The transition from persistence to nonsmooth-fold scenarios in relay control system

In many applications in different fields of science and engineering only two extreme values of the control variable can be easily applied, and a threshold on a suitable output variable is used to discriminate among the two control actions. This relay control introduces a discontinuity, so the closed-loop system is discontinuous piecewise smooth (called Filippov system). When a standard equilibrium attains the threshold, as a system or control parameter is varied, two generic scenarios are possible: the standard equilibrium turns into a pseudo-equilibrium on the discontinuity boundary (persistence), so a stationary solution persists trough the bifurcation; the collision and disappearance of the standard equilibrium and a coexisting pseudo-equilibrium (nonsmooth-fold). In this paper we analyze the degenerate situation separating these two scenarios, and we apply our results to a four-dimensional SISO system describing the ecological dynamics of a protected natural resource (a resource that cannot be harvested when below threshold). We show that while profitable exploitation is guaranteed (though often at the threshold) in the persistence scenario, the food chain collapses after a nonsmooth-fold.