The Unstable Set of a Periodic Orbit for Delayed Positive Feedback

摘要

In the paper [Large-amplitude periodic solutions for differential equations with delayed monotone positive feedback, JDDE 23 (2011), no. 4, 727-790], we have constructed large-amplitude periodic orbits for an equation with delayed monotone positive feedback. We have shown that the unstable sets of the large-amplitude periodic orbits constitute the global attractor besides spindle-like structures. In this paper we focus on a large-amplitude periodic orbit with two Floquet multipliers outside the unit circle, and we intend to characterize the geometric structure of its unstable set . We prove that is a three-dimensional -submanifold of the phase space and admits a smooth global graph representation. Within , there exist heteroclinic connections from to three different periodic orbits. These connecting sets are two-dimensional -submanifolds of and homeomorphic to the two-dimensional open annulus. They form -smooth separatrices in the sense that they divide the points of into three subsets according to their -limit sets.