{
 "artifacts": null,
 "category": "dynamics",
 "description": "Train a recurrent neural network or state-space model using a Poincare-style event loss: identify two consecutive latent alignment events and require the latent position and velocity at the second event to equal a transformed version of the first. Evaluate the Jacobian of this return map and penalize unstable non-neutral Floquet multipliers, producing long-horizon trajectories that are both periodic or symmetry-periodic and locally stable.",
 "formulas_latex": [
  "$$m_i\\ddot{\\mathbf q}_i=\\mathcal G\\sum_{j\\ne i}\\frac{m_im_j}{\\|\\mathbf q_j-\\mathbf q_i\\|^3}(\\mathbf q_j-\\mathbf q_i),$$",
  "$$P_\\theta(z)=S_\\alpha\\Phi_\\theta(\\tau(z),z),\\qquad h(z)=0,\\qquad P_\\theta(z^*)=z^*,$$",
  "$$\\mathcal L_{\\mathrm{return}}=\\|S_\\alpha\\Phi_\\theta(\\tau_2,z_1)-z_1\\|_2^2+\\|S_\\alpha\\Phi_\\theta(\\tau_2,z_2)-z_2\\|_2^2,$$",
  "$$\\mathcal L_{\\mathrm{Floquet}}=\\big[\\max(0,\\rho_{\\mathrm{nonneutral}}-1+\\varepsilon)\\big]^2,\\qquad \\rho_{\\mathrm{nonneutral}}=\\max_{\\lambda_i\\notin\\mathcal N}|\\lambda_i|,$$"
 ],
 "id": 3015,
 "implementation": "1. Integration point: use a GRU, neural ODE, or diagonal state-space model whose latent state is $z_k$ and whose output is predicted from $z_k$. Add the return loss to the ordinary next-step prediction loss. For a discrete model, define an event by a scalar observable $h(z_k)$ changing sign; for a cheap first version use $h(z)=z_1-c$ and linearly interpolate the crossing time between adjacent steps. Let $z_1$ and $z_2$ be two consecutive crossings, estimate the event-to-event map by unrolling the network, and use either a fixed symmetry $S_\\alpha$ or learn $\\alpha$ as a scalar constrained to $[0,2\\pi]$. 2. Pseudocode: initialize $z$; unroll until $h(z)$ crosses zero twice; save $z_1,z_2$; compute $r=S_\\alpha z_2-z_1$; add $\\|r\\|^2$ to the prediction loss; periodically compute Jacobian-vector products of the event map with autodiff; estimate its leading eigenvalues with 5--10 power or Arnoldi iterations; exclude the known phase direction and penalize $\\max(0,|\\lambda|-1+\\varepsilon)^2$. Backpropagate through the unroll for the return residual and update parameters. 3. Computed versus estimated: the paper supplies the event matching and rotated-monodromy mechanism; the neural Jacobian, event time, and Floquet multipliers are estimated numerically from autodiff and finite event interpolation. 4. First experiment: train a 2-layer GRU on noisy trajectories generated by a planar harmonic oscillator or a two-dimensional rotating limit-cycle system, comparing ordinary teacher forcing against prediction loss plus the return and Floquet terms. Sweep the return-loss coefficient and latent spectral radius. The quantitative prediction is that free-running error after $n$ events scales approximately as $\\rho_{\\mathrm{nonneutral}}^n$; models with measured $\\rho_{\\mathrm{nonneutral}}\u003c1$ should show geometric error decay, while long-horizon divergence should begin near the empirical boundary $\\rho_{\\mathrm{nonneutral}}=1$ within roughly 20%.",
 "math_summary": "Let the learned continuous-time or discretized latent dynamics be $\\dot z=f_\\theta(z,u)$ or $z_{k+1}=F_\\theta(z_k,u_k)$, with latent state $z\\in\\mathbb{R}^d$. Let $h(z)=0$ define an event section, such as an alignment condition, and let $\\tau(z)$ be the first positive event time after leaving the section. The event return map is $P_\\theta(z)=S_\\alpha\\Phi_\\theta(\\tau(z),z)$, where $\\Phi_\\theta$ is the model flow and $S_\\alpha$ is a known or learned symmetry transformation parameterized by phase or rotation angle $\\alpha$. A relative periodic orbit satisfies $P_\\theta(z^*)=z^*$. Its local stability is determined by eigenvalues of $DP_\\theta(z^*)$ restricted to the section; eigenvalues associated with time shift or continuous symmetry are neutral and should be projected out. The training mechanism is a return residual plus a spectral penalty on the remaining Floquet multipliers: $\\rho_{\\mathrm{nonneutral}}=\\max_i|\\lambda_i|\u003c1$ predicts exponential decay of perturbations at successive events.",
 "math_tags": [
  "dynamical-systems",
  "control-theory",
  "bifurcations",
  "numerical-analysis",
  "linear-algebra"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "world-model",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2609.01585",
  "arxiv_url": "https://arxiv.org/abs/2609.01585",
  "summary_what_math_gives_to_ml": "The paper provides a constructive event-section method for finding and continuing relative periodic orbits: instead of integrating over an unknown full period and rotation angle, it matches the state at two consecutive alignment events in a low-dimensional nonlinear system. Its transferable asset is a symmetry-aware return-map loss for recurrent or state-space neural networks, together with a Floquet-style stability test that removes neutral directions caused by phase and continuous symmetries. A practical implementation can train a model to realize repeatable latent trajectories and use continuation in the target period or rotation angle to reveal stability boundaries before long-horizon deployment.",
  "title": "Families of relative periodic orbits in the planar three-body problem via consecutive alignments",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 7,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "accuracy",
  "generalization"
 ],
 "title": "Alignment-Section Floquet Training for Recurrent Dynamics",
 "url": "https://synthcore.org/idea/3015/alignment-section-floquet-training-for-recurrent-dynamics",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}