arXiv:2608.12862 2026 Dynamics 1 ideas extracted · analyzed Sep 1, 2026
The paper characterizes topological shadowing for finite-dimensional linear dynamics through a hyperbolicity condition: eigenvalues must avoid the unit circle, so the system decomposes into stable and unstable modes without marginal directions. This provides a principled stability criterion for recurrent and state-space neural networks, where finite-precision arithmetic, truncated state updates, and noisy activations create pseudo-orbits rather than exact trajectories. The most practical transfer is a spectral-gap regularizer or projection that keeps transition eigenvalues at least a margin away from modulus one, followed by robustness tests under long-horizon perturbations.
Constrain a recurrent or state-space transition matrix so that its eigenvalues avoid a configurable annulus around the unit circle. This creates a stable/unstable decomposition and should reduce the accumulation of numerical, quantization, and activation-update errors over long sequences while preserving controlled long-term memory.