About Dulac's Problem in R3 for Perturbations of Linear Non-Degenerated Centers

Explore a 40-minute lecture on Dulac's problem in $\mathbb{R}^3$ for perturbations of linear non-degenerated centers, presented by María Martín Vega from the University of Valladolid at the Fields Institute. Delve into the intricacies of geometric and model theory, covering topics such as Poincaré First Return map, local sets of cycles, blowing up of singularities, and the analysis of two-dimensional systems. Examine the application of flow boxes, invariant surfaces, and the Poincaré map to understand the accumulation of cycles in the axis direction. Gain insights into the analytical approach to Dulac's problem and its implications for semihyperbolic cases.

Intro
Fixing notation
Introduction to Dulac's problem
Poincaré First Retum map
An example in
The local sets of cycles
Dulac's property for perturbations of linear non degenerated centers
Blowing up of the (un)stable manifold
Definition and analyticity of the 1 retum of Poincare
Conclusions for the semihyperbolic case
Strategy
Blowing up of the singularity
The total transform of the vector field in the coordinate chart C
Definition of a two dimensional system
Singularities of the two dimensional system
Application of the flow boxes to Cu
Compositions of blowing us
Final situations
Invariant surfaces
The analytic case
The Poincaré map
risa curve of fixed points
Accumulation of cycles in the axis direction
Final conclusion