Updated: 04/April/2012
This paper is on schedule. It is almost finished now, and has about 30 pages. To be put on arxiv tomorrow.
Updated: 19/03/2012
Section: Generic nilpotent singularities
(X,F) where F regular, X nilpotent:
F =F(y),
X = y d/dx + …
Since X(F) = 0 –> X = (y + …) d/dx
Normal forms: Takens (-Bogdanov), Gong in the integrable case
Assume y + … is generic in the expression of X
2 ways to write:
a) As Takens-Gong –> X = (y + g(x) ) d/dx where g(x) = ax^2 + odd_function (x), and assume that a \neq 0.
b) Normalize the set {X=0} (which is a parabol) –>
X = h(x,y). (y – x^2/2) d/dx
where h(x,y) is a smooth function, h(0) \neq 0
Question: which way is better to detect local invariants ?
The eigenvalue function on the curve S= {X= 0}
Using the second expression X = h(x,y). (y – x^2/2) d/dx,
we have E (q) = – x(q) h(q) for each q \in S ={X=0} = {y = x^2}
where E(q) means eigenvalue at q.
Question: up to equivalence, the behavior of the function h(x,y) is only important at the curve S ?
—-
This is another work in progress with my PhD student Minh. After finishing our paper on the geometry of Rn actions on n-manifolds, we now have time to write up this one.
To finish before: 15/April/20012
Here we want to study the properties and obtain a classification (up to continuous or smooth equivalence) of integrable systems of type (1,1), i.e. just 1 vector field and 1 first integral on a 2-dimensional surface.
0) Different equvalences to consider:
- local / semilocal / global
- continuous / smooth
- geometric (can multiply the vector field by a function) / exact ?
I) Local structure of singularities
- Nondegenerate / Morse ? Geometric linearization ? Smooth classification ? Poincare-Dulac normal form ? Problem of periods of the center ?
- Algebraically isolated singularities
- Singularities of the function ?
Remark Our singularities are singularities of the couple (X,f), and not of X alone.
* The function is regular: 1-dim family of hyperbolic vector fields in dim 1.
* The function is Morse: elliptic and hyperbolic
* The vector field is regular, but the function has a singularity (Morse-Bott function)
* Isolated singular point of the function, the vectoe field also vanishes at that point. Hamiltonianization of this situation ?
* Degenerate centers ?
* Multiplicity of the function ? (Important for the global picture)
II) Semi-local classification
- Hamiltonianization
- Regularized period functions
III) Global picture
- Global invarinats ?
* Marked singularity graph
* Monodormy ?! On each invariant curve ! (similar to our paper on Rn actions)
IV) Related works / References
- J Llibre & co ?
- Kudryavtseva ?
* Colin & Vey, Lemme de Morse isochore, 1979 ?
* Jaume Llibre, Clàudia Valls: Classification of the centers and their isochronicity for a class of polynomial differential systems of arbitrary degree, Advances in Mathematics (22 February 2011)
* Sergiy Maksymenko, Functions with isolated singularities on surfaces, arxiv 0806.4704
*Dufour Molino Toulet, Classification des systèmes intégrables en dimension 2 et invariants des modèles de Fomenko, CRAS, 1994
* Xianghong Gong, Annales Fourier 1995 (?) Integrable nilpotent case.
* Oshemkov Morse functions on two-dimensional surfaces. Encoding of singularities 1994
* A A Oshemkov and V V Sharko Classification of Morse-Smale flows on two-dimensional manifolds, 1998 Sb. Math. 189 1205
* Takens: Singularities of vector fields, Publ IHES 1974 (for the nipoent singularity)
* NT Zung: Nondegenerate singuarities … (2011, revised 2012)
* NT Zung: Linearization of smooth completely integrable vector fields (in preparation, 2012)
* NT Zung & NV Minh: Geometry of Rn-actions on n-manifolds, 2012
…
——
We’ll submit this paper to a special issue in Acta Math Vietnam
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