The bifurcation diagram of cubic polynomial vector fields on $\mathbb{C}\mathbb{P}^1$

TL;DR

This paper describes the bifurcation diagram of a family of cubic polynomial vector fields on the Riemann sphere, detailing the structure of stability regions and bifurcation surfaces in complex parameter space.

Contribution

It provides a detailed description of the bifurcation diagram for cubic polynomial vector fields on , including the conic structure and bifurcation surfaces, extending previous understanding of these dynamical systems.

Findings

Two stable regions separated by bifurcation surfaces

Bifurcations involve homoclinic connections of the pole at infinity

Description of bifurcation structure in  parameter space

Abstract

In this paper we give the bifurcation diagram of the family of cubic vector fields $\dot z=z^3+ \epsilon_1z+\epsilon_0$ 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.