A Globally Asymptotically Stable Planar Homogeneous Polynomial Vector Field With No Polynomial Lyapunov Function

What the math gives to ML

The paper gives a sharp counterexample showing that global asymptotic stability of a homogeneous polynomial vector field does not imply the existence of any positive-definite polynomial Lyapunov certificate, or even a locally real-analytic one. Its constructive asset is a degree-two homogeneous Lyapunov function that is globally C1 and radially unbounded but nonsmooth in higher derivatives at the origin. This suggests replacing polynomial or analytic Lyapunov critics for neural ODEs and recurrent dynamics with positively homogeneous, angularly parameterized C1 certificates. The homogeneity also gives a falsifiable algebraic decay law that can be monitored during training and rollout.