Codimension–one singularities of Constrained systems on [formula omitted

Abstract

This paper is concerned with the dynamics near an impasse singularity of Constrained systems A(x)x˙=F(x), where A(x) is a 3×3 smooth matrix-valued function and F(x) is a smooth vector field defined on R3. Impasse singularity can be either a tangential impasse point or an impasse equilibrium point. We present all the topological types and their respective normal forms of the codimension-one impasse singularities for a large class of Constrained systems on the three-dimensional space. We split the study of impasse equilibrium points into nonresonant and resonant cases. In the first case, the constrained system presents cubic impasse bifurcation, while for the other one occurs focus–node, saddle-node, or Hopf impasse bifurcations. The tangential impasse points present three types: Lips, Bec-to-bec, and Dove's tail singularities.