{
 "artifacts": null,
 "category": "dynamics",
 "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.",
 "formulas_latex": [
  "$$\\dot\\theta_i=\\Omega_i+\\kappa\\sum_{j=1}^{N}a_{ij}\\sin(\\theta_j-\\theta_i),\\qquad \\theta_i(t)=\\omega t+\\phi_i$$",
  "$$\\omega=\\Omega_i+\\kappa\\sum_{j=1}^{N}a_{ij}\\sin(\\phi_j-\\phi_i)\\quad\\forall i$$",
  "$$J_{ij}=\\kappa a_{ij}\\cos(\\phi_j-\\phi_i)\\ (j\\ne i),\\qquad J_{ii}=-\\kappa\\sum_j a_{ij}\\cos(\\phi_j-\\phi_i),\\qquad J\\mathbf{1}=0$$",
  "$$\\max_{\\lambda\\in\\operatorname{spec}(J),\\ \\lambda\\ne 0}\\operatorname{Re}(\\lambda)\u003c0\\ \\Longrightarrow\\ \\text{local exponential convergence modulo global phase}$$"
 ],
 "id": 2841,
 "implementation": "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.",
 "math_summary": "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|\u003c\\pi/2\\), the transverse dynamics contract locally. More generally, orbital stability requires every nonzero Jacobian eigenvalue to have negative real part.",
 "math_tags": [
  "dynamical-systems",
  "control-theory",
  "spectral-theory"
 ],
 "ml_areas": [
  "rnn",
  "world-model",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2608.29630",
  "arxiv_url": "https://arxiv.org/abs/2608.29630",
  "summary_what_math_gives_to_ml": "The paper provides a constructive theory of nonzero-frequency relative equilibria in directed Kuramoto networks, including existence conditions, orbital stability modulo the global phase direction, invariant neighborhoods, and exact discrete-Fourier stability tests for forward-neighbor graphs. The most transferable mechanism is a phase-based recurrent or state-space layer whose desired rotating solution is designed through an equitable partition or twisted profile, while common phase drift is treated as a neutral gauge direction rather than an instability. A second mechanism is mode-selective spectral shaping: explicit Fourier-mode factors predict whether a cyclic recurrent layer contracts or amplifies particular modes. These ideas are most credible for oscillatory RNNs, cyclic state-space models, and latent dynamical systems.",
  "title": "Existence and Stability of Dancing Equilibria in Asymmetric Kuramoto Networks",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 7,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "accuracy"
 ],
 "title": "Orbital-Stable Dancing RNN",
 "url": "https://synthcore.org/idea/2841/orbital-stable-dancing-rnn",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}