Vector fields on manifolds with boundary and reversibility — An expository account —

Abstract

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)∈R2:x>-0} whose topological unfolding is given by the quadratic three parameter family\(y\frac{\partial }{{\partial x}} + (x^2 + ax + c + \alpha x(x + b))\frac{\partial }{{\partial y}}\),\(\alpha = \pm 1\). This study can be applied to the analysis of the behavior of singularity of the germ of vector fieldX0(x, y)=(y, 2x(x4+x2y)) in the class of reversible vector fields.