# Attractor-Conditioned Floquet Stabilization

- ID: 2282
- Canonical URL: https://synthcore.org/idea/2282/attractor-conditioned-floquet-stabilization
- API JSON: https://synthcore.org/api/idea/2282.json
- API Markdown: https://synthcore.org/api/idea/2282.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2608.14203](https://arxiv.org/abs/2608.14203)
- Category: dynamics
- Solves: stability, accuracy
- ML areas: rnn, ssm, training-dynamics, optimizer
- Math tags: dynamical-systems, bifurcations, control-theory, linear-algebra
- Ratings: usefulness 8/10; difficulty 6/10; novelty 6/10

## Idea description

Add a rare-probe channel to a recurrent or state-space model and measure its local growth around every attractor reached by the same parameters. Penalize the worst attractor-conditioned growth rate, rather than checking stability only along one training trajectory, so a model cannot appear stable in one regime while exhibiting exploding perturbations in another. The method is especially appropriate for long-horizon RNNs, neural ODEs, and autonomous world models with recurrent hidden dynamics.

## Mathematical statement

Let the resident neural dynamics be $\dot h=F_{\theta}(h)$, with hidden state $h\in\mathbb{R}^{n}$ and parameters $\theta$. Introduce a small perturbation $z$ governed by the variational equation $\dot z=J_{\theta}(h(t))z$, where $J_{\theta}(h)=\partial F_{\theta}/\partial h$ and $\|z(0)\|=\varepsilon$. For an equilibrium $h^{*}$, local growth is determined by the largest real Jacobian eigenvalue $\lambda_{\mathrm{eq}}=\max\operatorname{Re}\operatorname{eig}(J_{\theta}(h^{*}))$. For a periodic attractor $h_{c}(t+T)=h_{c}(t)$ with period $T$, define the fundamental matrix $\Phi$ by $\dot\Phi=J_{\theta}(h_{c}(t))\Phi$, $\Phi(0)=I$; the largest Floquet exponent is $\lambda_{\mathrm{cyc}}=T^{-1}\log\rho(\Phi(T))$, where $\rho$ is spectral radius. Stability requires $\lambda<0$ (equivalently $\rho(\Phi(T))<1$). The rare-strain mechanism is the scalar special case $\dot q=[g(h(t))-\mu]q$, where $q$ is a rare perturbation, $g(h)$ is its weighted exposure, and $\mu$ is removal; then $\lambda_{\mathrm{eq}}=g(h^{*})-\mu$ and $\lambda_{\mathrm{cyc}}=T^{-1}\int_{0}^{T}[g(h_{c}(t))-\mu]dt$.

## Key formulas

- $$\dot q(t)=\left[\sum_{a=1}^{m}c_{a}u_{a}(t)-\mu\right]q(t),\qquad \lambda=\lim_{t\to\infty}\frac{1}{t}\log\frac{|q(t)|}{|q(0)|}.$$
- $$\lambda_{\mathrm{eq}}=\sum_{a=1}^{m}c_{a}u_{a}^{*}-\mu,\qquad \lambda_{\mathrm{cyc}}=\frac{1}{T}\int_{0}^{T}\left(\sum_{a=1}^{m}c_{a}u_{a}(t)-\mu\right)dt.$$
- $$\dot\Phi(t)=J_{\theta}(h_{c}(t))\Phi(t),\quad \Phi(0)=I,\qquad \lambda_{\mathrm{Floq}}=\frac{1}{T}\log\rho(\Phi(T)).$$
- $$\mathcal{L}_{\mathrm{stab}}=\frac{1}{K}\sum_{k=1}^{K}\left[\max\left(0,\widehat\lambda_{k}+\kappa\right)\right]^{2},\qquad \widehat\lambda_{k}=\frac{1}{L\Delta t}\log\frac{\|z_{L}^{(k)}\|}{\|z_{0}^{(k)}\|}.$$

## Implementation notes

Integrate this at the hidden-state transition of an autonomous RNN or neural state-space model. Use the resident update $h_{t+1}=F_{\theta}(h_t,x_t)$, or $\dot h=F_{\theta}(h,x)$ for a neural ODE, and maintain $K$ perturbed copies initialized near states collected from different long rollouts or different initial conditions. Pseudocode: (1) roll out $h_{t+1}=F_{\theta}(h_t,x_t)$ for burn-in; (2) cluster the post-burn-in states into candidate attractors using state-space distance or recurrence; (3) for each cluster representative $h^{(k)}$, draw $z_0\sim\mathcal N(0,I)$ and normalize it to norm $\varepsilon$; (4) propagate the resident and perturbed states, either by $z_{t+1}=J_tz_t$ using a Jacobian-vector product or by finite differences $z_{t+1}=F(h_t+\varepsilon z_t)-F(h_t)$; (5) renormalize $z_t$ periodically and accumulate log norm ratios; (6) add $\mathcal L_{\mathrm{stab}}$ to the task loss, with target margin $\kappa\ge0$. The paper supplies the mechanism that the sign can differ between attractors; Jacobians, attractor periods, and exponents are estimated from the current network. Start with a cheap experiment on a tanh GRU or 2-layer RNN trained for next-step prediction on a synthetic bistable oscillator dataset, comparing standard spectral normalization against this loss. Use identical parameters and two initial-state ensembles. The quantitative prediction is a sharp basin-dependent boundary: one attractor can have $\widehat\lambda_k>0$ while another has $\widehat\lambda_j<0$ before regularization. After stabilization, the maximum exponent should move below $-\kappa$ and long-horizon error should stop growing exponentially; measured perturbation norms should follow $\|z(t)\|\approx\|z(0)\|e^{\lambda_k t}$, with fitted slopes agreeing with finite-difference estimates within 20%.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: inconclusive
- Verdict: Built and numerically verified an attractor-conditioned rare-probe stabilization MVP. Finite-difference resident growth matched analytic Jacobian exponents within 1.3e-7 relative error, all probe stability boundaries matched mu=c*h to numerical precision, and worst-attractor optimization achieved the predicted -kappa margin while single-attractor monitoring left the unseen basin unstable. This confirms the mechanism, but no neural-network accuracy or FLOP win was tested.

### Mechanism check

- Verdict: Built and numerically verified an attractor-conditioned rare-probe stabilization MVP. Finite-difference resident growth matched analytic Jacobian exponents within 1.3e-7 relative error, all probe stability boundaries matched mu=c*h to numerical precision, and worst-attractor optimization achieved the predicted -kappa margin while single-attractor monitoring left the unseen basin unstable. This confirms the mechanism, but no neural-network accuracy or FLOP win was tested.
- Confidence: 9/10
- Limitations: This is a scalar continuous-time toy mechanism check. It does not test a learned RNN, GRU, neural ODE, periodic Floquet attractors, spectral normalization, task loss, long-horizon prediction accuracy, computational overhead, or learned attractor clustering.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.55035
- Paired wins: 5/8
- Benchmark verdict: no significant win

## Artifacts

- [README.md](https://synthcore.org/code/799/README.md)
- [bench_report.json](https://synthcore.org/code/799/bench_report.json)
- [experiment.py](https://synthcore.org/code/799/experiment.py)
- [report.md](https://synthcore.org/code/799/report.md)
- [report_bench_2026-09-01T105110.md](https://synthcore.org/code/799/report_bench_2026-09-01T105110.md)
- [results.json](https://synthcore.org/code/799/results.json)
- [stage2_bench.py](https://synthcore.org/code/799/stage2_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/799)

## Disclaimer

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