LAUSR

Abstract We consider the unidirectional cyclic system of delay differential equations x ˙ i ( t ) = g i ( x i ( t ) , x i + 1 ( t − τ i ) , t ) , 0 ≤ i ≤ N , where the indexes are taken modulo N + 1 , with N ∈ N 0 , τ i ∈ [ 0 , ∞ ) , τ : = ∑ i = 0 N τ i > 0 , and for all 0 ≤ i ≤ N , the feedback functions g i ( u , v , t ) are continuous in t ∈ R and C 1 in ( u , v ) ∈ R 2 , 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 τ. 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.