The Method of Holomorphic Foliations in Planar Periodic Systems: The Case of Riccati Equations

We develop a new method of investigation of periodic solutions of planar holomorphic polynomial differential equations with trigonometric coefficients. This method relies on analysis of geometry of leaves of certain holomorphic foliations in CP1×CP1. We concentrate our attention on the case of Riccati equations. We determine the dynamics of the non-autonomous equations, using some local invariants of holomorphic vector fields defining the foliations. In particular, new examples of Riccati equations without periodic solutions are presented.