Orbital-Stable Dancing RNN

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.