The unstable set of a periodic orbit for delayed positive feedback

Abstract

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 $\mathcal{O}_{p}$ with two Floquet multipliers outside the unit circle, and we intend to characterize the geometric structure of its unstable set $\mathcal{W}^{u}\left(\mathcal{O}_{p}\right)$. We prove that $\mathcal{W}^{u}\left(\mathcal{O}_{p}\right)$ is a three-dimensional $C^{1}$-submanifold of the phase space and admits a smooth global graph representation. Within $\mathcal{W}^{u}\left(\mathcal{O}_{p}\right)$, there exist heteroclinic connections from $\mathcal{O}_{p}$ to three different periodic orbits. These connecting sets are two-dimensional $C^{1}$-submanifolds of $\mathcal{W}^{u}\left(\mathcal{O}_{p}\right)$ and homeomorphic to the two-dimensional open annulus. They form $C^{1}$-smooth separatrices in the sense that they divide the points of $\mathcal{W}^{u}\left(\mathcal{O}_{p}\right)$ into three subsets according to their $\omega$-limit sets.