Homoclinic Symbolic Reservoir
Implementation & benchmark of arXiv:2607.09618 — Homoclinic Theorems for piecewise smooth vector fields
Source paper: Homoclinic Theorems for piecewise smooth vector fields arXiv:2607.09618 ⓘ · analyzed Aug 30, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Construct a periodically driven hybrid recurrent state-space model whose vector field is piecewise smooth across learned switching surfaces. Engineer a transverse homoclinic intersection around a hyperbolic recurrent state; the resulting shift-like invariant set provides a controllable symbolic reservoir for sequence prediction and long-horizon generation.
Formulas
Mathematical statement
Let the recurrent state be $z\in\mathbb{R}^{n}$ and phase $\theta\in\mathbb{S}^{1}$ with $\dot\theta=1$. Define switching surfaces $\Sigma_i=\{z:h_i(z)=0\}$ and piecewise vector fields $\dot z=X_j(\theta,z)$ in regions $A_j$. At every crossing, require $\nabla h_i(z_c)^{\mathsf T}X_j(\theta_c,z_c)\neq0$ to exclude tangencies and sliding. Let $F=\Phi^T_{\theta_0}$ be the stroboscopic time-$T$ map. If $p$ is a hyperbolic fixed point of $F$ and $q\in W^s(p)\cap W^u(p)$ is transverse, meaning $T_qW^s(p)\oplus T_qW^u(p)=\mathbb{R}^n$, then an invariant set $\Lambda$ contains a subsystem conjugate to the $j$-symbol Bernoulli shift: $\pi\circ F|_\Lambda=\sigma\circ\pi$. Consequently $h_{\mathrm{top}}(F|_\Lambda)\ge\log j$, or entropy rate at least $\log(j)/T$. In the neural adaptation, estimate and regularize the crossing margin, stable–unstable intersection distance, and tangent-space angle.
Implementation notes
(1) Integrate this at the recurrent-state level, before a task-specific readout. Use a two- or three-dimensional state $z_t$, phase $\theta_t=2\pi t/T$, two or more piecewise-affine or small-MLP vector-field branches, and switching surfaces $h_i(z)=a_i^{\mathsf T}z-b_i$. Use an event-aware ODE solver so branch changes occur at estimated zero crossings of $h_i$. (2) Pseudocode: integrate one period to compute $F(z)$; find a candidate fixed point $p$ by minimizing $\|F(p)-p\|^2$; estimate $J_F(p)$ by automatic differentiation or finite differences; retain candidates whose Jacobian has stable and unstable eigenspaces. Launch short forward and backward orbit segments from perturbations in these eigenspaces, minimize their closest approach distance to obtain $q$, and maximize the smallest singular value of the concatenated stable and unstable tangent bases. Add $L_{cross}=\sum_c\max(0,\delta-|\nabla h_i(z_c)^\mathsf{T}X_j(\theta_c,z_c)|)^2$ and a homoclinic penalty based on the forward/backward closest distance. Train a linear or shallow nonlinear readout from $z_t$. (3) The theorem is taken from the paper; fixed points, Jacobians, manifolds, crossing margins, tangent angles, symbolic partitions, and entropy are estimated numerically. (4) First experiment: compare this reservoir with a smooth tanh reservoir and an unstructured piecewise reservoir on delayed parity and periodic symbol-sequence prediction. Sweep the homoclinic-control coefficient and forcing amplitude. Encode each orbit by visited region and estimate block entropy $H_m/m$. The predicted signature is a sharp transition from near-zero entropy to a plateau approaching $\log j$ per stroboscopic step when transversality appears. The measured transition should agree within 20% with the first parameter where the stable/unstable intersection distance becomes small, their angle remains bounded away from zero, and the crossing margin exceeds $\delta$. Tangency or sliding should sharply reduce entropy and long-horizon prediction accuracy.
Verification
This idea has not been verified yet.
Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.
Artifacts
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