Exit from sliding in piecewise-smooth flows: Deterministic vs. determinacy-breaking

The collapse of flows onto hypersurfaces where their vector fields are discontinuous creates highly robust states called sliding modes. The way flows exit from such sliding modes can lead to complex and interesting behaviour about which little is currently known. Here, we examine the basic mechanisms by which a flow exits from sliding, either along a switching surface or along the intersection of two switching surfaces, with a view to understanding sliding and exit when many switches are involved. On a single switching surface, exit occurs via tangency of the flow to the switching surface. Along an intersection of switches, exit can occur at a tangency with a lower codimension sliding flow, or by a spiralling of the flow that exhibits geometric divergence (infinite steps in finite time). Determinacy-breaking can occur where a singularity creates a set-valued flow in an otherwise deterministic system, and we resolve such dynamics as far as possible by blowing up the switching surface into a switching layer. We show preliminary simulations exploring the role of determinacy-breaking events as organizing centres of local and global dynamics.