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 & Continuous Dynamical Systems - A, 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, (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/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, (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, ().

[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, (2006), 435. 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. 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.), (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, Center-stable manifolds for differential equations with state-dependent delays,, Discrete Contin. Dyn. Syst., 23 (2009), 1009. 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, (2010).

[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.

[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. 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.

[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 state-dependent 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 state-dependent 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/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, (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, ().

[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, (2006), 435. 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. 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.), (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, Center-stable manifolds for differential equations with state-dependent delays,, Discrete Contin. Dyn. Syst., 23 (2009), 1009. 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, (2010).

[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.

[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. 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.

[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 state-dependent 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 state-dependent delays,, J. Math. Sci. (N.Y.), 124 (2004), 5193. 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 & 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 2012, 32 (8) : 2701-2727. doi: 10.3934/dcds.2012.32.2701

[6]

Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038

[7]

István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773

[8]

Josef Diblík. Long-time behavior of positive solutions of a differential equation with state-dependent delay. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 31-46. doi: 10.3934/dcdss.2020002

[9]

Tibor Krisztin. A local unstable manifold for differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 993-1028. doi: 10.3934/dcds.2003.9.993

[10]

Hermann Brunner, Stefano Maset. Time transformations for state-dependent delay differential equations. Communications on Pure & Applied Analysis, 2010, 9 (1) : 23-45. doi: 10.3934/cpaa.2010.9.23

[11]

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

[12]

Hans-Otto Walther. On Poisson's state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 365-379. doi: 10.3934/dcds.2013.33.365

[13]

Jan Sieber. Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2607-2651. doi: 10.3934/dcds.2012.32.2607

[14]

F. M. G. Magpantay, A. R. Humphries. Generalised Lyapunov-Razumikhin techniques for scalar state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 85-104. doi: 10.3934/dcdss.2020005

[15]

Alexander Rezounenko. Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1547-1563. doi: 10.3934/dcdsb.2017074

[16]

Alexander Rezounenko. Viral infection model with diffusion and state-dependent delay: Stability of classical solutions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1091-1105. doi: 10.3934/dcdsb.2018143

[17]

Benjamin B. Kennedy. A state-dependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1633-1650. doi: 10.3934/dcdsb.2013.18.1633

[18]

Guo Lin, Haiyan Wang. Traveling wave solutions of a reaction-diffusion equation with state-dependent delay. Communications on Pure & Applied Analysis, 2016, 15 (2) : 319-334. doi: 10.3934/cpaa.2016.15.319

[19]

Benjamin B. Kennedy. A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 47-66. doi: 10.3934/dcdss.2020003

[20]

Qingwen Hu. A model of regulatory dynamics with threshold-type state-dependent delay. Mathematical Biosciences & Engineering, 2018, 15 (4) : 863-882. doi: 10.3934/mbe.2018039

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]