Maximal attractors for perturbations of unimodal maps near a homoclinic tangency
arXiv:2608.18761
2026
Dynamics
2 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper offers a transferable mechanism for constructing maximal attractors from trapping regions and for certifying hyperbolic behavior through invariant cone fields. The core neural-network application is to constrain recurrent or state-space transitions so that a compact hidden-state region maps strictly into its interior, while separately monitoring Jacobian expansion away from a critical set where the certificate is expected to fail. This gives falsifiable signatures: bounded long-horizon states, convergence of finite-time reachable sets, and a sharp change in tangent growth when trajectories enter the critical neighborhood.
Ideas from this paper
△ Mechanism confirmed, baseline not beaten
2026
Constrain a recurrent neural transition to map a compact learned-state region strictly into its interior, creating a neural analogue of the paper's maximal attractor. Unlike simple spectral normalization, this permits a nontrivial invariant set and can preserve task-relevant recurrent dynamics while preventing long-horizon state escape.
Useful7/10
Difficulty5/10
Novelty7/10
Unverified
2026
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.
Useful6/10
Difficulty7/10
Novelty8/10