A Strategy to Locate Fixed Points and Global Perturbations of ODE’s: Mixing Topology with Metric Conditions

Abstract

In this paper we discuss a topological treatment for the planar system $$\begin{aligned} z'=f(t,z)+g(t,z) \end{aligned}$$ (0.1) where \(f:\mathbb {R}\times \mathbb {R}^{2}\longrightarrow \mathbb {R}^{2}\) and \(g:\mathbb {R}\times \mathbb {R}^{2}\longrightarrow \mathbb {R}^{2}\) are \(T\)-periodic in time and \(g(t,z)\) is bounded. Namely, we study the effect of \(g(t,z)\) in two different frameworks: isochronous centers and time periodic systems having subharmonics. The main tool employed in the proofs consists of a topological strategy to locate fixed points in the class of orientation preserving embedding under the condition of some recurrence properties. Generally speaking, our topological result can be considered as an extension of the main result in Brown (Pac J Math 143:37–41, 1990) (concerning two cycles) to any recurrent point.