Non-deterministic chaos, and the two-fold singularity in piecewise smooth flows

A vector field is piecewise smooth if its value jumps across a hypersurface, and a two-fold singularity is a point where the flow is tangent to the hypersurface from both sides. Two-folds are generic in piecewise smooth systems of three or more dimensions. We derive the local dynamics of all possible two-folds in three dimensions, including nonlinear effects around certain bifurcations, finding that they admit a flow exhibiting chaotic but non-deterministic dynamics. In cases where the flow passes through the two-fold, upon reaching the singularity it is unique neither in forward nor backward time, meaning the causal link between inward and outward dynamics is severed. In one scenario this occurs recurrently. The resulting flow makes repeated, but non-periodic, excursions from the singularity, whose path and amplitude is not determined by previous excursions. We show that this behaviour is robust and has many of the properties associated with chaos. Local geometry reveals that the chaotic behaviour can be eliminated by varying a single parameter: the angular jump of the vector field across the two-fold.