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The Bifurcation Diagram of Cubic Polynomial Vector Fields on CP1

  • C. Rousseau (a1)

Abstract

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.

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Copyright

© Canadian Mathematical Society 2017 This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

References

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[1] Branner, B. and Dias, K., Classification of complex polynomial vector fields in one complex variable. J. Difference Equ. Appl. 16(2010), 463517. http://dx.doi.org/10.1080/10236190903251746
[2] Douady, A. and Sentenac, P., Champs de vecteurs polynomiaux sur C. Preprint, Paris 2005. (Both authors died before this important visionary work was officially published.)
[3] Hurtubise, J., Lambert, C., and Rousseau, C., Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity ofPoincaré rank k. Mosc. Math. J. 14(2014), no. 2, 309338, 427.
[4] Lavaurs, P., Systèmes dynamiques holomorphes: explosion de points périodiques paraboliques. Thesis, Université de Paris-Sud, 1989.
[5] Martinet, J., Remarques sur la bifurcation nœud-col dans le domaine complexe. Astérisque 15051(1987), 131-149.
[6] Mardesic, P., Roussarie, R., and Rousseau, C., Modulus of analytic classification for unfoldings of generic parabolic diffeomorphisms. Mosc. Math. J. 4(2004), 455502.
[7] Rousseau, C., Analytic moduli for unfoldings of germs of generic analytic diffeomorphisms with a codimension k parabolic point. Ergodic Theory Dynam. Systems 35(2015), no. 1, 274292. http://dx.doi.org/10.1017/etds.2013.37
[8] Rousseau, C. and Teyssier, L., Analytical moduli for unfoldings of saddle-node vector-fields. Mosc. Math. J. 8(2008), 547614.
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The Bifurcation Diagram of Cubic Polynomial Vector Fields on CP1

  • C. Rousseau (a1)

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