June  2016, 36(6): 3445-3461. doi: 10.3934/dcds.2016.36.3445

Local stability analysis of differential equations with state-dependent delay

1. 

Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany

Received  February 2015 Revised  September 2015 Published  December 2015

In the present article, we discuss some aspects of the local stability analysis for a class of abstract functional differential equations. This is done under smoothness assumptions which are often satisfied in the presence of a state-dependent delay. Apart from recapitulating the two classical principles of linearized stability and instability, we deduce the analogon of the Pliss reduction principle for the class of differential equations under consideration. This reduction principle enables to determine the local stability properties of a solution in the situation where the linearization does not have any eigenvalues with positive real part but at least one eigenvalue on the imaginary axis.
Citation: Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445
References:
[1]

H. Amann, Ordinary Differential Equations. An Introduction to Nonlinear Analysis (Translated from the German by G. Metzen), de Gruyter Studies in Mathematics, 13, Walter de Gruyter & Co., Berlin, 1990. doi: 10.1515/9783110853698.

[2]

P. Brunovský, A. Erdélyi and H.-O. Walther, On a model of a currency exchange rate - local stability and periodic solutions, J. Dynam. Differential Equations, 16 (2004), 393-432. doi: 10.1007/s10884-004-4285-1.

[3]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations. Functional, Complex, and Nonlinear Analysis, Applied Mathematical Sciences, 110, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.

[4]

P. Getto and M. Waurick, A differential equation with state-dependent delay from cell population biology, preprint, arXiv:1411.3097v1.

[5]

F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delay, in Hand. Differ. Equ.: Ordinary Differential Equations, 3, Elsevier/North-Holland, Amsterdam, 2006, 435-545. doi: 10.1016/S1874-5725(06)80009-X.

[6]

T. Krisztin, A local unstable manifold for differential equations with state-dependent delay, Discrete Contin. Dyn. Syst., 9 (2003), 993-1028. doi: 10.3934/dcds.2003.9.993.

[7]

T. Krisztin, $C^{1}$-smoothness of center manifolds for differential equations with state-dependent delay, in Nonlinear Dynamics and Evolution Equations (eds. H. Brunner et al.), Fields Inst. Commun., 48, Amer. Math. Soc., Providence, 2006, 213-226.

[8]

V. A. Pliss, A reduction principle in the theory of stability of motion (Russian), Izv. Akad. Nauk. SSSR Ser. Mat., 28 (1964), 1297-1324.

[9]

R. Qesmi and H.-O. Walther, Center-stable manifolds for differential equations with state-dependent delays, Discrete Contin. Dyn. Syst., 23 (2009), 1009-1033. doi: 10.3934/dcds.2009.23.1009.

[10]

E. Stumpf, On a differential equation with state-dependent delay: A global center-unstable manifold bordered by a periodic orbit, Doctoral dissertation, University of Hamburg, 2010. Available from: http://ediss.sub.uni-hamburg.de/volltexte/2010/4603.

[11]

E. Stumpf, The existence and $C^1$-smoothness of local center-unstable manifolds for differential equations with state-dependent delay, Rostock. Math. Kolloq., 66 (2011), 3-44. Available from http://www.math.uni-rostock.de/math/pub/romako/romako66.html.

[12]

E. Stumpf, On a differential equation with state-dependent delay: A center-unstable manifold connecting an equilibrium and a periodic orbit, J. Dynam. Differential Equations, 24 (2012), 197-248. doi: 10.1007/s10884-012-9245-6.

[13]

E. Stumpf, Attraction property of local center-unstable manifolds for differential equations with state-dependent delay, Electron. J. Qual. Theory Differ. Equ., 2015 (2015), 1-45. Available from: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=3010.

[14]

A. Vanderbauwhede, Centre manifolds, normal forms and elementary bifurcations, in Dynamics Reported, A series in dynamical systems and their applications, Vol. 2, Wiley, Chichester, 1989, 89-169.

