Parameter-Dependent Lyapunov Neural Dynamics

1. Integration point: use a residual RNN, neural ODE solver, or latent state-space block. Parameterize $P_\psi(\theta,t)=L_\psi(\theta,t)L_\psi(\theta,t)^\top+p_{\min}I$, where $L_\psi$ is lower triangular with positive diagonal. 2. Pseudocode: compute $z_{k+1}=z_k+h f_\phi(z_k,\theta_k,t_k)$; obtain $A_k$ explicitly for small states or through Jacobian-vector products and power iteration for large states; evaluate $P_k$ and $P_{k+1}$; compute the largest generalized eigenvalue of $(J_k^\top P_{k+1}J_k,e^{-2\alpha h}P_k)$; add its positive part to the task loss; update $\phi$ and $\psi$. 3. Computed from the mechanism: the Lyapunov inequality, decay rate $\alpha$, and metric positivity. Estimated empirically: local Jacobians, parameter derivatives of $P$, and minibatch averages over trajectories. 4. First experiment: train equal-width GRUs and residual MLP dynamics on sequence-copy and Lorenz-63 prediction tasks. Compare vanilla training, spectral normalization, and this penalty at equal optimizer steps; ablate both the Lyapunov term and parameter dependence of $P$. 5. Pre-registered signature: perturbation energy must satisfy $V_k\le e^{-2\alpha kh}V_0$ up to discretization error, and the measured contraction factor must cross one near $\lambda_{\max}(P_k^{-1/2}J_k^\top P_{k+1}J_kP_k^{-1/2})=e^{-2\alpha h}$. Reject the transfer if the empirical boundary differs by more than 20%, or if the Lyapunov residual is positive without reducing perturbation growth versus the ablation.