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Long-time behavior of positive solutions of a differential equation with state-dependent delay
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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 |
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 $ |
| $ q_t \mapsto (q(t),q(t - d(q_t))) $ |
| $ q $ |
Mathematics Subject Classification:
Primary: 34K13.
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
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
show all references
References:
| [1] |
B. B. Kennedy, The Poincaré-Bendixson theorem for a class of delay equations with state-dependent delay and monotonic feedback, preprint. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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