (1) to have a periodic solution with period \(p(\varepsilon)\) for all sufficiently small \(\varepsilon\) which satisfies \(| p(\varepsilon- p^*|\to 0\) as \(\varepsilon)\to 0\).