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Long-time behavior of positive solutions of a differential equation with state-dependent delay

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

 Department of Mathematics, Gettysburg College, 300 N. Washington St., Gettysburg, PA 17325, USA

Received  April 2017 Revised  July 2017 Published  January 2019

We consider a real-valued differential equation
 $\begin{equation*} x'(t) = f(x(t - d(x_t))), \end{equation*}$
with strictly monotonic negative feedback and state-dependent delay, that has a nontrivial periodic solution
 $q$
for which the planar map
 $q_t \mapsto (q(t),q(t - d(q_t)))$
is not injective on the orbit of
 $q$
in phase space. This solution demonstrates that Mallet-Paret and Sell's version of the Poincaré-Bendixson theorem for delay equations with constant delay and monotonic feedback does not carry over entirely to the state-dependent delay case.
Citation: Benjamin B. Kennedy. A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020003
##### References:
 [1] B. B. Kennedy, The Poincaré-Bendixson theorem for a class of delay equations with state-dependent delay and monotonic feedback, preprint.Google Scholar [2] T. Krisztin and O. Arino, The two-dimensional attractor of a differential equation with state-dependent delay, Journal of Dynamics and Differential Equations, 13 (2001), 453-522. doi: 10.1023/A:1016635223074. Google Scholar [3] J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags, I, Arch. Rational Mech. Anal., 120 (1992), 99-146. doi: 10.1007/BF00418497. Google Scholar [4] J. Mallet-Paret and G. R. Sell, Systems of differential delay equations: The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, Journal of Differential Equations, 125 (1996), 441-489. doi: 10.1006/jdeq.1996.0037. Google Scholar [5] H.-O. Walther, Algebraic-delay differential systems, state-dependent delay, and temporal order of reactions, Journal of Dynamics and Differential Equations, 21 (2009), 195-232. doi: 10.1007/s10884-009-9129-6. Google Scholar [6] H.-O. Walther, A homoclinic loop generated by variable delay, Journal of Dynamics and Differential Equations, 27 (2015), 1101-1139. doi: 10.1007/s10884-013-9333-2. Google Scholar

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##### References:
 [1] B. B. Kennedy, The Poincaré-Bendixson theorem for a class of delay equations with state-dependent delay and monotonic feedback, preprint.Google Scholar [2] T. Krisztin and O. Arino, The two-dimensional attractor of a differential equation with state-dependent delay, Journal of Dynamics and Differential Equations, 13 (2001), 453-522. doi: 10.1023/A:1016635223074. Google Scholar [3] J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags, I, Arch. Rational Mech. Anal., 120 (1992), 99-146. doi: 10.1007/BF00418497. Google Scholar [4] J. Mallet-Paret and G. R. Sell, Systems of differential delay equations: The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, Journal of Differential Equations, 125 (1996), 441-489. doi: 10.1006/jdeq.1996.0037. Google Scholar [5] H.-O. Walther, Algebraic-delay differential systems, state-dependent delay, and temporal order of reactions, Journal of Dynamics and Differential Equations, 21 (2009), 195-232. doi: 10.1007/s10884-009-9129-6. Google Scholar [6] H.-O. Walther, A homoclinic loop generated by variable delay, Journal of Dynamics and Differential Equations, 27 (2015), 1101-1139. doi: 10.1007/s10884-013-9333-2. Google Scholar
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