From two-dimensional continuous maps to one-dimensional discontinuous maps: a novel reduction explaining complex bifurcation structures in piecewise-linear families of maps

What the math gives to ML

The paper constructs an explicit first-return reduction from a two-dimensional piecewise-affine saddle dynamics to a discontinuous one-dimensional map with infinitely many geometrically scaled linear pieces. The transferable asset is not the specific border-collision application, but the combination of logarithmic state partitioning, exact event-driven iteration counts, self-similar parameter scaling, and a controlled approximation error. This suggests neural modules that route or transform activations according to their logarithmic magnitude while sharing parameters across scales. A second opportunity is an event-driven recurrent layer that analytically skips long runs of nearly linear dynamics instead of evaluating every intermediate step.