Universal unfolding of a singularity of a symmetric vector field with 7-jet C∞-equivalent to y ∂/∂x+(±x3 ±x6y) ∂/∂y
- 387 Downloads
In this paper we study the codimension 4 singularity at the origin for symmetric vector fields with nilpotent linear part and 7-jet C∞-equivalent to y ∂/∂x+(ax3+bx6y) ∂/∂y, a,b ≠ 0. For this we introduce the universal unfolding of the singularity and derive its bifurcation diagram. The methods are classical and make an extensive use of properties of elliptic integrals.
KeywordsVector Field Singular Point Hopf Bifurcation Bifurcation Diagram Algebraic Curve
Unable to display preview. Download preview PDF.
- S. N. Chow, C. Li and D. Wang, Center manifolds, normal forms and bifurcations of vector fields, book, in preparation.Google Scholar
- F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter families of planar vector fields, unfoldings of saddle, focus and elliptic singularities with nilpotent linear parts, preprint, 1989.Google Scholar
- A Jebrane and R. Mourtada, Cyclicité finie des lacets doubles, preprint (Dijon), 1990.Google Scholar
- C. Li and C. Rousseau, Codimension 2 symmetric homoclinic bifurcations and application to 1:2 resonance, to appear in Canadian J. Math.Google Scholar
- R. Roussarie, On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields, Bol. Soc. Bras. Mat., 17, 67–101.Google Scholar
- C. Rousseau, Codimension 1 and 2 bifurcations of fixed points of diffeomorphisms and of periodic orbits of vector fields, to appear in Annales Mathématiques du Québec.Google Scholar
- C. Rousseau and H. Zoladek, Zeroes of complete elliptic integrals in real domain, preprint, 1989.Google Scholar
- F. Takens, Forced oscillations and bifurcations, in Applications of global analysis I, Comm. Math. Ins. Rijksuniveersiteit Utrecht, (1974), 1–59.Google Scholar