Periodic orbits in piecewise Riccati vector fields

Abstract

In this paper a class of planar vector fields is characterized and their periodic orbits are studied. The class under study is formed by vector fields that can be written as Riccati ordinary differential equations. The characterization is stated in terms of some homogeneity conditions on the vector fields. The study of periodic orbits is done through Poincaré maps defined around monodromic points. In particular, some properties of the so-called Möbius transformations are considered. In the study, both smooth and discontinuous vector fields defined by sectors are studied.Theorem 2.2 characterizes the Riccati vector fields and Theorem 3.4 proves that the Poincaré maps of such vector fields are Möbius transformations. In Theorem 4.1 a sharp upper bound of two periodic orbits is stated. The polynomial case is considered in Theorems 4.3 and 4.4 for which existence, upper bounds, and hyperbolicity of periodic orbits are stated.