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Parameter estimation by quasilinearization in differential equations with statedependent delays
A statedependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions
1.  Department of Mathematics, Gettysburg College, Gettysburg, PA 173251484 
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 finitedimensional map.
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, SpringerVerlag, New York, 1995. doi: 10.1007/9781461242062. 
[2] 
Ferenc Hartung, Differentiability of solutions with respect to the initial data in differential equations with statedependent delays, Journal of Dynamics and Differential Equations, 23 (2011), 843884. doi: 10.1007/s1088401192181. 
[3] 
Ferenc Hartung, Tibor Krisztin, HansOtto Walther and Jianhong Wu, Functional differential equations with statedependent delays: Theory and applications, in "Handbook of Differential Equations: Ordinary Differential Equations. Vol. III" (eds. A. Cañada, P. Dràbek and A. Fonda), Elsevier/NorthHolland, Amsterdam, (2006), 435545. doi: 10.1016/S18745725(06)80009X. 
[4] 
Anatoli F. Ivanov and Jérôme Losson, Stable rapidly oscillating solutions in delay equations with negative feedback, Differential and Integral Equations, 12 (1999), 811832. 
[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), 268282. 
[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), 166. 
[7] 
Tibor Krisztin and Ovide Arino, The twodimensional attractor of a differential equation with statedependent delay, Journal of Dynamics and Differential Equations, 13 (2001), 453522. doi: 10.1023/A:1016635223074. 
[8] 
Y. Kuang and H. L. Smith, Slowly oscillating periodic solutions of autonomous statedependent delay equations, Nonlinear Analysis: Theory, Methods, & Applications, 19 (1992), 855872. doi: 10.1016/0362546X(92)90055J. 
[9] 
John MalletParet and Roger D. Nussbaum, Boundary layer phenomena for differentialdelay equations with statedependent time lags, I, Arch. Rational Mech. Anal., 120 (1999), 99146. doi: 10.1007/BF00418497. 
[10] 
John MalletParet, Roger D. Nussbaum and Panagiotis Paraskevopoulous, Periodic solutions for functionaldifferential equations with multiple statedependent time lags, Topological Methods in Nonlinear Analysis, 3 (1994), 101162. 
[11] 
John MalletParet and HansOtto 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), 201238. doi: 10.1007/s1088400690684. 
[13] 
HansOtto Walther, Density of Slowly Oscillating Solutions of $x'(t) = f(x(t1))$, Journal of Mathematical Analysis and Applications, 79 (1981), 127140. doi: 10.1016/0022247X(81)900147. 
[14] 
HansOtto Walther, Stable periodic motion of a system with statedependent delay, Differential and Integral Equations, 15 (2002), 923944. 
[15] 
HansOtto Walther, The solution manifold and $C^1$ smoothness for differential equations with statedependent delay, Journal of Differential Equations, 195 (2003), 4665. 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), 259280. doi: 10.1016/00220396(92)90032I. 
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, SpringerVerlag, New York, 1995. doi: 10.1007/9781461242062. 
[2] 
Ferenc Hartung, Differentiability of solutions with respect to the initial data in differential equations with statedependent delays, Journal of Dynamics and Differential Equations, 23 (2011), 843884. doi: 10.1007/s1088401192181. 
[3] 
Ferenc Hartung, Tibor Krisztin, HansOtto Walther and Jianhong Wu, Functional differential equations with statedependent delays: Theory and applications, in "Handbook of Differential Equations: Ordinary Differential Equations. Vol. III" (eds. A. Cañada, P. Dràbek and A. Fonda), Elsevier/NorthHolland, Amsterdam, (2006), 435545. doi: 10.1016/S18745725(06)80009X. 
[4] 
Anatoli F. Ivanov and Jérôme Losson, Stable rapidly oscillating solutions in delay equations with negative feedback, Differential and Integral Equations, 12 (1999), 811832. 
[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), 268282. 
[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), 166. 
[7] 
Tibor Krisztin and Ovide Arino, The twodimensional attractor of a differential equation with statedependent delay, Journal of Dynamics and Differential Equations, 13 (2001), 453522. doi: 10.1023/A:1016635223074. 
[8] 
Y. Kuang and H. L. Smith, Slowly oscillating periodic solutions of autonomous statedependent delay equations, Nonlinear Analysis: Theory, Methods, & Applications, 19 (1992), 855872. doi: 10.1016/0362546X(92)90055J. 
[9] 
John MalletParet and Roger D. Nussbaum, Boundary layer phenomena for differentialdelay equations with statedependent time lags, I, Arch. Rational Mech. Anal., 120 (1999), 99146. doi: 10.1007/BF00418497. 
[10] 
John MalletParet, Roger D. Nussbaum and Panagiotis Paraskevopoulous, Periodic solutions for functionaldifferential equations with multiple statedependent time lags, Topological Methods in Nonlinear Analysis, 3 (1994), 101162. 
[11] 
John MalletParet and HansOtto 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), 201238. doi: 10.1007/s1088400690684. 
[13] 
HansOtto Walther, Density of Slowly Oscillating Solutions of $x'(t) = f(x(t1))$, Journal of Mathematical Analysis and Applications, 79 (1981), 127140. doi: 10.1016/0022247X(81)900147. 
[14] 
HansOtto Walther, Stable periodic motion of a system with statedependent delay, Differential and Integral Equations, 15 (2002), 923944. 
[15] 
HansOtto Walther, The solution manifold and $C^1$ smoothness for differential equations with statedependent delay, Journal of Differential Equations, 195 (2003), 4665. 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), 259280. doi: 10.1016/00220396(92)90032I. 
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