Homoclinic Theorems for piecewise smooth vector fields

What the math gives to ML

The paper extends the Birkhoff–Smale homoclinic theorem to periodically forced piecewise-smooth vector fields with crossing-only switching and no sliding. Its transferable mechanism is a constructive route from a transverse homoclinic intersection to an invariant subsystem conjugate to a finite- or countable-symbol Bernoulli shift, yielding positive entropy and structured symbolic memory. A neural implementation can use a hybrid recurrent state-space model with piecewise-affine vector-field branches, a learned switching surface, and a periodic forcing phase. The key falsifiable prediction is that symbolic entropy and sequence capacity increase sharply when a transverse homoclinic intersection appears, while tangencies or sliding destroy this behavior.