Noninvertibility and Bifurcation Phenomena in a Four-Partitions Piecewise Linear Map

What the math gives to ML

The paper provides a concrete two-dimensional piecewise-linear dynamical system whose qualitative behavior is organized by computable Flip, Border Collision, and Fold Border Collision bifurcations. Its most transferable asset is a recurrent cell with explicit quadrant-wise Jacobians, Schur stability inequalities, and parameter surfaces at which fixed points lose stability or cross activation-region boundaries. This can be used conservatively as a stability-certified recurrent module, or deliberately as a low-dimensional multi-attractor memory and reservoir whose periodic regimes are selected by parameter continuation. The transfer is most compelling for RNNs, state-space models, and compact recurrent controllers rather than generic feed-forward networks.