DCDS
Simple umbilic points on surfaces immersed in $\R^4$
Carlos Gutierrez Víctor Guíñez
Discrete & Continuous Dynamical Systems - A 2003, 9(4): 877-900 doi: 10.3934/dcds.2003.9.877
We study local problems around simple umbilic points of surfaces immersed in $\mathbb R^4$ such as finite determinacy and versal unfoldings.
keywords: quartic differential forms. umbilic points Lines of curvature smooth immersions
DCDS
Versal Unfoldings for rank--2 singularities of positive quadratic differential forms: The remaining case
Víctor Guíñez Eduardo Sáez
Discrete & Continuous Dynamical Systems - A 2005, 12(5): 887-904 doi: 10.3934/dcds.2005.12.887
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.
keywords: singular points foliations Quadratic differential forms versal unfoldings
DCDS
Quartic differential forms and transversal nets with singularities
Carlos Gutierrez Víctor Guíñez Alvaro Castañeda
Discrete & Continuous Dynamical Systems - A 2010, 26(1): 225-249 doi: 10.3934/dcds.2010.26.225
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.
keywords: Quartic Differential Forms nets.
DCDS
Hopf bifurcation at infinity for planar vector fields
Begoña Alarcón Víctor Guíñez Carlos Gutierrez
Discrete & Continuous Dynamical Systems - A 2007, 17(2): 247-258 doi: 10.3934/dcds.2007.17.247
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.
keywords: Hopf bifurcation vector field Poincaré index. singular points

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