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# Article

 Title: On the instability of linear nonautonomous delay systems (English) Author: Naulin, Raúl Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 53 Issue: 3 Year: 2003 Pages: 497-514 Summary lang: English . Category: math . Summary: 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. (English) Keyword: Liapounov instability Keyword: $h$-instability Keyword: instability of delay equations Keyword: nonconstant delays MSC: 34D05 MSC: 34D09 MSC: 34D20 MSC: 34K06 MSC: 34K20 idZBL: Zbl 1080.34543 idMR: MR2000048 . Date available: 2009-09-24T11:03:48Z Last updated: 2020-07-03 Stable URL: http://hdl.handle.net/10338.dmlcz/127818 . Reference: [1] E. A.  Coddington and N. Levinson: Theory of Ordinary Differential Equations.McGill-Hill, New York, 1975. Reference: [2] K. L. Cooke: Functional differential equations close to differential equations.Bull. Amer. Math. Soc. 72 (1966), 285–288. Zbl 0151.10401, MR 0221047, 10.1090/S0002-9904-1966-11494-5 Reference: [3] W. A.  Coppel: On the stability of ordinary differential equations.J.  London Math. Soc. 39 (1964), 255–260. Zbl 0128.08205, MR 0164094, 10.1112/jlms/s1-39.1.255 Reference: [4] W. A. Coppel: Dichotomies in Stability Theory. Lecture Notes in Mathematics Vol.  629.Springer Verlag, Berlin, 1978. MR 0481196 Reference: [5] L. E.  Elsgoltz and S. B. Norkin: Introduction to the Theory of Differential Equations with Deviating Arguments.Nauka, Moscow, 1971. (Russian) MR 0352646 Reference: [6] J.  Gallardo and M. Pinto: Asymptotic integration of nonautonomous delay-differential systems.J.  Math. Anal. Appl. 199 (1996), 654–675. MR 1386598, 10.1006/jmaa.1996.0168 Reference: [7] K.  Gopalsamy: Stability and Oscillations in Delay Differential Equations of Populations Dynamics.Kluwer, Dordrecht, 1992. MR 1163190 Reference: [8] I.  Győri and M. Pituk: Stability criteria for linear delay differential equations.Differential Integral Equations 10 (1997), 841–852. MR 1741755 Reference: [9] N.  Rouche, P. Habets and M. Laloy: Stability Theory by Liapounov’s Second Method. App. Math. Sciences  22.Springer, Berlin, 1977. MR 0450715, 10.1007/978-1-4684-9362-7 Reference: [10] J. K. Hale: Theory of Functional Differential Equations.Springer-Verlag, New York, 1977. Zbl 0352.34001, MR 0508721 Reference: [11] J. K.  Hale and S. M.  Verduyn Lunel: Introduction to Functional Differential Equations.Springer-Verlag, New York, 1993. MR 1243878 Reference: [12] R. Naulin: Instability of nonautonomous differential systems.Differential Equations Dynam. Systems 6 (1998), 363–376. Zbl 0992.34034, MR 1664030 Reference: [13] R.  Naulin: Weak dichotomies and asymptotic integration of nonlinear differential systems.Nonlinear Studies 5 (1998), 201–218. Zbl 0918.34013, MR 1652618 Reference: [14] R.  Naulin: Functional analytic characterization of a class of dichotomies.Unpublished work (1999). Reference: [15] R.  Naulin and M. Pinto: Roughness of $(h,k)$-dichotomies.J. Differential Equations 118 (1995), 20–35. MR 1329401, 10.1006/jdeq.1995.1065 Reference: [16] R.  Naulin and M. Pinto: Admissible perturbations of exponential dichotomy roughness.J. Nonlinear Anal. TMA 31 (1998), 559–571. MR 1487846 Reference: [17] R. Naulin and M. Pinto: Projections for dichotomies in linear differential equations.Appl. Anal. 69 (1998), 239–255. MR 1706475, 10.1080/00036819808840660 Reference: [18] M.  Pinto: Non autonomous semilinear differential systems: Asymptotic behavior and stable manifolds.Preprint (1997). Reference: [19] M.  Pinto: Dichotomy and asymptotic integration.Contributions USACH (1992), 13–22. .

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