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The Bifurcation Diagram of Cubic Polynomial Vector Fields on $\mathbb{C}\mathbb{P}^1$

Open Access article
  Published:2017-02-21
 Printed: Jun 2017
  • C. Rousseau,
    Department of mathematics and statistics, University of Montreal, C.P. 6128, succ. centre-ville, Montreal, Quebec, H3C 3J7
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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.
Keywords: complex polynomial vector field, bifurcation diagram, Douady-Sentenac invariant complex polynomial vector field, bifurcation diagram, Douady-Sentenac invariant
MSC Classifications: 34M45, 32G34 show english descriptions Differential equations on complex manifolds
Moduli and deformations for ordinary differential equations (e.g. Knizhnik-Zamolodchikov equation) [See also 34Mxx]
34M45 - Differential equations on complex manifolds
32G34 - Moduli and deformations for ordinary differential equations (e.g. Knizhnik-Zamolodchikov equation) [See also 34Mxx]
 

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