# 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 and 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, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2. [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. doi: 10.1007/s10884-011-9218-1. [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. doi: 10.1016/S1874-5725(06)80009-X. [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. doi: 10.1023/A:1016635223074. [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. doi: 10.1016/0362-546X(92)90055-J. [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. doi: 10.1007/BF00418497. [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. doi: 10.1007/s10884-006-9068-4. [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. doi: 10.1016/0022-247X(81)90014-7. [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. doi: 10.1016/j.jde.2003.07.001. [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. doi: 10.1016/0022-0396(92)90032-I.

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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, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2. [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. doi: 10.1007/s10884-011-9218-1. [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. doi: 10.1016/S1874-5725(06)80009-X. [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. doi: 10.1023/A:1016635223074. [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. doi: 10.1016/0362-546X(92)90055-J. [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. doi: 10.1007/BF00418497. [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. doi: 10.1007/s10884-006-9068-4. [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. doi: 10.1016/0022-247X(81)90014-7. [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. doi: 10.1016/j.jde.2003.07.001. [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. doi: 10.1016/0022-0396(92)90032-I.
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