Global Basin Reorganization Without Local Bifurcation in an Asymmetric Gradient System

Linear stability analysis around fixed points is the standard tool for understanding transitions in dynamical systems. We study a factor-coupled gradient system with asymmetric cumulative coupling whose escape probability varies with the asymmetry parameter γ across γ ≈ 0.20–0.40. Analytical derivation of the linearized dynamics shows that all eigenvalues of the interior fixed point remain positive for all γ > −1 — with det M = α⁶(1+γ)⁴/2 > 0 — so the fixed point is a source at every γ and no local bifurcation exists. Multi-seed simulations (27,500 trajectories) show that the γ-dependence is instead a continuous, seed-robust reorganization of basin volumes. Earlier single-seed sampling made this shift appear as a sharp threshold; higher-count sampling in the companion analysis (doi:10.5281/zenodo.18754845) resolves it as a stable landscape with no universal γ*. Local stability is γ-invariant; all γ-dependence of outcomes lives in the basin geometry. α(d,β) = 1 + d/(d+β), β = 6/23.--Author: James E. Dunn — Independent Researcher, Hydrogen Lifecycle Research ProgrammeORCID: https://orcid.org/0009-0005-2679-6574Corpus: https://zenodo.org/search?q=creators.orcid:0009-0005-2679-6574License: CC BY 4.0 International