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Stable periodic solutions for delay equations with positive feedback - a computer-assisted proof

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  • We study the delay equation $\dot{x}(t)=-\mu x(t)+f(x(t-1))$ with $\mu>0$ and a nonmonotone $C^1$-function $f$ obeying $x f(x)>0$ (positive feedback) outside a small neighbourhood of zero. By means of a computer-assisted method we prove the existence of asymptotically orbitally stable periodic solutions. The main idea behind our proof is the reduction of the infinite-dimensional dynamics to a finite-dimensional map. In particular, for two classes of nonlinearities $f$ we construct two types of solutions, the dynamics of which is reduced to a one- and a two-dimensional map, respectively.
    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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