Vector Fields on Manifolds with Boundary and Reversibility - An Expository Account

Ronaldo Garcia & Marco Antonio Teixeira

Dedicated to Jorge Sotomayor on his 60th birthday

In this paper an expository account on singularities of reversible vector fields on manifolds and boundary singularities is presented. Also we present the bifurcation diagram of a boundary cusp of codimension three, i.e, a Bogdanov-Takens singular point in the boundary of the semi plane $\{(x,y)\in {\R}^2:\; x\geq 0\}$ whose topological unfolding is given by the quadratic three parameter family $y\frac{\partial}{\partial x}+(x^2+ax+c+ \alpha y(x+b))\frac{\partial}{% \partial y}, \;\; \alpha =\pm 1$.
This study can be applied to the analysis of the behavior of singularity of the germ of vector field $% X_{0}(x,y)=(y,2x(x^4+x^2y))$ in the class of reversible vector
fields.