On the instability of linear nonautonomous delay systems

Raúl Naulin

Czechoslovak Mathematical Journal (2003)

  • Volume: 53, Issue: 3, page 497-514
  • ISSN: 0011-4642

Abstract

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The unstable properties of the linear nonautonomous delay system x ' ( t ) = A ( t ) x ( t ) + B ( t ) x ( t - r ( t ) ) , with nonconstant delay r ( t ) , are studied. It is assumed that the linear system y ' ( t ) = ( A ( t ) + B ( t ) ) y ( t ) is unstable, the instability being characterized by a nonstable manifold defined from a dichotomy to this linear system. The delay r ( t ) is assumed to be continuous and bounded. Two kinds of results are given, those concerning conditions that do not include the properties of the delay function r ( t ) and the results depending on the asymptotic properties of the delay function.

How to cite

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Naulin, Raúl. "On the instability of linear nonautonomous delay systems." Czechoslovak Mathematical Journal 53.3 (2003): 497-514. <http://eudml.org/doc/30794>.

@article{Naulin2003,
abstract = {The unstable properties of the linear nonautonomous delay system $x^\{\prime \}(t)=A(t)x(t)+B(t)x(t-r(t))$, with nonconstant delay $r(t)$, are studied. It is assumed that the linear system $y^\{\prime \}(t)=(A(t)+B(t))y(t)$ is unstable, the instability being characterized by a nonstable manifold defined from a dichotomy to this linear system. The delay $r(t)$ is assumed to be continuous and bounded. Two kinds of results are given, those concerning conditions that do not include the properties of the delay function $r(t)$ and the results depending on the asymptotic properties of the delay function.},
author = {Naulin, Raúl},
journal = {Czechoslovak Mathematical Journal},
keywords = {Liapounov instability; $h$-instability; instability of delay equations; nonconstant delays; Lyapounov instability; -instability; instability of delay equations; nonconstant delays},
language = {eng},
number = {3},
pages = {497-514},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the instability of linear nonautonomous delay systems},
url = {http://eudml.org/doc/30794},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Naulin, Raúl
TI - On the instability of linear nonautonomous delay systems
JO - Czechoslovak Mathematical Journal
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 53
IS - 3
SP - 497
EP - 514
AB - The unstable properties of the linear nonautonomous delay system $x^{\prime }(t)=A(t)x(t)+B(t)x(t-r(t))$, with nonconstant delay $r(t)$, are studied. It is assumed that the linear system $y^{\prime }(t)=(A(t)+B(t))y(t)$ is unstable, the instability being characterized by a nonstable manifold defined from a dichotomy to this linear system. The delay $r(t)$ is assumed to be continuous and bounded. Two kinds of results are given, those concerning conditions that do not include the properties of the delay function $r(t)$ and the results depending on the asymptotic properties of the delay function.
LA - eng
KW - Liapounov instability; $h$-instability; instability of delay equations; nonconstant delays; Lyapounov instability; -instability; instability of delay equations; nonconstant delays
UR - http://eudml.org/doc/30794
ER -

References

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  1. Theory of Ordinary Differential Equations, McGill-Hill, New York, 1975. (1975) 
  2. 10.1090/S0002-9904-1966-11494-5, Bull. Amer. Math. Soc. 72 (1966), 285–288. (1966) Zbl0151.10401MR0221047DOI10.1090/S0002-9904-1966-11494-5
  3. 10.1112/jlms/s1-39.1.255, J.  London Math. Soc. 39 (1964), 255–260. (1964) Zbl0128.08205MR0164094DOI10.1112/jlms/s1-39.1.255
  4. Dichotomies in Stability Theory. Lecture Notes in Mathematics Vol.  629, Springer Verlag, Berlin, 1978. (1978) MR0481196
  5. Introduction to the Theory of Differential Equations with Deviating Arguments, Nauka, Moscow, 1971. (Russian) (1971) MR0352646
  6. 10.1006/jmaa.1996.0168, J.  Math. Anal. Appl. 199 (1996), 654–675. (1996) MR1386598DOI10.1006/jmaa.1996.0168
  7. Stability and Oscillations in Delay Differential Equations of Populations Dynamics, Kluwer, Dordrecht, 1992. (1992) MR1163190
  8. Stability criteria for linear delay differential equations, Differential Integral Equations 10 (1997), 841–852. (1997) MR1741755
  9. 10.1007/978-1-4684-9362-7, Springer, Berlin, 1977. (1977) MR0450715DOI10.1007/978-1-4684-9362-7
  10. Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. (1977) Zbl0352.34001MR0508721
  11. Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. (1993) MR1243878
  12. Instability of nonautonomous differential systems, Differential Equations Dynam. Systems 6 (1998), 363–376. (1998) Zbl0992.34034MR1664030
  13. Weak dichotomies and asymptotic integration of nonlinear differential systems, Nonlinear Studies 5 (1998), 201–218. (1998) Zbl0918.34013MR1652618
  14. Functional analytic characterization of a class of dichotomies, Unpublished work (1999). (1999) 
  15. 10.1006/jdeq.1995.1065, J. Differential Equations 118 (1995), 20–35. (1995) MR1329401DOI10.1006/jdeq.1995.1065
  16. Admissible perturbations of exponential dichotomy roughness, J. Nonlinear Anal. TMA 31 (1998), 559–571. (1998) MR1487846
  17. 10.1080/00036819808840660, Appl. Anal. 69 (1998), 239–255. (1998) MR1706475DOI10.1080/00036819808840660
  18. Non autonomous semilinear differential systems: Asymptotic behavior and stable manifolds, Preprint (1997). (1997) 
  19. Dichotomy and asymptotic integration, Contributions USACH (1992), 13–22. (1992) 

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