From discontinuity to non-determinism in dynamical systems

A recent study into the geometry underlying discontinuities in dynamics revealed some surprises. The problems of interest are fundamental, things like: frictional sticking, electronic switching, protein activation and neuron spiking. When a discontinuity occurs at some threshold value in a system of differential equations, the solutions that result might not be unique. Besides the myriad cute models from applications, we want to know what discontinuities really tell us about dynamics in the real world. Non-unique solutions are easily dismissed as unphysical, yet they tell us something about the extreme behaviour made possible in the limit as a sudden change becomes almost discontinuous. Initially unique solutions may become multi-valued, revealing extreme sensitivity to initial conditions, a breakdown of determinism, yet the possible outcomes lie in a well-defined set: an "explosion". An intriguing connection between discontinuities and singularly perturbations is revealed by studying the so-called two-fold singularities and canards, borrowing ideas from nonstandard analysis along the way. The outcomes have been seen in superconductor experiments, are possible in control circuits, they are hidden in plain sight in the dynamics of friction, impacts, and neuron spiking, and they lead to non-deterministic forms of chaos.