On the slow passage through a Hopf in generalized Shishkova systems: Exponential asymptotics and maximal delay

What the math gives to ML

The paper provides a geometric mechanism for delayed instability when a slowly varying complex-conjugate eigenpair crosses the imaginary axis: the attracting slow manifold persists beyond the instantaneous Hopf point, and its eventual loss is controlled by exponentially small splitting between attracting and repelling slow manifolds. The transferable asset is an integrated-growth criterion rather than an instantaneous spectral-radius test. In neural optimization, this suggests ramping a learning-rate, gain, or continuation parameter through a complex stability boundary while tracking accumulated real-part growth, thereby distinguishing harmless post-boundary delay from genuine divergence. The prediction is a measurable exit threshold determined by the area under the leading real eigenvalue curve.