A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback

Abstract

We consider a real-valued differential equation \begin{document}$ \begin{equation*} x'(t) = f(x(t - d(x_t))), \end{equation*} $\end{document} with strictly monotonic negative feedback and state-dependent delay, that has a nontrivial periodic solution \begin{document}$ q $\end{document} for which the planar map \begin{document}$ q_t \mapsto (q(t),q(t - d(q_t))) $\end{document} is not injective on the orbit of \begin{document}$ q $\end{document} in phase space. This solution demonstrates that Mallet-Paret and Sell's version of the Poincare-Bendixson theorem for delay equations with constant delay and monotonic feedback does not carry over entirely to the state-dependent delay case.