Attractor-merging Crises and Intermittency in Reservoir Computing

The paper documents, via simulations, an attractor‑merging crisis in binary‑feedback ESNs as the global gain ρ̂ increases, and observes post‑crisis intermittency with a power‑law decay of the mean residence time 〈τ〉 with respect to ρ̂ − ρ̂c; the onset is plausibly explained as a collision between the expanding attractor and the preimage set M of the decision boundary H = {wout^⊤x = 0} (with a computable surrogate S of “switching points”) in SM Sec. IV . Analogous behavior is shown for a closed‑loop (double‑Lorenz) setting, along with a caveat that increasing ρ̂ does not always induce a crisis, as other bifurcations can occur . The model’s solution offers a crisis‑theoretic sketch consistent with the narrative (basin‑boundary/preimage contact, unstable periodic‑orbit mediator, and power‑law intermittency) but asserts extra smoothness and genericity statements (e.g., that Mρ̂ varies smoothly and that an AMC must occur in any generic one‑parameter family) that are not established for these high‑dimensional, piecewise‑smooth maps. Hence, both accounts are insightful but incomplete: the paper lacks a full proof and relies on “likely” mechanisms, while the model includes technically doubtful smoothness claims and leaves key hypotheses (hyperbolicity, transversality, continuity of attractor families) implicit.