Period Homeostasis Near Hopf Bifurcation

What the math gives to ML

The paper provides a constructive mechanism for making the period of stable oscillations locally insensitive to an input. Near a nondegenerate Hopf bifurcation, Lyapunov-Schmidt reduction produces a low-dimensional amplitude-phase system whose period T(lambda, I) can be differentiated with respect to parameters. Hopf and infinitesimal period-homeostasis conditions define intersecting parameter curves, giving a quantitative operating-point map between steady, oscillatory, and input-sensitive regimes. This can transfer to recurrent or state-space neural networks by adding a learnable oscillatory core and training its parameters toward dT/dI = 0, producing temporal representations whose frequency is robust to changes in input magnitude.