Structural Stability of the Two-Fold Singularity

At a two-fold singularity, the velocity vector of a flow switches discontinuously across a codimension one switching manifold, between two directions that both lie tangent to the manifold. Particularly intricate dynamics arises when the local flow curves towards the switching manifold from both sides, a case referred to as the Teixeira singularity. The flow locally performs two different actions: it winds around the singularity by crossing repeatedly through, and passes through the singularity by sliding along, the switching manifold. The case when the number of rotations around the singularity is infinite has been analysed in detail. Here we study the case when the flow makes a finite – but previously unknown – number of rotations around the singularity between incidents of sliding. We show that the solution is remarkably simple: the maximum and minimum number of rotations made anywhere in the flow differs only by one, and increases incrementally with a single parameter: the angular jump in the flow direction across the switching manifold, at the singularity.