# Equal-amplitude synchronized oscillator modes

- ID: 1108
- Canonical URL: https://synthcore.org/idea/1108/equal-amplitude-synchronized-oscillator-modes
- API JSON: https://synthcore.org/api/idea/1108.json
- API Markdown: https://synthcore.org/api/idea/1108.md
- Verification status: unverified
- Source: [arXiv:2607.18337](https://arxiv.org/abs/2607.18337)
- Category: dynamics
- Solves: stability, accuracy, sample-efficiency
- ML areas: rnn, ssm, training-dynamics, regularization
- Math tags: dynamical-systems, bifurcations, graph-theory, harmonic-analysis, control-theory
- Ratings: usefulness 6/10; difficulty 6/10; novelty 7/10

## 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.

## 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.

## Key formulas

- $$A_j^2=\sum_{l=1}^{n}\left[(c_{1l}^{j})^2+(c_{2l}^{j})^2\right],\qquad A_j^2=A_k^2\ \text{or}\ A_k=0.$$
- $$\mathcal{L}_{\mathrm{amp}}=\frac{1}{|\mathcal A|^2}\sum_{j,k\in\mathcal A}(A_j^2-A_k^2)^2+\lambda_0\sum_j\min(A_j^2,\varepsilon),$$
- $$\dot\theta_j=\omega_j+\kappa\sum_kW_{jk}\sin(\theta_k-\theta_j),\qquad \theta_j=\operatorname{atan2}(c_2^j,c_1^j),$$
- $$\kappa\lambda_2(L)>\Delta\omega\ \Longrightarrow\ \text{expected phase locking},$$

## 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 $A_j\approx0$.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.