Blow-up Parameter Landscapes for Polynomial Dynamical Systems
arXiv:2607.14269
2026
Dynamics
2 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper develops a constructive method for detecting finite-time blow-up in parameterized polynomial dynamical systems by compactifying infinity and analyzing the resulting boundary flow. Its transferable asset is the reduction of large-state behavior to the highest-degree homogeneous component: directional equilibria at infinity satisfy F_n(u) = alpha u, and the sign of alpha together with angular stability predicts runaway trajectories. This can be used in polynomial neural ODEs, recurrent models, state-space models, and deep equilibrium systems as an instability atlas and as a runtime burst controller. The main falsifiable prediction is that hidden-state divergence begins near parameter values where an attracting direction at infinity acquires positive radial growth.
Ideas from this paper
✗ Failed on benchmark
2026
For a neural ODE, residual flow, or deep equilibrium model with a dominant polynomial component, compute the directional dynamics induced by its highest-degree homogeneous term on the unit sphere. Penalize or reject parameter regions containing radially growing attracting directions, preventing finite-time activation blow-up while preserving nonlinear dynamics in safe directions.
Useful8/10
Difficulty6/10
Novelty8/10
△ Mechanism confirmed, baseline not beaten
2026
Use the paper's distinction between radial attraction and tangential instability at infinity to detect impending hidden-state bursts before they cause numerical failure. When the state approaches a radially growing directional equilibrium, temporarily add radial damping or switch to a bounded fallback update, then restore the original dynamics after angular ejection.
Useful7/10
Difficulty5/10
Novelty7/10