On time transformations for differential equations with state-dependent delay

Alexander Rezounenko

Open Mathematics (2014)

  • Volume: 12, Issue: 2, page 298-307
  • ISSN: 2391-5455

Abstract

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Systems of differential equations with state-dependent delay are considered. The delay dynamically depends on the state, i.e. is governed by an additional differential equation. By applying the time transformations we arrive to constant delay systems and compare the asymptotic properties of the original and transformed systems.

How to cite

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Alexander Rezounenko. "On time transformations for differential equations with state-dependent delay." Open Mathematics 12.2 (2014): 298-307. <http://eudml.org/doc/269109>.

@article{AlexanderRezounenko2014,
abstract = {Systems of differential equations with state-dependent delay are considered. The delay dynamically depends on the state, i.e. is governed by an additional differential equation. By applying the time transformations we arrive to constant delay systems and compare the asymptotic properties of the original and transformed systems.},
author = {Alexander Rezounenko},
journal = {Open Mathematics},
keywords = {State-dependent delay; Time transformations; state-dependent delay; time transformations},
language = {eng},
number = {2},
pages = {298-307},
title = {On time transformations for differential equations with state-dependent delay},
url = {http://eudml.org/doc/269109},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Alexander Rezounenko
TI - On time transformations for differential equations with state-dependent delay
JO - Open Mathematics
PY - 2014
VL - 12
IS - 2
SP - 298
EP - 307
AB - Systems of differential equations with state-dependent delay are considered. The delay dynamically depends on the state, i.e. is governed by an additional differential equation. By applying the time transformations we arrive to constant delay systems and compare the asymptotic properties of the original and transformed systems.
LA - eng
KW - State-dependent delay; Time transformations; state-dependent delay; time transformations
UR - http://eudml.org/doc/269109
ER -

References

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  1. [1] Arino O., Hadeler K.P., Hbid M.L., Existence of periodic solutions for delay differential equations with state dependent delay, J. Differential Equations, 1998, 144(2), 263–301 http://dx.doi.org/10.1006/jdeq.1997.3378 
  2. [2] Brunner H., Maset S., Time transformations for delay differential equations, Discrete Contin. Dyn. Syst., 2009, 25(3), 751–775 http://dx.doi.org/10.3934/dcds.2009.25.751 Zbl1187.34093
  3. [3] Brunner H., Maset S., Time transformations for state-dependent delay differential equations, Commun. Pure Appl. Anal., 2010, 9(1), 23–45 http://dx.doi.org/10.3934/cpaa.2010.9.23 Zbl1194.34135
  4. [4] Chepyzhov V.V., Vishik M.I., Appendix: Non-authonomous dynamical systems and their attractors, In: Vishik M.I., Asymptotic Behaviour of Solutions of Evolutionary Equations, Lezioni Lincee, Cambridge University Press, Cambridge, 1992 
  5. [5] Chueshov I.D., Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Univ. Lektsii Sovrem. Mat., ACTA, Kharkov, 1999 (in Russian); English translation available at http://www.emis.de/monographs/Chueshov/ Zbl1100.37046
  6. [6] Diekmann O., van Gils S.A., Verduyn Lunel S.M., Walther H.-O., Delay Equations, Appl. Math. Sci., 110, Springer, New York, 1995 http://dx.doi.org/10.1007/978-1-4612-4206-2 
  7. [7] Hale J., Theory of Functional Differential Equations, 2nd ed., Appl. Math. Sci., 3, Springer, Heidelberg-New York, 1977 http://dx.doi.org/10.1007/978-1-4612-9892-2 
  8. [8] Hale J.K., Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., 25, American Mathematical Society, Providence, 1988 Zbl0642.58013
  9. [9] Hartung F., Krisztin T., Walther H.-O., Wu J., Functional differential equations with state-dependent delays: Theory and applications, In: Handbook of Differential Equations: Ordinary Differential Equations, III, Handb. Differ. Equ., Elsevier/North Holland, 2006, Amsterdam, 435–545 http://dx.doi.org/10.1016/S1874-5725(06)80009-X 
  10. [10] Krisztin T., A local unstable manifold for differential equations with state-dependent delay, Discrete Contin. Dyn. Syst., 2003, 9(4), 993–1028 http://dx.doi.org/10.3934/dcds.2003.9.993 Zbl1048.34123
  11. [11] Mallet-Paret J., Nussbaum R.D., Paraskevopoulos P., Periodic solutions for functional-differential equations with multiple state-dependent time lags, Topol. Methods Nonlinear Anal., 1994, 3(1), 101–162 Zbl0808.34080
  12. [12] Rezounenko A.V., Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions, Nonlinear Anal., 2009, 70(11), 3978–3986 http://dx.doi.org/10.1016/j.na.2008.08.006 Zbl1163.35494
  13. [13] Rezounenko A.V., A condition on delay for differential equations with discrete state-dependent delay, J. Math. Anal. Appl., 2012, 385(1), 506–516 http://dx.doi.org/10.1016/j.jmaa.2011.06.070 Zbl1242.34136
  14. [14] Temam R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci., 68, Springer, New York, 1988 http://dx.doi.org/10.1007/978-1-4684-0313-8 
  15. [15] Vorotnikov V.I., Rumyantsev V.V., Stability and Control with Respect to Part of the Coordinates of the Phase Vector of Dynamical Systems: Theory, Methods and Applications, Nauchnyi Mir, Moscow, 2001 (in Russian) Zbl1016.34052
  16. [16] Walther H.-O., The solution manifold and C 1-smoothness for differential equations with state-dependent delay, J. Differential Equations, 2003, 195(1), 46–65 http://dx.doi.org/10.1016/j.jde.2003.07.001 Zbl1045.34048

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