On Limit cycles for Bernoulli and Riccati differential equations

Abstract

This thesis presents a comprehensive study of planar polynomial differential systems, which are fundamental in the qualitative analysis of differential equations. Among the dynamic behaviors of interest are limit cycles closed periodic solutions that characterize long term system behavior and stability. The central problem lies in proving the existence, number, and stability of such cycles, especially in specific cases such as Bernoulli and Riccati equations .Within this framework, the thesis reviews key preliminary notions such as vector fields, equilibrium points, invariant curves, and Darboux integrability, alongside analytical tools like the Poincaré map and the Hartman Grobman theorem for classifying behavior near critical points. The core contribution consists in studying and interpreting the results from Clàudia Valls’ article [44], where I reformulated and simplified the theoretical proofs concerning rational limit cycles, and enriched them with illustrative examples and diagrams aimed at enhancing understanding