Loss of determinacy at small scales, with application to multiple time-scale and nonsmooth dynamics

A singularity is described that creates a forward time loss of determinacy in a two-timescale system, in the limit where the timescale separation is large. We describe how the situation can arise in a dynamical system of two fast variables and three slow variables or parameters, with weakly coupling between the fast variables. A wide set of initial conditions enters the ε-neighbourhood of the singularity, and explodes back out of it to fill a large region of phase space, all in finite time. The scenario has particular significance in application to piecewisesmooth systems, where it arises in the blow up of dynamics at a discontinuity and is followed by abrupt re-collapse of solutions to ‘hide’ the loss of determinacy, and yet leave behind a remnant of it in the global dynamics. This constitutes a generalization of a ‘micro-slip’ phenomenon found recently in spring-coupled blocks, whereby coupled oscillators undergo unpredictable stick-slipstick sequences instigated by a higher codimension form of the singularity. The indeterminacy is localised to brief slips events, but remains evident in the indeterminate sequencing of nearsimultaneous slips of multiple blocks.