# Contractive Floquet return map

- ID: 2094
- Canonical URL: https://synthcore.org/idea/2094/contractive-floquet-return-map
- API JSON: https://synthcore.org/api/idea/2094.json
- API Markdown: https://synthcore.org/api/idea/2094.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2608.10738](https://arxiv.org/abs/2608.10738)
- Category: regularization
- Solves: stability, accuracy, sample-efficiency
- ML areas: ssm, world-model, regularization, training-dynamics
- Math tags: dynamical-systems, control-theory, geometry, numerical-analysis
- Ratings: usefulness 7/10; difficulty 6/10; novelty 7/10

## Idea description

For systems with a repeating orbit, train a periodic neural dynamical model together with a return map whose transverse deviations contract after each period. Enforce and measure orbital contraction rather than requiring phase-aligned pointwise trajectories to remain close, allowing phase drift while suppressing divergence across many cycles.

## Mathematical statement

The transferable object is the stroboscopic return map \(P\), defined by advancing a state by one period \(\hat T\): \(P(x)=\widehat\Phi_{\Theta}(\hat T;x)\). A contraction certificate on a neighborhood \(U\) is \(\|P(x)-P(y)\|\le q\|x-y\|\) for all \(x,y\in U\), with \(0\le q<1\). If \(x^\star\) is a fixed point satisfying \(P(x^\star)=x^\star\), then deviations contract as \(\|P^n(x)-x^\star\|\le q^n\|x-x^\star\|\). With one-period defect \(\delta\), the perturbed recursion is \(e_{n+1}\le qe_n+\delta\), giving \(e_n\le q^ne_0+\delta(1-q^n)/(1-q)\). For periodic oscillators, phase error should be removed by measuring distance to the learned orbit \(\Gamma_\Theta\): \(d_{orb}(x,\Gamma_\Theta)=\min_{\phi}\|x-\Gamma_\Theta(\phi)\|_2\). The paper's obstruction motivates this orbital rather than pointwise certificate.

## Key formulas

- $$P(x)=\widehat{\Phi}_{\Theta}(\hat T;x),\qquad P(x^\star)=x^\star,$$
- $$\|P(x)-P(y)\|\le q\|x-y\|,\qquad 0\le q<1,$$
- $$e_{n+1}\le q e_n+\delta\quad\Longrightarrow\quad e_n\le q^n e_0+\delta\frac{1-q^n}{1-q},$$
- $$d_{\mathrm{orb}}(x_n,\Gamma_\Theta)=\min_{\phi\in[0,2\pi)}\|x_n-\Gamma_\Theta(\phi)\|_2.$$

## Implementation notes

Use this in a periodic latent ODE or recurrent world model for locomotion, seasonal signals, rotating systems, or periodic control tasks. Parameterize the vector field with explicit period \(\hat T\), feeding \((\sin(2\pi t/\hat T),\cos(2\pi t/\hat T))\) to the network. Learn a nominal periodic trajectory \(\Gamma_\Theta(\phi)\) and define the one-period map \(P_\Theta(x)=\operatorname{ODESolve}(f_\Theta,x,[0,\hat T])\). Add \(L_{contract}=\max(0,\|P_\Theta(x)-P_\Theta(y)\|_2-q_{target}\|x-y\|_2)^2\) for nearby pairs sampled around the orbit, with \(q_{target}<1\). Estimate the empirical contraction factor using \(\hat q=\max_b\|P(x_b)-P(y_b)\|_2/\|x_b-y_b\|_2\), or estimate the largest local Jacobian singular value by power iteration through \(J_P(x)\). Estimate the defect after phase alignment as \(\hat\delta=\operatorname{Quantile}_q d_{orb}(P(x_b),\Gamma_\Theta)\). During inference, monitor \(\hat q^ne_0+\hat\delta(1-\hat q^n)/(1-\hat q)\). The paper supplies the contraction-plus-defect recursion; the batch estimates, orbit matching, and penalty are engineering approximations. First experiment: train on noisy pendulum or Van der Pol cycles and compare an unconstrained periodic neural ODE against this model over 100--1000 periods. Evaluate phase-invariant orbital RMSE, pointwise RMSE, estimated \(\hat q\), and the fraction of rollouts remaining inside the learned tube. Success means bounded orbital error with measured \(\hat q<1\), even when pointwise phase error grows.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: inconclusive
- Verdict: Built a reproducible toy contractive Floquet return-map experiment in floquet_toy.py. All three mechanism predictions were confirmed: the q=1 stability transition and exponential rates matched exactly, the defect floor matched delta/(1-q) with relative error below 5e-16, and transverse orbital error contracted despite substantial phase-sensitive pointwise error. In the secondary noisy 1000-cycle comparison, the contractive map remained bounded while the q=1.02 baseline diverged, though this is not evidence from a learned neural ODE.

### Mechanism check

- Verdict: Built a reproducible toy contractive Floquet return-map experiment in floquet_toy.py. All three mechanism predictions were confirmed: the q=1 stability transition and exponential rates matched exactly, the defect floor matched delta/(1-q) with relative error below 5e-16, and transverse orbital error contracted despite substantial phase-sensitive pointwise error. In the secondary noisy 1000-cycle comparison, the contractive map remained bounded while the q=1.02 baseline diverged, though this is not evidence from a learned neural ODE.
- Confidence: 9/10
- Limitations: The experiment uses an analytically specified affine return map rather than training a neural ODE or recurrent world model on noisy oscillator data. It does not test optimization behavior, learned orbit quality, computational cost, multidimensional nonlinear Jacobians, or robustness beyond the sampled scalar transverse dynamics.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.328
- Paired wins: 6/8
- Benchmark verdict: no significant win

## Artifacts

- [README.md](https://synthcore.org/code/841/README.md)
- [bench_floquet.py](https://synthcore.org/code/841/bench_floquet.py)
- [bench_report.json](https://synthcore.org/code/841/bench_report.json)
- [floquet_toy.py](https://synthcore.org/code/841/floquet_toy.py)
- [report.md](https://synthcore.org/code/841/report.md)
- [report_bench_2026-09-01T114948.md](https://synthcore.org/code/841/report_bench_2026-09-01T114948.md)
- [results.json](https://synthcore.org/code/841/results.json)
- [Download all files as ZIP](https://synthcore.org/download/841)

## Disclaimer

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