Hopf-like bifurcations induced by hysteresis and time-delay near monodromic tangential singularities
arXiv:2608.10581
2026
Dynamics
1 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper gives a constructive return-map mechanism showing that hysteresis or time delay added to a piecewise-smooth system with a stable monodromic tangential singularity generically creates a local attracting limit cycle. Its transferable asset is an explicit displacement-map balance, D(y, mu) approximately equal to V_2n y^(2n) plus delta kappa_td mu divided by y^(2|k_R-k_L|), which predicts both the existence and amplitude of delay-induced oscillations. A neural optimizer can use this relation to estimate a maximum safe stale-gradient delay or hysteresis width, and deliberately permit a small predicted cycle near saddles or plateaus before removing the delay.
Ideas from this paper
✗ Failed on benchmark
2026
Represent training near a switching condition as two locally smooth optimizer modes, such as low- and high-momentum updates or two preconditioners, with a delayed gate. Estimate the leading return-map coefficient and use the paper's scaling law to cap the delay or hysteresis width before an attracting optimization oscillation becomes large. The controller can also intentionally permit a small predicted cycle near saddles or plateaus, then remove the delay as soon as the measured cycle amplitude…
Useful7/10
Difficulty6/10
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