The two-fold singularity of nonsmooth flows: Leading order dynamics in nn-dimensions

• The first fully explicit derivation of the normal form of two-fold singularity is given.
• The normal form is extended to many dimensions for the first time.
• We review the classification of two-fold dynamics as it holds in many dimensions.
• We discuss the implications of the singularity in higher dimensions.
• The general role of the two-fold singularity in canard-type behaviour is given.

A discontinuity in a system of ordinary differential equations can create a flow that slides along the discontinuity locus. Prior to sliding, the flow may have collapsed onto the discontinuity, making the reverse flow non-unique, as happens when dry-friction causes objects to stick. Alternatively, a flow may slide along the discontinuity before escaping it at some indeterminable time, implying non-uniqueness in forward time. At a two-fold singularity these two behaviours are brought together, so that a single point may have multiple possible futures as well as histories. Two-folds are a generic consequence of discontinuities in three or more dimensions, and play an important role in both local and global dynamics. Despite this, until now nothing was known about two-fold singularities in systems of more than 3 dimensions. Here, the normal form of the two-fold is extended to higher dimensions, where we show that much of its lower dimensional dynamics survives.