# Orbital-Stable Dancing RNN

- ID: 2841
- Canonical URL: https://synthcore.org/idea/2841/orbital-stable-dancing-rnn
- API JSON: https://synthcore.org/api/idea/2841.json
- API Markdown: https://synthcore.org/api/idea/2841.md
- Verification status: unverified
- Source: [arXiv:2608.29630](https://arxiv.org/abs/2608.29630)
- Category: dynamics
- Solves: stability, accuracy
- ML areas: rnn, world-model, training-dynamics
- Math tags: dynamical-systems, control-theory, spectral-theory
- Ratings: usefulness 7/10; difficulty 6/10; novelty 7/10

## Idea description

Construct a recurrent layer whose hidden states evolve as directed phase oscillators with a prescribed nonzero common frequency and fixed phase offsets. Train task-relevant dynamics in the quotient space that removes the global phase-shift direction, so a rotating latent representation is not incorrectly penalized as unstable.

## Mathematical statement

Let \(z_i=r_i e^{\mathrm{i}\theta_i}\) be the complex hidden unit for node \(i\), and let \(a_{ij}\geq 0\) be a directed coupling weight. The phase dynamics are \(\dot\theta_i=\Omega_i+\kappa\sum_j a_{ij}\sin(\theta_j-\theta_i)\). A dancing equilibrium is \(\theta_i(t)=\omega t+\phi_i\), where the phase offsets satisfy \(\omega=\Omega_i+\kappa\sum_j a_{ij}\sin(\phi_j-\phi_i)\) for every \(i\). The linearization has a neutral global-shift eigenvector \(\mathbf{1}\). If the graph is strongly connected and every active edge has \(|\phi_j-\phi_i|<\pi/2\), the transverse dynamics contract locally. More generally, orbital stability requires every nonzero Jacobian eigenvalue to have negative real part.

## Key formulas

- $$\dot\theta_i=\Omega_i+\kappa\sum_{j=1}^{N}a_{ij}\sin(\theta_j-\theta_i),\qquad \theta_i(t)=\omega t+\phi_i$$
- $$\omega=\Omega_i+\kappa\sum_{j=1}^{N}a_{ij}\sin(\phi_j-\phi_i)\quad\forall i$$
- $$J_{ij}=\kappa a_{ij}\cos(\phi_j-\phi_i)\ (j\ne i),\qquad J_{ii}=-\kappa\sum_j a_{ij}\cos(\phi_j-\phi_i),\qquad J\mathbf{1}=0$$
- $$\max_{\lambda\in\operatorname{spec}(J),\ \lambda\ne 0}\operatorname{Re}(\lambda)<0\ \Longrightarrow\ \text{local exponential convergence modulo global phase}$$

## Implementation notes

Use this as a recurrent phase layer in an oscillatory RNN or latent world model. Represent each hidden channel by \(z_i=r_i\exp(\mathrm{i}\theta_i)\), keep amplitudes fixed initially, and update phases with Euler integration. At each step compute \(d_{ij}=\theta_j-\theta_i\), then \(v_i=\Omega_i+\kappa\sum_j a_{ij}\sin(d_{ij})\), and finally \(\theta_i\leftarrow\theta_i+h v_i\). Feed \([r_i\cos\theta_i,r_i\sin\theta_i]\) to the task head. Use a strongly connected directed ring with skip edges, parameterizing couplings as \(a_{ij}=\operatorname{softplus}(b_{ij})\). Penalize edge phase differences beyond \(\pi/2\), or initialize offsets \(\phi_i\) with all differences below \(\pi/3\). Estimate the rotating solution by averaging \(v_i\) over time. Compute the Jacobian from the displayed formula, remove its zero phase mode, and log the largest transverse real eigenvalue. The existence and angle condition come from the paper; the learned Jacobian and basin size are empirical. First test a 32-unit model on sequential MNIST and noisy sine-wave prediction against tanh and unconstrained complex RNNs. Sweep \(\kappa\) and timestep \(h\). The predicted signature is a stability boundary when an edge angle crosses \(\pi/2\) or the transverse spectral abscissa reaches zero. Below the boundary, perturbation norms should decay as \(\exp(\alpha t)\), where \(\alpha\) is the largest transverse real eigenvalue.

## Disclaimer

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