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
Global dynamics for symmetric planar maps
Begoña Alarcón Sofia B. S. D. Castro Isabel S. Labouriau
Discrete & Continuous Dynamical Systems - A 2013, 33(6): 2241-2251 doi: 10.3934/dcds.2013.33.2241
We consider sufficient conditions to determine the global dynamics for equivariant maps of the plane with a unique fixed point which is also hyperbolic. When the map is equivariant under the action of a compact Lie group, it is possible to describe the local dynamics. In particular, if the group contains a reflection, there is a line invariant by the map. This allows us to use results based on the theory of free homeomorphisms to describe the global dynamical behaviour. We briefly discuss the case when reflections are absent, for which global dynamics may not follow from local dynamics near the unique fixed point.
keywords: symmetry local and global dynamics. Planar embedding

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