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### Open Access Journals

DCDS

We study local problems around simple umbilic points of surfaces immersed in $\mathbb R^4$
such as finite determinacy and versal unfoldings.

DCDS

We complete the local study of rank--2 singular points of positive
quadratic differential forms on oriented two--dimensional manifolds.
We associate to each positive quadratic differential form
$\omega$ defined on an oriented two--dimensional manifold $M$
two transversal one--dimensional foliations
$f_1(\omega)$ and $f_2(\omega)$ with common set of singular points. This study was begun in [Gut-Gui] for a generic class of singularities called

*simple*, and continued in [Gui-Sa] for those non--simple rank--2 singular points called of*type C*. Taking into account the classification of [Gui3], the only rank--2 singular points which remain to be studied are those of type E($\lambda$), for $\lambda\geq 1 $. We undertake the local study of the remaining case under a non--flatness condition on the positive quadratic differential form at the singular point.
DCDS

We consider a class $ \mathcal{Q}(M) \,$ consisting of smooth
quartic differential forms which are defined on an oriented
two-manifold $ M $, to each of which we associate a pair of
transversal nets with common singularities. These quartic forms are
related to geometric objects such as curvature lines, asymptotic
lines of surfaces immersed in $\R^4.$ Local problems around the
rank-2 singular points of the elements of $ \mathcal{Q}(M) \,$,
such as stability, normal forms, finite determinacy, versal
unfoldings, are studied in [2]. Here we make a similar study
for a rank-1 singular point that is analogous to the saddle-node
singularity of vector fields.

DCDS

We study, from a new point of view, families of planar vector
fields without singularities $ \{ X_{\mu}$ : $-\varepsilon < \mu <
\varepsilon\} $ defined on the complement of an open ball centered
at the origin such that, at $\mu=0$, infinity changes from repellor
to attractor, or vice versa. We also study a sort of local stability
of some $C^1$ planar vector fields around infinity.

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