Configurations of periodic orbits for equations with delayed positive feedback

Abstract

We consider scalar delay differential equations of the formx˙(t)=−μx(t)+f(x(t−1)), where μ>0 and f is a nondecreasing C1-function. If χ is a fixed point of fμ:R∋u↦f(u)/μ∈R with fμ′(χ)>1, then [−1,0]∋s↦χ∈R is an unstable equilibrium. A periodic solution is said to have large amplitude if it oscillates about at least two fixed points χ−<χ+ of fμ with fμ′(χ−)>1 and fμ′(χ+)>1. We investigate what type of large-amplitude periodic solutions may exist at the same time when the number of such fixed points (and hence the number of unstable equilibria) is an arbitrary integer N≥2. It is shown that the number of different configurations equals the number of ways in which N symbols can be parenthesized. The location of the Floquet multipliers of the corresponding periodic orbits is also discussed.