Normal form method of center-focus problem in piecewise-smooth systems and algorithm design

What the math gives to ML

The paper provides a constructive normal-form algorithm for piecewise-smooth monodromic systems, where the first nonzero normal-form coefficient is algebraically equivalent to the corresponding Lyapunov constant and determines local focus stability and degeneracy order. Its transferable asset is a computable local certificate for whether a switched dynamical system spirals toward or away from an equilibrium, without evaluating cumbersome orbit integrals. A neural-network analogue is a state-dependent switched optimizer whose branches are selected using online estimates of the first nonzero radial normal-form coefficient. The resulting controller predicts a sharp stability boundary: the selected training dynamics should contract locally when the coefficient is negative and diverge when it is positive.