Infinite Number of Stable Periodic Solutions for an Equation with Negative Feedback

For all μ > 0, a locally Lipschitz continuous map f with xf (x) > 0, x ∈ R\{0}, is constructed, such that the scalar equation ẋ (t) = −μx (t)−f (x (t− 1)) with delayed negative feedback has an infinite number of periodic orbits. All periodic solutions defining these orbits oscillate slowly around 0 in the sense that they admit at most one sign change in each interval of length of 1. Moreover, if f is continuously differentiable, then the periodic orbits are hyperbolic and stable. In this example f is not bounded, but the Lipschitz constants for the restrictions of f to certain intervals are small. Based on this property, an infinite sequence of contracting return maps is given. Their fixed points are the initial segments of the periodic solutions.