Optimal Control of Saddle Node Bifurcations

What the math gives to ML

The paper provides a nonstandard control mechanism for nonautonomous saddle-node systems: every forcing trajectory is assigned a unique critical rate separating safe tracking from rate-induced overshoot. This converts a constrained optimal-control problem into unconstrained minimization through a rate function, and an approximation of that function yields a closed-form control. The transferable neural-network idea is to treat a scalar training or inference order parameter, such as learning rate, activation gain, recurrent state, or optimizer momentum, as a slowly forced saddle-node normal form, and adapt the schedule before crossing its predicted critical-rate boundary. The key falsifiable signature is a sharp transition from bounded tracking to overshoot or divergence as the schedule rate exceeds the estimated critical rate.