  An anti-lock braking system (ABS) is the primary motivation for this
research. The ABS controller switches the actions of charging and discharging
valves in the hydraulic actuator of the brake cylinder based on the wheels'
angular speed and acceleration. The controller is, therefore, modeled by
discontinuous differential equations where two smooth vector fields are
separated by a switching manifold S. The goal of the controller is to maximize
the tire-road friction force during braking (and, in particular, to prevent the
wheel lock-up). Since the optimal slip L of the wheel is known rather
approximately, the actual goal of the controller is to achieve such a switching
strategy that makes the dynamics converging to a limit cycle surrounding the
region of prospective values of L. In this paper we show that the required
limit cycle can be obtained as a bifurcation from a point x0 of S when a
suitable parameter D crosses 0. The point x0 turns out to be a fold-fold
singularity (the vector fields on the two sides of S are tangent one another at
x0) and the parameter D measures the deviation of the switching manifold from a
hyperplane. The proposed result is based on an extension of the classical fold
bifurcation theory available for smooth maps. Construction of a suitable smooth
map is a crucial step of the proof.
