-
Previous Article
Modified Schmidt games and non-dense forward orbits of partially hyperbolic systems
- DCDS Home
- This Issue
-
Next Article
Inheriting of chaos in uniformly convergent nonautonomous dynamical systems on the interval
Local stability analysis of differential equations with state-dependent delay
1. | Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany |
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. |
[1] |
Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169 |
[2] |
Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687 |
[3] |
Ismael Maroto, Carmen NÚÑez, Rafael Obaya. Dynamical properties of nonautonomous functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3939-3961. doi: 10.3934/dcds.2017167 |
[4] |
Ferenc Hartung, Janos Turi. Linearized stability in functional differential equations with state-dependent delays. Conference Publications, 2001, 2001 (Special) : 416-425. doi: 10.3934/proc.2001.2001.416 |
[5] |
A. R. Humphries, O. A. DeMasi, F. M. G. Magpantay, F. Upham. Dynamics of a delay differential equation with multiple state-dependent delays. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2701-2727. doi: 10.3934/dcds.2012.32.2701 |
[6] |
István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773 |
[7] |
Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038 |
[8] |
Jitai Liang, Ben Niu, Junjie Wei. Linearized stability for abstract functional differential equations subject to state-dependent delays with applications. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6167-6188. doi: 10.3934/dcdsb.2019134 |
[9] |
Josef Diblík. Long-time behavior of positive solutions of a differential equation with state-dependent delay. Discrete and Continuous Dynamical Systems - S, 2020, 13 (1) : 31-46. doi: 10.3934/dcdss.2020002 |
[10] |
Tibor Krisztin. A local unstable manifold for differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 993-1028. doi: 10.3934/dcds.2003.9.993 |
[11] |
Qingwen Hu, Bernhard Lani-Wayda, Eugen Stumpf. Preface: Delay differential equations with state-dependent delays and their applications. Discrete and Continuous Dynamical Systems - S, 2020, 13 (1) : i-i. doi: 10.3934/dcdss.20201i |
[12] |
Hermann Brunner, Stefano Maset. Time transformations for state-dependent delay differential equations. Communications on Pure and Applied Analysis, 2010, 9 (1) : 23-45. doi: 10.3934/cpaa.2010.9.23 |
[13] |
Matthias Büger, Marcus R.W. Martin. Stabilizing control for an unbounded state-dependent delay equation. Conference Publications, 2001, 2001 (Special) : 56-65. doi: 10.3934/proc.2001.2001.56 |
[14] |
Hans-Otto Walther. On Poisson's state-dependent delay. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 365-379. doi: 10.3934/dcds.2013.33.365 |
[15] |
Jan Sieber. Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2607-2651. doi: 10.3934/dcds.2012.32.2607 |
[16] |
F. M. G. Magpantay, A. R. Humphries. Generalised Lyapunov-Razumikhin techniques for scalar state-dependent delay differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (1) : 85-104. doi: 10.3934/dcdss.2020005 |
[17] |
Alexander Rezounenko. Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1547-1563. doi: 10.3934/dcdsb.2017074 |
[18] |
Alexander Rezounenko. Viral infection model with diffusion and state-dependent delay: Stability of classical solutions. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1091-1105. doi: 10.3934/dcdsb.2018143 |
[19] |
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 |
[20] |
Guo Lin, Haiyan Wang. Traveling wave solutions of a reaction-diffusion equation with state-dependent delay. Communications on Pure and Applied Analysis, 2016, 15 (2) : 319-334. doi: 10.3934/cpaa.2016.15.319 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]