arXiv:2607.18337 2026 Dynamics 2 ideas extracted · analyzed Aug 30, 2026
The paper analyzes coupled Van der Pol oscillators through harmonic coefficients and identifies an invariant amplitude structure, with each oscillator satisfying a fixed-radius relation such as $(c_{1m}^{k})^{2}+(c_{2m}^{k})^{2}=r^{2}$. Its most transferable mechanism is nonlinear negative damping: small amplitudes are amplified, large amplitudes are attenuated, producing a stable limit cycle rather than unconstrained recurrent growth or decay. The coupled coefficient equations also suggest a mode-selection rule in which nonzero oscillators tend toward equal squared amplitudes, $A_j^2=A_k^2$, while some modes may collapse to zero. These mechanisms can be implemented as oscillator-inspired recurrent or state-space layers with explicit amplitude monitors and testable synchronization transitions.
Replace an unconstrained linear recurrent update with a two-dimensional oscillator state per hidden feature and use amplitude-dependent damping: negative damping below a target radius and positive damping above it. The cell should preserve phase information over long sequences while preventing hidden-state explosion or collapse.
Use multiple oscillator modes with weak phase coupling and regularize their active amplitudes toward a common squared amplitude. This transfers the paper's conclusion that coupled nonzero modes satisfy $A_j^2=A_k^2$ or that a mode collapses to zero, producing a controllable mixture of synchronized persistent modes and suppressed modes.