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Longtime behavior of positive solutions of a differential equation with statedependent delay
A periodic solution with nonsimple oscillation for an equation with statedependent delay and strictly monotonic negative feedback
Department of Mathematics, Gettysburg College, 300 N. Washington St., Gettysburg, PA 17325, USA 
$ \begin{equation*} x'(t) = f(x(t  d(x_t))), \end{equation*} $ 
$ q $ 
$ q_t \mapsto (q(t),q(t  d(q_t))) $ 
$ q $ 
References:
[1] 
B. B. Kennedy, The PoincaréBendixson theorem for a class of delay equations with statedependent delay and monotonic feedback, preprint. 
[2] 
T. Krisztin and O. Arino, The twodimensional attractor of a differential equation with statedependent delay, Journal of Dynamics and Differential Equations, 13 (2001), 453522. doi: 10.1023/A:1016635223074. 
[3] 
J. MalletParet and R. D. Nussbaum, Boundary layer phenomena for differentialdelay equations with statedependent time lags, I, Arch. Rational Mech. Anal., 120 (1992), 99146. doi: 10.1007/BF00418497. 
[4] 
J. MalletParet and G. R. Sell, Systems of differential delay equations: The PoincaréBendixson theorem for monotone cyclic feedback systems with delay, Journal of Differential Equations, 125 (1996), 441489. doi: 10.1006/jdeq.1996.0037. 
[5] 
H.O. Walther, Algebraicdelay differential systems, statedependent delay, and temporal order of reactions, Journal of Dynamics and Differential Equations, 21 (2009), 195232. doi: 10.1007/s1088400991296. 
[6] 
H.O. Walther, A homoclinic loop generated by variable delay, Journal of Dynamics and Differential Equations, 27 (2015), 11011139. doi: 10.1007/s1088401393332. 
show all references
References:
[1] 
B. B. Kennedy, The PoincaréBendixson theorem for a class of delay equations with statedependent delay and monotonic feedback, preprint. 
[2] 
T. Krisztin and O. Arino, The twodimensional attractor of a differential equation with statedependent delay, Journal of Dynamics and Differential Equations, 13 (2001), 453522. doi: 10.1023/A:1016635223074. 
[3] 
J. MalletParet and R. D. Nussbaum, Boundary layer phenomena for differentialdelay equations with statedependent time lags, I, Arch. Rational Mech. Anal., 120 (1992), 99146. doi: 10.1007/BF00418497. 
[4] 
J. MalletParet and G. R. Sell, Systems of differential delay equations: The PoincaréBendixson theorem for monotone cyclic feedback systems with delay, Journal of Differential Equations, 125 (1996), 441489. doi: 10.1006/jdeq.1996.0037. 
[5] 
H.O. Walther, Algebraicdelay differential systems, statedependent delay, and temporal order of reactions, Journal of Dynamics and Differential Equations, 21 (2009), 195232. doi: 10.1007/s1088400991296. 
[6] 
H.O. Walther, A homoclinic loop generated by variable delay, Journal of Dynamics and Differential Equations, 27 (2015), 11011139. doi: 10.1007/s1088401393332. 
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