Chaos in the fractional-order Hala system: closed-form stability thresholds and a quench–order chaos map

Fractional-order chaotic flows are commonly assumed to need unstable equilibria, which through Matignon’s stability theorem fixes a minimum commensurate order for double-scroll chaos. We test this minimum order in the fractional-order Hala system, a Lorenz- or Chen type flow with the quench −γ(x2+y2)z, whose equilibria are known exactly for every order. We derive closed form Hopf thresholds, γH = (293 − √65761)/26244 (Lorenz host) and (181 − √631)/7938 (Chen host), and hence the Matignon boundary qc(γ) in the plane of quench factor γ and Caputo order q. Chaos is mapped over this plane with a full-memory predictor–corrector and detected with the 0–1 test and twin-trajectory divergence, both valid for systems with memory. On the Lorenz host chaos persists below the Matignon boundary in 45 of 256 cells: the chaotic region is the Matignon region translated by (0.68±0.09)×10−3 in γ, the width of the integer-order window in which chaos coexists with stable equilibria after a subcritical Hopf bifurcation. Near the boundary this chaos survives the longest horizon tested, with Lyapunov exponents comparable to unquenched chaos; deeper below, it is a transient whose lifetime falls with depth. On the Chen host fractional memory extends chaos from γ ≈ 4.5 ×10−3 at integer order to 1.65 × 10−2, where it ends on the Matignon boundary. The minimum order is sharp when the quench acts only through the equilibria, and it fails by a predictable margin when the integer-order flow has a coexistence window.