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.

[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.

[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.

[4]

Anatoli F. Ivanov and Jérôme Losson, Stable rapidly oscillating solutions in delay equations with negative feedback,, Differential and Integral Equations, 12 (1999), 811.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 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. 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.

[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.

[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.

show all references

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.

[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.

[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.

[4]

Anatoli F. Ivanov and Jérôme Losson, Stable rapidly oscillating solutions in delay equations with negative feedback,, Differential and Integral Equations, 12 (1999), 811.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 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. 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.

[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.

[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.

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