Skip to main content
Log in

More on Linearized Stability for Neutral Equations with State-Dependent Delays

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

We prove a principle of linearized stability for semiflows generated by neutral functional differential equations of the form

$$ x'(t)=g(\partial\,x_t,x_t). $$

The state space is a closed subset in a manifold of C 2-functions. Applications include equations with state-dependent delay, as for example

$$ x'(t)=A(x'(t+d(x(t))))+f(x(t+r(x(t)))) $$

with nonlinear functions \({A:\mathbb{R}\to\mathbb{R}}\) and \({d:\mathbb{R}\to(-h,0), f:\mathbb{R}\to\mathbb{R}, r:\mathbb{R}\to[-h,0]}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arino O., Sanchez E.: A saddle point theorem for functional state-dependent delay equations. Discret. Contin. Dyn. Syst. 12, 687–722 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bartha M.: On stability properties for neutral differential equations with state-dependent delay. Differ. Equ. Dyn. Syst. 7, 197–220 (1999)

    MathSciNet  MATH  Google Scholar 

  3. Brunovský P., Erdélyi A., Walther H.O.: On a model of a currency exchange rate—local stability and periodic solutions. J. Dyn. Differ. Equ. 16, 393–432 (2004)

    Article  MATH  Google Scholar 

  4. Hale, J.K., Meyer, K.R.: A class of functional equations of neutral type. Mem. Am. Math. Soc. 76 (1967)

  5. Hale J.K., Verduyn Lunel S.M.: Introduction to Functional Differential Equations. Springer, New York (1993)

    MATH  Google Scholar 

  6. Hartung F.: Linearized stability for a class of neutral functional differential equations with state-dependent delays. Nonlinear Anal. Theory Methods Appl. 69, 1629–1643 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hartung F., Krisztin T., Walther H.O., Wu J.: Functional differential equations with state-dependent delay: theory and applications. In: Canada, A., Drabek, P., Fonda, A. (eds) Handbook Of Differential Equations, Ordinary Differential Equations, pp. 435–545. Elsevier Science B. V., North Holland, Amsterdam (2006)

    Chapter  Google Scholar 

  8. Henry D.: Linear autonomous neutral functional differential equations. J. Differ. Equ. 15, 106–128 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  9. Krishnan H.P.: Existence of unstable manifolds for a certain class of delay differential equations. Electron. J. Differ. Equ. 32, 1–13 (2002)

    MathSciNet  Google Scholar 

  10. Krisztin T.: A local unstable manifold for differential equations with state-dependent delay. Discret. Contin. Dyn. Syst. 9, 993–1028 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Krisztin, T.: C 1-smoothness of center manifolds for differential equations with state-dependent delays. In: Nonlinear Dynamics and Evolution Equations, Fields Institute Communications 48, 213–226 (2006)

  12. Mallet-Paret J., Nussbaum R.D., Paraskevopoulos P.: Periodic solutions for functional differential equations with multiple state-dependent time lags. Topol. Methods Nonlinear Anal. 3, 101–162 (1994)

    MathSciNet  MATH  Google Scholar 

  13. Qesmi R., Walther H.O.: Center-stable manifolds for differential equations with state-dependent delay. Discret. Contin. Dyn. Syst. 23, 1009–1033 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Stumpf, E.: On a differential equation with state-dependent delay: A global center-unstable manifold bordered by a periodic orbit. Doctoral dissertation, Universität Hamburg (2010)

  15. Walther H.O.: The solution manifold and C 1-smoothness of solution operators for differential equations with state dependent delay. J. Differ. Equ. 195, 46–65 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Walther, H.O.: Smoothness properties of semiflows for differential equations with state dependent delay. Russian. In: Proceedings of the International Conference on Differential and Functional Differential Equations, Moscow, 2002, vol. 1, pp. 40–55, Moscow State Aviation Institute (MAI), Moscow 2003. English version: Journal of the Mathematical Sciences 124, 5193–5207 (2004)

  17. Walther H.O.: Convergence to square waves in a price model with delay. Discret. Contin. Dyn. Syst. 13, 1325–1342 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Walther H.O.: Bifurcation of periodic solutions with large periods for a delay differential equation. Annali di Matematica Pura ed Applicata 185, 577–611 (2006)

    Article  MathSciNet  Google Scholar 

  19. Walther, H.O.: Semiflows for neutral equations with state-dependent delays. Preprint (2009)

  20. Walther H.O.: Linearized stability for semiflows generated by a class of neutral equations, with applications to state-dependent delays. J. Differ. Equ. Dyn. Syst. 22, 439–462 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hans-Otto Walther.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Walther, HO. More on Linearized Stability for Neutral Equations with State-Dependent Delays. Differ Equ Dyn Syst 19, 315–333 (2011). https://doi.org/10.1007/s12591-011-0093-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-011-0093-3

Keywords

Navigation