Global dynamics for planar vector fields with a star node and homogeneous nonlinearities

Begoña Alarcón (UFF)

We study the global dynamics of vector fields of the form u0 = u + Q(u), where 6= 0 and Q(u) is a planar homogeneous polynomial vector field. We discuss the number and stability of equilibrium points, both in the plane and in the circle at infinity in the Poicaré compactification. We obtain conditions for the existence of a globally attracting poly-cycle, thus extending previous results on the existence of limit cycles. A more detailed analysis is done for symmetric vector fields.

This is joint work with Isabel Labouriau and Sofia Castro, University of Porto (Portugal). Referências:

**[1] BB. Alarcón, S.B.S.D. Castro and I. S. Labouriau, Global planar dynamics with a star node and contracting nonlinearity. Research in the Mathematical Sciences, 11:21, 1-22, 2024. **

**[2]BB. Alarcón, S.B.S.D Castro and I. S. Labouriau, Discrete Symmetric Planar Dynamics. Global planar dynamics with star nodes: beyond Hilbert’s 16th problem , https://arxiv.org/pdf/2106.07516.pdf.**