Stable periodic orbits for the Mackey-Glass equation

[摘要] We study the classical Mackey-Glass delay differential equation x'(t) = -ax(t) + bf(n)(x(t - 1)) where a, b, n are positive reals, and f(n)(xi) = xi/[1 + xi(n)] for xi >= 0. As a limiting (n -> infinity) case we also consider the discontinuous equation x'(t)=-ax(t)+bf(x(t - 1)) where f(xi) = xi for xi [0, 1), f(1) = 1/2, and f(xi) = 0 for xi > 1. First, for certain parameter values b > a > 0, an orbitally asymptotically stable periodic orbit is constructed for the discontinuous equation. Then it is shown that for large values of n, and with the same parameters a, b, the Mackey-Glass equation also has an orbitally asymptotically stable periodic orbit near to the periodic orbit of the discontinuous equation. Although the obtained periodic orbits are stable, their projections R (sic) t bar right arrow (x(t), (x(t - 1))) is an element of R-2 can be complicated. (C) 2021 The Author(s). Published by Elsevier Inc.

[发布日期] 2021-09-25

[关键词] Mackey-Glass equation;Limiting nonlinearity;Periodic orbit;Return map;Computer-assisted proof