Critical-Set Cone Monitor
Implementation & benchmark of arXiv:2608.18761 — Maximal attractors for perturbations of unimodal maps near a homoclinic tangency
Source paper: Maximal attractors for perturbations of unimodal maps near a homoclinic tangency arXiv:2608.18761 ⓘ · analyzed Sep 1, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Add a Jacobian cone-field regularizer to recurrent dynamics so that tangent directions expand and remain aligned with an unstable cone outside a designated critical neighborhood. The network is not forced to be uniformly expanding: the regularizer is disabled near the critical set, allowing controlled bifurcation-like behavior while exposing where long-horizon sensitivity changes.
Formulas
Mathematical statement
For a differentiable map f and Jacobian Df_x, let C^u_(x,rho) and C^s_(x,rho) be unstable and stable cones with aperture parameter rho in (0,1]. Let C_delta be a critical neighborhood of radius delta, and let K be a compact state region. The paper's hyperbolicity mechanism is strict cone invariance and expansion while an orbit avoids C_delta: Df_x^i(C^u_(x,rho)) is contained in the interior of C^u_(f^i(x),rho), and ||Df_x^i v|| >= c0 theta exp(c1 i) ||v|| for constants c0,c1 > 0 and distortion factor theta. In a neural model, use a unit tangent q_x and impose one-step expansion ||Df_x q_x|| >= exp(gamma), plus alignment of the normalized vector Df_x q_x with q_(f(x)). The critical set prevents imposing this condition where the theorem does not provide it.
Implementation notes
1. Exact integration point: use a recurrent transition h_(t+1) = f_theta(h_t) and apply the regularizer only when h_t is outside C_delta = {h: ||h - h_c||_2 < delta}. Let q_h be a unit tangent direction, initialized randomly and updated by tangent propagation. 2. Pseudocode: sample h from replay trajectories; compute the Jacobian-vector product v = J_h q_h using automatic differentiation; set q_new = v divided by (||v|| + 1e-8); calculate expansion_loss = softplus(gamma - log(||v|| + 1e-8)); calculate angle_loss from the angle between q_new and the stored tangent direction at f_theta(h); if ||h - h_c|| >= delta, add expansion_loss + beta times angle_loss to the task loss; otherwise add zero; update q_h to q_new after detaching it from the graph. 3. Taken from the paper: the critical-set exclusion, invariant-cone condition, and exponential tangent expansion. Estimated empirically: Jacobian-vector products, cone angles, finite-time Lyapunov growth, and the critical-set center h_c. 4. First cheap experiment: train a 32-unit RNN on Lorenz or Mackey-Glass prediction, comparing task-only training against task plus cone regularization. Sweep gamma and delta, and measure g_i = (1/i) log(||J_(h_(i-1)) ... J_(h_0) q_0|| / ||q_0||). The predicted signature is a growth plateau near gamma for trajectories avoiding C_delta, followed by a reproducible drop or sign change after critical-set entry. Increasing delta should shorten the certified expansion interval, while the plateau level should remain approximately unchanged.
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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