# On the Universal Unfolding of Vector Fields in One Variable: A Proof of Kostov’s Theorem

@article{Klime2020OnTU, title={On the Universal Unfolding of Vector Fields in One Variable: A Proof of Kostov’s Theorem}, author={Martin Klime{\vs} and Christiane Rousseau}, journal={Qualitative Theory of Dynamical Systems}, year={2020} }

In this note we present variants of Kostov’s theorem on a versal deformation of a parabolic point of a complex analytic 1-dimensional vector field. First we provide a self-contained proof of Kostov’s theorem, together with a proof that this versal deformation is indeed universal. We then generalize to the real analytic and formal cases, where we show universality, and to the $${\mathcal {C}}^\infty $$
C
∞
case, where we show that only versality is possible.

## One Citation

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- 2020

When are two germs of analytic systems conjugate or orbitally equivalent under an analytic change of coordinates in the neighborhood of a singular point? A way to answer is to use normal forms. But…

## References

SHOWING 1-10 OF 25 REFERENCES

### MODULUS OF ANALYTIC CLASSIFICATION FOR UNFOLDINGS OF GENERIC PARABOLIC DIFFEOMORPHISMS

- Mathematics
- 2004

In this paper we give a complete modulus of analytic classi- fication under weak equivalence for generic analytic 1-parameter unfold- ings of dieomorphisms with a generic parabolic point. The modulus…

### Analytical Moduli for Unfoldings of Saddle-Node Vector Fields

- Mathematics
- 2007

In this paper we consider germs of k-parameter generic families of analytic 2-dimensional vector fields unfolding a saddle-node of codimension k and we give a complete modulus of analytic…

### Formal classification of unfoldings of parabolic diffeomorphisms

- MathematicsErgodic Theory and Dynamical Systems
- 2008

Abstract We provide a complete system of invariants for the formal classification of unfoldings φ(x,x1,…,xn)=(f(x,x1,…,xn),x1,…,xn) of complex analytic germs of diffeomorphisms at $({\mathbb C},0)$…

### Analytic moduli for unfoldings of germs of generic analytic diffeomorphisms with a codimension $k$ parabolic point

- MathematicsErgodic Theory and Dynamical Systems
- 2013

Abstract In this paper we provide a complete modulus of analytic classification for germs of generic analytic families of diffeomorphisms which unfold a parabolic fixed point of codimension $k$. We…

### Modulus of analytic classification for unfoldings of resonant diffeomorphisms

- Mathematics
- 2006

We provide a complete system of analytic invariants for unfoldings of non-linearizable resonant complex analytic diffeomorphisms as well as its geometrical interpretation. In order to fulfill this…

### Remarks on topological algebras

- Mathematics
- 2007

The note complements “topological” aspects of the chiral algebras story from [BD]. In its first section (which has a whiff of [G] in it) we show that the basic chiral algebra format (chiral…

### VERSAL DEFORMATIONS OF DIFFERENTIAL FORMS OF REAL DEGREE ON THE REAL LINE

- Mathematics
- 1991

A normal form is given for families of differential forms of nonzero real degree on the real line (in particular, a normal form for families of vector fields on the line), which depend smoothly on…

### Singularity Theory II Classification and Applications

- Mathematics
- 1993

In the first volume of this survey (Arnol’d et al. (1988), hereafter cited as “EMS 6”) we acquainted the reader with the basic concepts and methods of the theory of singularities of smooth mappings…

### Lectures on Analytic Differential Equations

- Mathematics
- 2007

Normal forms and desingularization Singular points of planar analytic vector fields Local and global theory of linear systems Functional moduli of analytic classification of resonant germs and their…

### The classification of critical points, caustics and wave fronts

- Mathematics
- 1985

Part I. Basic concepts.- The simplest examples.- The classes Sigma^ I .- The quadratic differential of a map.- The local algebra of a map and the Weierstrass preparation theorem.- The local…