Neutral Entry--Exit Cycles with Quadratic Grazing: Uniform Return Reduction and Local Two-Parameter Bifurcations

What the math gives to ML

The paper provides a constructive local normal form for a neutral return map combining a smooth quadratic displacement with a one-sided quadratic-grazing term, Delta(q,p)=S(q,p)+q_+^{3/2}K(sqrt(q_+),q,p). Its strongest transferable asset is the use of exact constant and linear displacement coefficients A=Delta(0,p) and B=Delta_q(0,p) as rank-two coordinates, together with explicit fold curves and cycle-count and stability chambers. This can be transferred to recurrent or implicit neural networks by treating a scalar latent coordinate as a penetration coordinate and monitoring the local return map around a fixed point. The resulting diagnostic predicts sharp transitions in the number and stability of latent fixed points, enabling a bifurcation-aware regularizer or a controller that keeps training away from undesirable fold and grazing boundaries.