Absence of small solutions and existence of Morse decomposition for a cyclic system of delay differential equations

We consider the unidirectional cyclic system of delay differential equations where the indexes are taken modulo N 1, with N E No, 2l E [0, Do), t := ENi 0 2l > 0, and for all 0 < i < N, the feedback functions gi (u, v, t) are continuous in t E R and C1 in (u, v) E R2, and each of them satisfies either a positive or a negative feedback condition in the delayed term. We show that all components of a superexponential solution (i.e. nonzero solutions that converge to zero faster than any exponential function) must have infinitely many sign -changes on any interval of length 7. As a corollary we obtain that if a backwards -bounded global pullback attractor exists, then it does not contain any superexponential solutions. In the autonomous case we also prove that the global attractor possesses a Morse decomposition that is based on a discrete Lyapunov function. This generalizes former results by Mallet-Paret (1988) [28] and Polner (2002) [37] in which the scalar case was studied. 2020 Elsevier Inc. All rights reserved.