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Local stability analysis of differential equations with statedependent 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, (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. doi: 10.1007/s1088400442851. 
[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, (1995). doi: 10.1007/9781461242062. 
[4] 
P. Getto and M. Waurick, A differential equation with statedependent delay from cell population biology,, preprint, (). 
[5] 
F. Hartung, T. Krisztin, H.O. Walther and J. Wu, Functional differential equations with statedependent delay,, in Hand. Differ. Equ.: Ordinary Differential Equations, (2006), 435. doi: 10.1016/S18745725(06)80009X. 
[6] 
T. Krisztin, A local unstable manifold for differential equations with statedependent delay,, Discrete Contin. Dyn. Syst., 9 (2003), 993. doi: 10.3934/dcds.2003.9.993. 
[7] 
T. Krisztin, $C^{1}$smoothness of center manifolds for differential equations with statedependent delay,, in Nonlinear Dynamics and Evolution Equations (eds. H. Brunner et al.), (2006), 213. 
[8] 
V. A. Pliss, A reduction principle in the theory of stability of motion (Russian),, Izv. Akad. Nauk. SSSR Ser. Mat., 28 (1964), 1297. 
[9] 
R. Qesmi and H.O. Walther, Centerstable manifolds for differential equations with statedependent delays,, Discrete Contin. Dyn. Syst., 23 (2009), 1009. doi: 10.3934/dcds.2009.23.1009. 
[10] 
E. Stumpf, On a differential equation with statedependent delay: A global centerunstable manifold bordered by a periodic orbit,, Doctoral dissertation, (2010). 
[11] 
E. Stumpf, The existence and $C^1$smoothness of local centerunstable manifolds for differential equations with statedependent delay,, Rostock. Math. Kolloq., 66 (2011), 3. 
[12] 
E. Stumpf, On a differential equation with statedependent delay: A centerunstable manifold connecting an equilibrium and a periodic orbit,, J. Dynam. Differential Equations, 24 (2012), 197. doi: 10.1007/s1088401292456. 
[13] 
E. Stumpf, Attraction property of local centerunstable manifolds for differential equations with statedependent delay,, Electron. J. Qual. Theory Differ. Equ., 2015 (2015), 1. 
[14] 
A. Vanderbauwhede, Centre manifolds, normal forms and elementary bifurcations,, in Dynamics Reported, (1989), 89. 
[15] 
H.O. Walther, The solution manifold and $C^{1}$smoothness for differential equations with statedependent delay,, J. of Differential Equations, 195 (2003), 46. doi: 10.1016/j.jde.2003.07.001. 
[16] 
H.O. Walther, Smoothness properties of semiflows for differential equations with statedependent delays,, J. Math. Sci. (N.Y.), 124 (2004), 5193. 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, (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. doi: 10.1007/s1088400442851. 
[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, (1995). doi: 10.1007/9781461242062. 
[4] 
P. Getto and M. Waurick, A differential equation with statedependent delay from cell population biology,, preprint, (). 
[5] 
F. Hartung, T. Krisztin, H.O. Walther and J. Wu, Functional differential equations with statedependent delay,, in Hand. Differ. Equ.: Ordinary Differential Equations, (2006), 435. doi: 10.1016/S18745725(06)80009X. 
[6] 
T. Krisztin, A local unstable manifold for differential equations with statedependent delay,, Discrete Contin. Dyn. Syst., 9 (2003), 993. doi: 10.3934/dcds.2003.9.993. 
[7] 
T. Krisztin, $C^{1}$smoothness of center manifolds for differential equations with statedependent delay,, in Nonlinear Dynamics and Evolution Equations (eds. H. Brunner et al.), (2006), 213. 
[8] 
V. A. Pliss, A reduction principle in the theory of stability of motion (Russian),, Izv. Akad. Nauk. SSSR Ser. Mat., 28 (1964), 1297. 
[9] 
R. Qesmi and H.O. Walther, Centerstable manifolds for differential equations with statedependent delays,, Discrete Contin. Dyn. Syst., 23 (2009), 1009. doi: 10.3934/dcds.2009.23.1009. 
[10] 
E. Stumpf, On a differential equation with statedependent delay: A global centerunstable manifold bordered by a periodic orbit,, Doctoral dissertation, (2010). 
[11] 
E. Stumpf, The existence and $C^1$smoothness of local centerunstable manifolds for differential equations with statedependent delay,, Rostock. Math. Kolloq., 66 (2011), 3. 
[12] 
E. Stumpf, On a differential equation with statedependent delay: A centerunstable manifold connecting an equilibrium and a periodic orbit,, J. Dynam. Differential Equations, 24 (2012), 197. doi: 10.1007/s1088401292456. 
[13] 
E. Stumpf, Attraction property of local centerunstable manifolds for differential equations with statedependent delay,, Electron. J. Qual. Theory Differ. Equ., 2015 (2015), 1. 
[14] 
A. Vanderbauwhede, Centre manifolds, normal forms and elementary bifurcations,, in Dynamics Reported, (1989), 89. 
[15] 
H.O. Walther, The solution manifold and $C^{1}$smoothness for differential equations with statedependent delay,, J. of Differential Equations, 195 (2003), 46. doi: 10.1016/j.jde.2003.07.001. 
[16] 
H.O. Walther, Smoothness properties of semiflows for differential equations with statedependent delays,, J. Math. Sci. (N.Y.), 124 (2004), 5193. doi: 10.1023/B:JOTH.0000047253.23098.12. 
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