arXiv:2607.05829 2026 Dynamics 1 ideas extracted · analyzed Aug 30, 2026
The paper provides a concrete way to quantify when a periodically switched linear operator behaves like its averaged operator: the leading discrepancy is controlled by pairwise noncommutators, not merely by the norms or stability of the individual modes. This transfers naturally to a periodic switched state-space layer whose latent transition alternates among several learned generators, where average stability alone can otherwise hide unstable order-dependent dynamics. The most useful implementation is to regularize commutators during training and certify the actual cycle transition with a sampled quadratic Lyapunov inequality, giving both a trainable stability surrogate and an inference-time contraction check.
Build a state-space layer whose latent dynamics use a fixed cyclic schedule of learned generators instead of a single generator. Penalize pairwise commutator norms so that the true ordered cycle remains close to the averaged flow, while periodically checking a quadratic Lyapunov contraction condition on the exact cycle transition.