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| 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 the scalar delay equation \[
\dot{x}\left(t\right)=-\mu x\left(t\right)+f\left(x\left(t-1\right)\right),
\]where $\mu>0$ and $f$ is a strictly increasing, continuously differentiable nonlinearity (positive feedback case). We have shown that the unstable sets of the large-amplitude periodic orbits constitute the global attractor besides the so called spindle-like structures.
In this talk 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 show 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.
The research of Gabriella Vas was supported by the European Union and the State of Hungary, co-financed by the European Social Fund in the framework of TAMOP-4.2.4.A/2-11/1-2012-0001 'National Excellence Program'. |
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