# Supercritical Hopf Latent Cell

- ID: 2719
- Canonical URL: https://synthcore.org/idea/2719/supercritical-hopf-latent-cell
- API JSON: https://synthcore.org/api/idea/2719.json
- API Markdown: https://synthcore.org/api/idea/2719.md
- Verification status: unverified
- Source: [arXiv:2608.24200](https://arxiv.org/abs/2608.24200)
- Category: dynamics
- Solves: stability, accuracy
- ML areas: rnn, ssm, world-model, training-dynamics
- Math tags: dynamical-systems, bifurcations, control-theory, stability
- Ratings: usefulness 6/10; difficulty 5/10; novelty 6/10

## Idea description

Replace an unconstrained recurrent hidden-state channel with a two-dimensional oscillator constrained to the supercritical Hopf normal form. A learned control parameter can place the channel below threshold for decaying dynamics or above threshold for sustained periodic dynamics, while the cubic term bounds the amplitude and prevents recurrent-state explosion.

## Mathematical statement

The center-manifold dynamics near a supercritical Hopf bifurcation are represented by a complex state z = r exp(i phi) satisfying dz/dt = (mu + i omega)z - (a + i b)|z|²z. Here mu is the distance from the bifurcation, omega is the linear angular frequency, a is the nonlinear amplitude-saturation coefficient, and b controls the amplitude-dependent frequency shift. With a > 0, the polar equations are dr/dt = mu r - a r³ and dphi/dt = omega - b r². The zero-amplitude state is stable when mu < 0. When mu > 0, it becomes unstable and a stable limit cycle appears at r* = sqrt(mu/a). Linearization around this cycle gives radial relaxation rate approximately 2 mu. These onset, amplitude, and decay-rate relations are the quantitative mechanisms transferred to the neural recurrent cell.

## Key formulas

- dz/dt = (mu + i omega) z - (a + i b)|z|² z
- dr/dt = mu r - a r³,    dphi/dt = omega - b r²
- r* = 0 for mu <= 0;    r* = sqrt(mu/a) for mu > 0, with a > 0
- z_(k+1) = z_k + h[(mu_k + i omega_k)z_k - (a_k + i b_k)|z_k|²z_k + B x_k]

## Implementation notes

1. Integration point: add one or more two-dimensional oscillator channels to an RNN, state-space model, or world model. For input x_k and hidden state h_k, a small controller computes mu_k and omega_k. The real and imaginary parts of z_k are concatenated with x_k and passed to the remaining neural update. Parameterize a_k = softplus(alpha_k) + epsilon to guarantee positive saturation. 2. Pseudocode: `mu, omega = controller(x, h); a = softplus(alpha) + eps; forcing = B @ x; z = z + hstep*((mu + 1j*omega)*z - (a + 1j*b)*abs(z)**2*z + forcing); h = recurrent_core(concat(x, real(z), imag(z)))`. For large steps, use a smaller oscillator step or an implicit radial update. 3. Taken from the paper: the Hopf normal form, threshold mu = 0, condition a > 0, cycle amplitude sqrt(mu/a), and radial relaxation rate 2 mu. Estimated empirically: the input-dependent controller, effective frequency, forcing scale, and discretization error. 4. First experiment: compare a vanilla GRU with this Hopf-GRU on synthetic damped sinusoids, sustained sinusoids, noisy oscillators, and regime-switching signals; then test a small ECG or motion-forecasting dataset. Sweep constant mu values through zero with a = 1 and no forcing. Measure asymptotic amplitude and radial relaxation. The falsifiable prediction is negligible amplitude for mu < 0, amplitude r approximately sqrt(mu/a) for mu > 0, and a log radial-error slope near -2 mu. The measured threshold and amplitude law should agree within 20 percent. Also compare hidden-state norm variance and long-horizon forecast error against the unconstrained GRU.

## Disclaimer

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