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
