Geodesic
The Bifurcation Diagram of Cubic Polynomial Vector Fields on CP1
Canadian mathematical bulletin, Tome 60 (2017) no. 2, pp. 381-401

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper we give the bifurcation diagram of the family of cubic vector fields $\dot{z}=$ ${{z}^{3}}+{{\epsilon }_{1}}z+{{\epsilon }_{0}}$ for $z\in \mathbb{C}{{\mathbb{P}}^{1}}$ , depending on the values of ${{\epsilon }_{1}},{{\epsilon }_{0}}\in \mathbb{C}$ . The bifurcation diagram is in ${{\mathbb{R}}^{^{4}}}$ , but its conic structure allows describing it for parameter values in ${{\mathbb{S}}^{3}}$ . There are two open simply connected regions of structurally stable vector fields separated by surfaces corresponding to bifurcations of homoclinic connections between two separatrices of the pole at infinity. These branch from the codimension 2 curve of double singular points. We also explain the bifurcation of homoclinic connection in terms of the description of Douady and Sentenac of polynomial vector fields.
DOI : 10.4153/CMB-2016-095-3
Mots-clés : 34M45, 32G34, complex polynomial vector field, bifurcation diagram, Douady-Sentenac invariant. 
Rousseau, C. The Bifurcation Diagram of Cubic Polynomial Vector Fields on CP1. Canadian mathematical bulletin, Tome 60 (2017) no. 2, pp. 381-401. doi: 10.4153/CMB-2016-095-3
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