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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

A state-dependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions
Pages: 1633 - 1650, Issue 6, August 2013

doi:10.3934/dcdsb.2013.18.1633      Abstract        References        Full text (505.0K)           Related Articles

Benjamin B. Kennedy - Department of Mathematics, Gettysburg College, Gettysburg, PA 17325-1484, United States (email)

1 O. Diekmann, S. A. Van Gils, S. M. Verduyn Lunel and H.-O. Walther, "Delay Equations. Funtional, Complex, and Nonlinear Analysis," Applied Mathematical Sciences, 110, Springer-Verlag, New York, 1995.       
2 Ferenc Hartung, Differentiability of solutions with respect to the initial data in differential equations with state-dependent delays, Journal of Dynamics and Differential Equations, 23 (2011), 843-884.       
3 Ferenc Hartung, Tibor Krisztin, Hans-Otto Walther and Jianhong Wu, Functional differential equations with state-dependent delays: Theory and applications, in "Handbook of Differential Equations: Ordinary Differential Equations. Vol. III" (eds. A. Cañada, P. Dràbek and A. Fonda), Elsevier/North-Holland, Amsterdam, (2006), 435-545.       
4 Anatoli F. Ivanov and Jérôme Losson, Stable rapidly oscillating solutions in delay equations with negative feedback, Differential and Integral Equations, 12 (1999), 811-832.       
5 James L. Kaplan and James A. Yorke, On the stability of a periodic solution of a differential delay equation, SIAM Journal of Mathematical Analysis, 6 (1975), 268-282.       
6 Benjamin Kennedy, Stability and instability for periodic solutions of delay equations with "steplike" feedback, Electronic Journal of Qualitative Theory of Differential Equations, Proceedings of the 9th Colloquium on the Qualitative Theory of Differential Equations, 8 (2011), 1-66.
7 Tibor Krisztin and Ovide Arino, The two-dimensional attractor of a differential equation with state-dependent delay, Journal of Dynamics and Differential Equations, 13 (2001), 453-522.       
8 Y. Kuang and H. L. Smith, Slowly oscillating periodic solutions of autonomous state-dependent delay equations, Nonlinear Analysis: Theory, Methods, & Applications, 19 (1992), 855-872.       
9 John Mallet-Paret and Roger D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags, I, Arch. Rational Mech. Anal., 120 (1999), 99-146.       
10 John Mallet-Paret, Roger D. Nussbaum and Panagiotis Paraskevopoulous, Periodic solutions for functional-differential equations with multiple state-dependent time lags, Topological Methods in Nonlinear Analysis, 3 (1994), 101-162.       
11 John Mallet-Paret and Hans-Otto Walther, Rapid oscillations are rare in scalar systems governed by monotone negative feedback with a time lag, preprint.
12 D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, Journal of Dynamics and Differential Equations, 20 (2008), 201-238.       
13 Hans-Otto Walther, Density of Slowly Oscillating Solutions of $x'(t) = -f(x(t-1))$, Journal of Mathematical Analysis and Applications, 79 (1981), 127-140.       
14 Hans-Otto Walther, Stable periodic motion of a system with state-dependent delay, Differential and Integral Equations, 15 (2002), 923-944.       
15 Hans-Otto Walther, The solution manifold and $C^1$ smoothness for differential equations with state-dependent delay, Journal of Differential Equations, 195 (2003), 46-65.       
16 Xianwen Xie, The multiplier equation and its application to $S$-solutions of a differential delay equation, Journal of Differential Equations, 95 (1992), 259-280.       

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