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

Abstract

We consider the unidirectional cyclic system of delay differential equationsx˙i(t)=gi(xi(t),xi+1(t−τi),t),0≤i≤N, where the indexes are taken modulo N+1, with N∈N0, τi∈[0,∞), τ:=∑i=0Nτi>0, and for all 0≤i≤N, the feedback functions gi(u,v,t) are continuous in t∈R and C1 in (u,v)∈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 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.