American Institute of Mathematical Sciences

Advanced
July  2009, 23(3): 1009-1033. doi: 10.3934/dcds.2009.23.1009

Center-stable manifolds for differential equations with state-dependent delays

Redouane Qesmi 1, and  Hans-Otto Walther 2,

1. 

Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada M3J 1P3, Canada
2. 

Mathematisches Institut, Universität Gieβen, Arndtstr. 2, 35392 Gieβen

Received  November 2007 Revised  June 2008 Published  November 2008

Consider the functional differential equation (FDE) $\dot{x}(t)=f(x_t)$ with $f$ defined on an open subset of the space $C^1=C^1([-h,0],\R^n)$. Under mild smoothness assumptions, which are designed for the application to differential equations with state-dependent delays, the FDE generates a semiflow on a submanifold of $C^1$ with continuously differentiable time-$t$-maps. We show that at a stationary point continuously differentiable local center-stable manifolds of the semiflow exist. The proof uses results of Chen, Hale and Tan and of Krisztin about invariant manifolds of time-$t$-maps and their invariance properties with respect to the semiflow.
Keywords: functional differential equation, Center-stable manifold, state-dependent delay.
Mathematics Subject Classification: Primary: 34K19; Secondary: 37L10.
Citation: Redouane Qesmi, Hans-Otto Walther. Center-stable manifolds for differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 1009-1033. doi: 10.3934/dcds.2009.23.1009
[1]

Tibor Krisztin. A local unstable manifold for differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 993-1028. doi: 10.3934/dcds.2003.9.993
[2]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & 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 & Continuous Dynamical Systems, 2017, 37 (7) : 3939-3961. doi: 10.3934/dcds.2017167
[4]

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
[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, 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]

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
[8]

Josef Diblík. Long-time behavior of positive solutions of a differential equation with state-dependent delay. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : 31-46. doi: 10.3934/dcdss.2020002
[9]

Qingwen Hu, Bernhard Lani-Wayda, Eugen Stumpf. Preface: Delay differential equations with state-dependent delays and their applications. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : i-i. doi: 10.3934/dcdss.20201i
[10]

Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445
[11]

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
[12]

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
[13]

Jitai Liang, Ben Niu, Junjie Wei. Linearized stability for abstract functional differential equations subject to state-dependent delays with applications. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6167-6188. doi: 10.3934/dcdsb.2019134
[14]

Hans-Otto Walther. On Poisson's state-dependent delay. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 365-379. doi: 10.3934/dcds.2013.33.365
[15]

István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773
[16]

Jan Sieber. Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2607-2651. doi: 10.3934/dcds.2012.32.2607
[17]

F. M. G. Magpantay, A. R. Humphries. Generalised Lyapunov-Razumikhin techniques for scalar state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : 85-104. doi: 10.3934/dcdss.2020005
[18]

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
[19]

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
[20]

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, 2020, 13 (1) : 47-66. doi: 10.3934/dcdss.2020003

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (68)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy