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Keywords:
Liapounov instability; $h$-instability; instability of delay equations; nonconstant delays
Liapounov instability; $h$-instability; instability of delay equations; nonconstant delays
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.
References:
[1] E. A. Coddington and N. Levinson: Theory of Ordinary Differential Equations. McGill-Hill, New York, 1975.
[2] K. L. Cooke: Functional differential equations close to differential equations. Bull. Amer. Math. Soc. 72 (1966), 285–288. DOI 10.1090/S0002-9904-1966-11494-5 | MR 0221047 | Zbl 0151.10401
[3] W. A. Coppel: On the stability of ordinary differential equations. J. London Math. Soc. 39 (1964), 255–260. DOI 10.1112/jlms/s1-39.1.255 | MR 0164094 | Zbl 0128.08205
[4] W. A. Coppel: Dichotomies in Stability Theory. Lecture Notes in Mathematics Vol. 629. Springer Verlag, Berlin, 1978. MR 0481196
[5] L. E. Elsgoltz and S. B. Norkin: Introduction to the Theory of Differential Equations with Deviating Arguments. Nauka, Moscow, 1971. (Russian) MR 0352646
[6] J. Gallardo and M. Pinto: Asymptotic integration of nonautonomous delay-differential systems. J. Math. Anal. Appl. 199 (1996), 654–675. DOI 10.1006/jmaa.1996.0168 | MR 1386598
[7] K. Gopalsamy: Stability and Oscillations in Delay Differential Equations of Populations Dynamics. Kluwer, Dordrecht, 1992. MR 1163190
[8] I. Győri and M. Pituk: Stability criteria for linear delay differential equations. Differential Integral Equations 10 (1997), 841–852. MR 1741755
[9] N. Rouche, P. Habets and M. Laloy: Stability Theory by Liapounov’s Second Method. App. Math. Sciences 22. Springer, Berlin, 1977. DOI 10.1007/978-1-4684-9362-7 | MR 0450715
[10] J. K. Hale: Theory of Functional Differential Equations. Springer-Verlag, New York, 1977. MR 0508721 | Zbl 0352.34001
[11] J. K. Hale and S. M. Verduyn Lunel: Introduction to Functional Differential Equations. Springer-Verlag, New York, 1993. MR 1243878
[12] R. Naulin: Instability of nonautonomous differential systems. Differential Equations Dynam. Systems 6 (1998), 363–376. MR 1664030 | Zbl 0992.34034
[13] R. Naulin: Weak dichotomies and asymptotic integration of nonlinear differential systems. Nonlinear Studies 5 (1998), 201–218. MR 1652618 | Zbl 0918.34013
[14] R. Naulin: Functional analytic characterization of a class of dichotomies. Unpublished work (1999).
[15] R. Naulin and M. Pinto: Roughness of $(h,k)$-dichotomies. J. Differential Equations 118 (1995), 20–35. DOI 10.1006/jdeq.1995.1065 | MR 1329401
[16] R. Naulin and M. Pinto: Admissible perturbations of exponential dichotomy roughness. J. Nonlinear Anal. TMA 31 (1998), 559–571. MR 1487846
[17] R. Naulin and M. Pinto: Projections for dichotomies in linear differential equations. Appl. Anal. 69 (1998), 239–255. DOI 10.1080/00036819808840660 | MR 1706475
[18] M. Pinto: Non autonomous semilinear differential systems: Asymptotic behavior and stable manifolds. Preprint (1997).
[19] M. Pinto: Dichotomy and asymptotic integration. Contributions USACH (1992), 13–22.
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