Abstract
We prove a principle of linearized stability for semiflows generated by neutral functional differential equations of the form
The state space is a closed subset in a manifold of C 2-functions. Applications include equations with state-dependent delay, as for example
with nonlinear functions \({A:\mathbb{R}\to\mathbb{R}}\) and \({d:\mathbb{R}\to(-h,0), f:\mathbb{R}\to\mathbb{R}, r:\mathbb{R}\to[-h,0]}\).
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References
Arino O., Sanchez E.: A saddle point theorem for functional state-dependent delay equations. Discret. Contin. Dyn. Syst. 12, 687–722 (2005)
Bartha M.: On stability properties for neutral differential equations with state-dependent delay. Differ. Equ. Dyn. Syst. 7, 197–220 (1999)
Brunovský P., Erdélyi A., Walther H.O.: On a model of a currency exchange rate—local stability and periodic solutions. J. Dyn. Differ. Equ. 16, 393–432 (2004)
Hale, J.K., Meyer, K.R.: A class of functional equations of neutral type. Mem. Am. Math. Soc. 76 (1967)
Hale J.K., Verduyn Lunel S.M.: Introduction to Functional Differential Equations. Springer, New York (1993)
Hartung F.: Linearized stability for a class of neutral functional differential equations with state-dependent delays. Nonlinear Anal. Theory Methods Appl. 69, 1629–1643 (2008)
Hartung F., Krisztin T., Walther H.O., Wu J.: Functional differential equations with state-dependent delay: theory and applications. In: Canada, A., Drabek, P., Fonda, A. (eds) Handbook Of Differential Equations, Ordinary Differential Equations, pp. 435–545. Elsevier Science B. V., North Holland, Amsterdam (2006)
Henry D.: Linear autonomous neutral functional differential equations. J. Differ. Equ. 15, 106–128 (1974)
Krishnan H.P.: Existence of unstable manifolds for a certain class of delay differential equations. Electron. J. Differ. Equ. 32, 1–13 (2002)
Krisztin T.: A local unstable manifold for differential equations with state-dependent delay. Discret. Contin. Dyn. Syst. 9, 993–1028 (2003)
Krisztin, T.: C 1-smoothness of center manifolds for differential equations with state-dependent delays. In: Nonlinear Dynamics and Evolution Equations, Fields Institute Communications 48, 213–226 (2006)
Mallet-Paret J., Nussbaum R.D., Paraskevopoulos P.: Periodic solutions for functional differential equations with multiple state-dependent time lags. Topol. Methods Nonlinear Anal. 3, 101–162 (1994)
Qesmi R., Walther H.O.: Center-stable manifolds for differential equations with state-dependent delay. Discret. Contin. Dyn. Syst. 23, 1009–1033 (2009)
Stumpf, E.: On a differential equation with state-dependent delay: A global center-unstable manifold bordered by a periodic orbit. Doctoral dissertation, Universität Hamburg (2010)
Walther H.O.: The solution manifold and C 1-smoothness of solution operators for differential equations with state dependent delay. J. Differ. Equ. 195, 46–65 (2003)
Walther, H.O.: Smoothness properties of semiflows for differential equations with state dependent delay. Russian. In: Proceedings of the International Conference on Differential and Functional Differential Equations, Moscow, 2002, vol. 1, pp. 40–55, Moscow State Aviation Institute (MAI), Moscow 2003. English version: Journal of the Mathematical Sciences 124, 5193–5207 (2004)
Walther H.O.: Convergence to square waves in a price model with delay. Discret. Contin. Dyn. Syst. 13, 1325–1342 (2005)
Walther H.O.: Bifurcation of periodic solutions with large periods for a delay differential equation. Annali di Matematica Pura ed Applicata 185, 577–611 (2006)
Walther, H.O.: Semiflows for neutral equations with state-dependent delays. Preprint (2009)
Walther H.O.: Linearized stability for semiflows generated by a class of neutral equations, with applications to state-dependent delays. J. Differ. Equ. Dyn. Syst. 22, 439–462 (2010)
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Walther, HO. More on Linearized Stability for Neutral Equations with State-Dependent Delays. Differ Equ Dyn Syst 19, 315–333 (2011). https://doi.org/10.1007/s12591-011-0093-3
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DOI: https://doi.org/10.1007/s12591-011-0093-3