Equal-amplitude synchronized oscillator modes
Implementation & benchmark of arXiv:2607.18337 — Coupled Van der Pol Networks
Source paper: Coupled Van der Pol Networks arXiv:2607.18337 ⓘ · analyzed Aug 30, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
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.
Formulas
Mathematical statement
For mode $j$, let $c^j=(c_1^j,c_2^j)$ denote its coefficient vectors and define $A_j^2=\sum_l[(c_{1l}^{j})^2+(c_{2l}^{j})^2]$. The paper's coupled cubic equations contain cross-mode terms proportional to $c_m^k A_j^2$ and self-mode cubic terms, with the stated consequence that active modes satisfy $A_j^2=A_k^2$ or a mode has $A_k=0$. Implement this using an amplitude-consensus penalty over active modes, together with a gate or sparsity term that allows inactive modes. For phase variables $\theta_j=\operatorname{atan2}(c_2^j,c_1^j)$, use graph coupling $\dot\theta_j=\omega_j+\kappa\sum_kW_{jk}\sin(\theta_k-\theta_j)$, where $\omega_j$ is intrinsic frequency, $W$ is a nonnegative coupling graph, and $\kappa$ is coupling strength.
Implementation notes
1. Integration point: insert $M$ oscillator modes into an SSM or recurrent backbone. Each mode has vectors $c_1^j,c_2^j\in\mathbb{R}^n$; decode by concatenating modes or summing their projections. Add $\lambda_A\mathcal L_{\mathrm{amp}}$ to the task loss, with a smooth active gate so consensus is applied only to modes that are not being suppressed. 2. Pseudocode: compute A2[j] = (c1[j]2 + c2[j]2).sum(); compute smooth gates g[j] = sigmoid(q*(A2[j]-tau)); add lambda_A * mean(g[j]*g[k]*(A2[j]-A2[k])**2) over pairs; compute phases with atan2; add the phase drift kappa * sum(W[j,k]*sin(theta[k]-theta[j])) to each mode update. 3. Taken from the paper: the alternatives $A_j^2=A_k^2$ or $A_k=0$ and the cubic cross-mode amplitude interactions. The synchronization threshold is an adapted diagnostic; estimate $\Delta\omega$ from learned frequencies and $\lambda_2(L)$ from the graph Laplacian. 4. First cheap experiment: compare uncoupled oscillator modes, consensus-coupled modes, and a GRU on permuted sequential MNIST and synthetic multisine forecasting. Sweep $\kappa$ across $0$ to $2\Delta\omega/\lambda_2(L)$ and measure phase order $R_\theta=|M^{-1}\sum_je^{i\theta_j}|$, amplitude coefficient of variation, number of suppressed modes, and validation error. The falsifiable prediction is a transition toward high $R_\theta$ near $\kappa_c\approx\Delta\omega/\lambda_2(L)$, accompanied by decreasing amplitude dispersion among active modes; modes below the activation threshold should converge toward $A_j\approx0$.
Verification
This idea has not been verified yet.
Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.
Artifacts
Artifacts unavailable.