Existence of a nontrivial slowly oscillating periodic solution for the cyclic delay system

Consider system

with nonlinearities $f_j$, $1\le j\le n$, satisfying all the hypotheses stated in the Introduction. Let $a_0$ and $a_1$ be the constants defined in Lemmas

and

$$, respectively, and suppose that$$

a>\max{a_0,a_1}.

$$**Periodic-solution conjecture.** Under these assumptions, system$$

has a nontrivial slowly oscillating periodic solution.

The conjecture predicts the existence of a periodic solution once the parameter $a$ exceeds the thresholds controlling the relevant real and purely imaginary eigenvalues. The supplied text gives numerical evidence involving pairs of complex conjugate characteristic roots, but does not state a resolution of the conjecture.