A Boundary Cusp Singular Point and Reversible Vector Fields on the Plane

In this paper we describe the bifurcation diagram of a boundary cusp of codimension three, i.e, a Bogdanov-Takens singular point in the boundary of the semi plane $\{(x,y)\in {\mathbb R}^2:\; x\geq 0\}$. This study is 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. We classify the generic three parameter families of reversible vector fields $X_{a,b,c}$ with $(a,b,c) \in ({\mathbb R^3},0)$ and $X_{a,b,c}=X_0$.