Abstract
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.
Keywords
- Vector Field
- Singular Point
- Hopf Bifurcation
- Bifurcation Diagram
- Algebraic Curve
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This work was supported by NSERC and FCAR.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
V. I. Arnold, Loss of stability of self-oscillations close to resonances and versal deformations of equivariant vector fields, Funct. Anal. Appl., 11, (1977), 1–10.
V. I. Arnold, Geometrical methods in the theory of ordinary differential equations, Springer Verlag, New York, Heidelberg, Berlin, 1983 (Russian original 1978).
J. Carr, Applications of centre manifold theory, Springer Verlag, New York, Heidelberg, Berlin, 1981.
S. N. Chow, C. Li and D. Wang, Center manifolds, normal forms and bifurcations of vector fields, book, in preparation.
F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergodic theory and dynamical systems, 7, (1987), 375–413.
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.
E. I. Horozov, Versal deformations of equivariant vector fields under symmetries of order 2 and 3, Trudy Seminar Petrovskii, 5, (1979), 163–192.
A Jebrane and R. Mourtada, Cyclicité finie des lacets doubles, preprint (Dijon), 1990.
C. Li and C. Rousseau, A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: the cusp of order 4, J. Differential Equations, 79, (1989), 132–167.
C. Li and C. Rousseau, Codimension 2 symmetric homoclinic bifurcations and application to 1:2 resonance, to appear in Canadian J. Math.
G. S. Petrov, Elliptic integrals and their nonoscillation, Funct. Anal. Appl., 20, (1986), 37–40.
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.
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.
C. Rousseau and H. Zoladek, Zeroes of complete elliptic integrals in real domain, preprint, 1989.
F. Takens, Forced oscillations and bifurcations, in Applications of global analysis I, Comm. Math. Ins. Rijksuniveersiteit Utrecht, (1974), 1–59.
F. Takens, Singularities of vector fields, Publ. Math. I.H.E.S., 43, (1974), 47–100.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer-Verlag
About this paper
Cite this paper
Rousseau, C. (1990). Universal unfolding of a singularity of a symmetric vector field with 7-jet C∞-equivalent to y ∂/∂x+(±x3 ±x6y) ∂/∂y. In: Françoise, JP., Roussarie, R. (eds) Bifurcations of Planar Vector Fields. Lecture Notes in Mathematics, vol 1455. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085399
Download citation
DOI: https://doi.org/10.1007/BFb0085399
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-53509-6
Online ISBN: 978-3-540-46722-9
eBook Packages: Springer Book Archive
