BISTABILITY, BIFURCATIONS AND CHAOS IN THE MACKEY-GLASS EQUATION

The paper is an explicitly numerical bifurcation study: it identifies a cusp of folds of periodic orbits around (n, τ) ≈ (15.16, 1.1) with ensuing bistability between two stable cycles, and later between a cycle and a chaotic attractor; it documents a SNOCA-type destruction of chaos at τ = 2 near n ≈ 11.21485; and it finds a window near (n, τ) ≈ (8.1, 2.6) where subcritical period-doublings coexist with chaos. The authors emphasize that rigorous chaos for Mackey–Glass remains elusive, and they rely on DDE-BIFTOOL continuation and Lyapunov-exponent computations to support their claims, not a proof. See their explicit statement that a rigorous proof remains elusive, their location of the cusp and bistability, the subcritical PD window near n = 8.1, τ ≈ 2.6, and the SNOCA scenario at τ = 2 with fold at n = 11.21485 . The model likewise presents a reduction strategy (center-manifold/normal-form conditions for cusp/flip and a topological horseshoe for chaos) but performs no rigorous validation. Both sources therefore point to the problem as open at the stated cutoff; the model’s outline is plausible but incomplete, and the paper does not claim a theorem.