The bifurcation diagram of cubic polynomial vector fields on CP¹

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In this paper we give the bifurcation diagram of the family of cubic vector fields z=z³+ ε₁z+ε₀ for z∈ CP¹, depending on the values of ε₁,ε₀. The bifurcation diagram is in R⁴, but its conic structure allows describing it for parameter values in S³. 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.