# American Institute of Mathematical Sciences

August  2013, 18(6): 1633-1650. doi: 10.3934/dcdsb.2013.18.1633

## A state-dependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions

 1 Department of Mathematics, Gettysburg College, Gettysburg, PA 17325-1484

Received  December 2011 Revised  April 2012 Published  March 2013

We consider state-dependent delay equations of the form $x'(t) = f(x(t - d(x(t))))$ where $d$ is smooth and $f$ is smooth, bounded, nonincreasing, and satisfies the negative feedback condition $xf(x) < 0$ for $x \neq 0$. We identify a special family of such equations each of which has a rapidly oscillating" periodic solution $p$. The initial segment $p_0$ of $p$ is the fixed point of a return map $R$ that is differentiable in an appropriate setting.
We show that, although all the periodic solutions $p$ we consider are unstable, the stability can be made arbitrarily mild in the sense that, given $\epsilon > 0$, we can choose $f$ and $d$ such that the spectral radius of the derivative of $R$ at $p_0$ is less than $1 + \epsilon$. The spectral radii are computed via a semiconjugacy of $R$ with a finite-dimensional map.
Citation: Benjamin B. Kennedy. A state-dependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1633-1650. doi: 10.3934/dcdsb.2013.18.1633
##### References:
 [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 (1995). doi: 10.1007/978-1-4612-4206-2. Google Scholar [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. doi: 10.1007/s10884-011-9218-1. Google Scholar [3] Ferenc Hartung, Tibor Krisztin, Hans-Otto Walther and Jianhong Wu, Functional differential equations with state-dependent delays: Theory and applications,, in, (2006), 435. doi: 10.1016/S1874-5725(06)80009-X. Google Scholar [4] Anatoli F. Ivanov and Jérôme Losson, Stable rapidly oscillating solutions in delay equations with negative feedback,, Differential and Integral Equations, 12 (1999), 811. Google Scholar [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. Google Scholar [6] Benjamin Kennedy, Stability and instability for periodic solutions of delay equations with "steplike" feedback,, Electronic Journal of Qualitative Theory of Differential Equations, 8 (2011), 1. Google Scholar [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. doi: 10.1023/A:1016635223074. Google Scholar [8] Y. Kuang and H. L. Smith, Slowly oscillating periodic solutions of autonomous state-dependent delay equations,, Nonlinear Analysis: Theory, 19 (1992), 855. doi: 10.1016/0362-546X(92)90055-J. Google Scholar [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. doi: 10.1007/BF00418497. Google Scholar [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. Google Scholar [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., (). Google Scholar [12] D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions,, Journal of Dynamics and Differential Equations, 20 (2008), 201. doi: 10.1007/s10884-006-9068-4. Google Scholar [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. doi: 10.1016/0022-247X(81)90014-7. Google Scholar [14] Hans-Otto Walther, Stable periodic motion of a system with state-dependent delay,, Differential and Integral Equations, 15 (2002), 923. Google Scholar [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. doi: 10.1016/j.jde.2003.07.001. Google Scholar [16] Xianwen Xie, The multiplier equation and its application to $S$-solutions of a differential delay equation,, Journal of Differential Equations, 95 (1992), 259. doi: 10.1016/0022-0396(92)90032-I. Google Scholar

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##### References:
 [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 (1995). doi: 10.1007/978-1-4612-4206-2. Google Scholar [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. doi: 10.1007/s10884-011-9218-1. Google Scholar [3] Ferenc Hartung, Tibor Krisztin, Hans-Otto Walther and Jianhong Wu, Functional differential equations with state-dependent delays: Theory and applications,, in, (2006), 435. doi: 10.1016/S1874-5725(06)80009-X. Google Scholar [4] Anatoli F. Ivanov and Jérôme Losson, Stable rapidly oscillating solutions in delay equations with negative feedback,, Differential and Integral Equations, 12 (1999), 811. Google Scholar [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. Google Scholar [6] Benjamin Kennedy, Stability and instability for periodic solutions of delay equations with "steplike" feedback,, Electronic Journal of Qualitative Theory of Differential Equations, 8 (2011), 1. Google Scholar [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. doi: 10.1023/A:1016635223074. Google Scholar [8] Y. Kuang and H. L. Smith, Slowly oscillating periodic solutions of autonomous state-dependent delay equations,, Nonlinear Analysis: Theory, 19 (1992), 855. doi: 10.1016/0362-546X(92)90055-J. Google Scholar [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. doi: 10.1007/BF00418497. Google Scholar [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. Google Scholar [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., (). Google Scholar [12] D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions,, Journal of Dynamics and Differential Equations, 20 (2008), 201. doi: 10.1007/s10884-006-9068-4. Google Scholar [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. doi: 10.1016/0022-247X(81)90014-7. Google Scholar [14] Hans-Otto Walther, Stable periodic motion of a system with state-dependent delay,, Differential and Integral Equations, 15 (2002), 923. Google Scholar [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. doi: 10.1016/j.jde.2003.07.001. Google Scholar [16] Xianwen Xie, The multiplier equation and its application to $S$-solutions of a differential delay equation,, Journal of Differential Equations, 95 (1992), 259. doi: 10.1016/0022-0396(92)90032-I. Google Scholar
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