[15]

H.-O. Walther, The solution manifold and $C^{1}$-smoothness for differential equations with state-dependent delay, J. of Differential Equations, 195 (2003), 46-65. doi: 10.1016/j.jde.2003.07.001.

[16]

H.-O. Walther, Smoothness properties of semiflows for differential equations with state-dependent delays, J. Math. Sci. (N.Y.), 124 (2004), 5193-5207. doi: 10.1023/B:JOTH.0000047253.23098.12.

show all references

References:
[1]

H. Amann, Ordinary Differential Equations. An Introduction to Nonlinear Analysis (Translated from the German by G. Metzen), de Gruyter Studies in Mathematics, 13, Walter de Gruyter & Co., Berlin, 1990. doi: 10.1515/9783110853698.

[2]

P. Brunovský, A. Erdélyi and H.-O. Walther, On a model of a currency exchange rate - local stability and periodic solutions, J. Dynam. Differential Equations, 16 (2004), 393-432. doi: 10.1007/s10884-004-4285-1.

[3]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations. Functional, Complex, and Nonlinear Analysis, Applied Mathematical Sciences, 110, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.

[4]

P. Getto and M. Waurick, A differential equation with state-dependent delay from cell population biology, preprint, arXiv:1411.3097v1.

[5]

F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delay, in Hand. Differ. Equ.: Ordinary Differential Equations, 3, Elsevier/North-Holland, Amsterdam, 2006, 435-545. doi: 10.1016/S1874-5725(06)80009-X.

[6]

T. Krisztin, A local unstable manifold for differential equations with state-dependent delay, Discrete Contin. Dyn. Syst., 9 (2003), 993-1028. doi: 10.3934/dcds.2003.9.993.

[7]

T. Krisztin, $C^{1}$-smoothness of center manifolds for differential equations with state-dependent delay, in Nonlinear Dynamics and Evolution Equations (eds. H. Brunner et al.), Fields Inst. Commun., 48, Amer. Math. Soc., Providence, 2006, 213-226.

[8]

V. A. Pliss, A reduction principle in the theory of stability of motion (Russian), Izv. Akad. Nauk. SSSR Ser. Mat., 28 (1964), 1297-1324.

[9]

R. Qesmi and H.-O. Walther, Center-stable manifolds for differential equations with state-dependent delays, Discrete Contin. Dyn. Syst., 23 (2009), 1009-1033. doi: 10.3934/dcds.2009.23.1009.

[10]

E. Stumpf, On a differential equation with state-dependent delay: A global center-unstable manifold bordered by a periodic orbit, Doctoral dissertation, University of Hamburg, 2010. Available from: http://ediss.sub.uni-hamburg.de/volltexte/2010/4603.

[11]

E. Stumpf, The existence and $C^1$-smoothness of local center-unstable manifolds for differential equations with state-dependent delay, Rostock. Math. Kolloq., 66 (2011), 3-44. Available from http://www.math.uni-rostock.de/math/pub/romako/romako66.html.

[12]

E. Stumpf, On a differential equation with state-dependent delay: A center-unstable manifold connecting an equilibrium and a periodic orbit, J. Dynam. Differential Equations, 24 (2012), 197-248. doi: 10.1007/s10884-012-9245-6.

[13]

E. Stumpf, Attraction property of local center-unstable manifolds for differential equations with state-dependent delay, Electron. J. Qual. Theory Differ. Equ., 2015 (2015), 1-45. Available from: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=3010.

[14]

A. Vanderbauwhede, Centre manifolds, normal forms and elementary bifurcations, in Dynamics Reported, A series in dynamical systems and their applications, Vol. 2, Wiley, Chichester, 1989, 89-169.

[15]

H.-O. Walther, The solution manifold and $C^{1}$-smoothness for differential equations with state-dependent delay, J. of Differential Equations, 195 (2003), 46-65. doi: 10.1016/j.jde.2003.07.001.

[16]

H.-O. Walther, Smoothness properties of semiflows for differential equations with state-dependent delays, J. Math. Sci. (N.Y.), 124 (2004), 5193-5207. doi: 10.1023/B:JOTH.0000047253.23098.12.

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