The limiting version of the Mackey–Glass delay differential equation x0(t) =?ax(t) + b f(x(t ? 1)) is considered where a, b are positive reals, and f(ξ) = ξ forξ ∈ [0, 1), f(1) = 1/2, and f(ξ) = 0 for ξ > 1. For every a > 0 we prove theexistence of an ε0 = ε0(a) > 0 so that for all b ∈ (a, a + ε0) there exists a periodicsolution p = p(a, b) : R → (0, ∞) with minimal period ω(a, b) such that ω(a, b) → ∞as b → a+. A consequence is that, for each a > 0, b ∈ (a, a + ε0(a)) and sufficientlylarge n, the classical Mackey–Glass equation y0(t) = ?ay(t) + by(t ? 1)/[1 + yn(t ? 1)]has an orbitally asymptotically stable periodic orbit, as well, close to the periodic orbitof the limiting equation.
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