# Asymptotic behaviour of solutions of real two-dimensional differential system with nonconstant delay

Archivum Mathematicum (2009)

- Volume: 045, Issue: 3, page 223-236
- ISSN: 0044-8753

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topRebenda, Josef. "Asymptotic behaviour of solutions of real two-dimensional differential system with nonconstant delay." Archivum Mathematicum 045.3 (2009): 223-236. <http://eudml.org/doc/250553>.

@article{Rebenda2009,

abstract = {In this article, stability and asymptotic properties of solutions of a real two-dimensional system $x^\{\prime \}(t) = \mathbf \{A\} (t) x(t) + \mathbf \{B\} (t) x (\tau (t)) + \mathbf \{h\} (t, x(t), x(\tau (t)))$ are studied, where $\mathbf \{A\}$, $\mathbf \{B\}$ are matrix functions, $\mathbf \{h\}$ is a vector function and $\tau (t) \le t$ is a nonconstant delay which is absolutely continuous and satisfies $\lim \limits _\{t \rightarrow \infty \} \tau (t) = \infty $. Generalization of results on stability of a two-dimensional differential system with one constant delay is obtained using the methods of complexification and Lyapunov-Krasovskii functional and some new corollaries and examples are presented.},

author = {Rebenda, Josef},

journal = {Archivum Mathematicum},

keywords = {stability; asymptotic behaviour; differential system; nonconstant delay; Lyapunov method; stability; asymptotic behaviour; differential system; nonconstant delay; Lyapunov method},

language = {eng},

number = {3},

pages = {223-236},

publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},

title = {Asymptotic behaviour of solutions of real two-dimensional differential system with nonconstant delay},

url = {http://eudml.org/doc/250553},

volume = {045},

year = {2009},

}

TY - JOUR

AU - Rebenda, Josef

TI - Asymptotic behaviour of solutions of real two-dimensional differential system with nonconstant delay

JO - Archivum Mathematicum

PY - 2009

PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno

VL - 045

IS - 3

SP - 223

EP - 236

AB - In this article, stability and asymptotic properties of solutions of a real two-dimensional system $x^{\prime }(t) = \mathbf {A} (t) x(t) + \mathbf {B} (t) x (\tau (t)) + \mathbf {h} (t, x(t), x(\tau (t)))$ are studied, where $\mathbf {A}$, $\mathbf {B}$ are matrix functions, $\mathbf {h}$ is a vector function and $\tau (t) \le t$ is a nonconstant delay which is absolutely continuous and satisfies $\lim \limits _{t \rightarrow \infty } \tau (t) = \infty $. Generalization of results on stability of a two-dimensional differential system with one constant delay is obtained using the methods of complexification and Lyapunov-Krasovskii functional and some new corollaries and examples are presented.

LA - eng

KW - stability; asymptotic behaviour; differential system; nonconstant delay; Lyapunov method; stability; asymptotic behaviour; differential system; nonconstant delay; Lyapunov method

UR - http://eudml.org/doc/250553

ER -

## References

top- Kalas, J., Asymptotic behaviour of a two-dimensional differential systems with nonconstant delay, accepted in Math. Nachr.
- Kalas, J., Baráková, L., 10.1016/S0022-247X(02)00023-9, J. Math. Anal. Appl. 269 (2002), 278–300. (2002) Zbl1008.34064MR1907886DOI10.1016/S0022-247X(02)00023-9
- Ráb, M., Kalas, J., Stability of dynamical systems in the plane, Differential Integral Equations 3 (1990), 124–144. (1990) MR1014730
- Rebenda, J., Asymptotic properties of solutions of real two-dimensional differential systems with a finite number of constant delays, Mem. Differential Equations Math. Phys. 41 (2007), 97–114. (2007) Zbl1157.34356MR2391945
- Rebenda, J., Stability of the trivial solution of real two-dimensional differential system with nonconstant delay, In 6. matematický workshop - sborník, FAST VUT Brno 2007, 2007, 49–50 (abstract). Fulltext available at http://math.fce.vutbr.cz/~pribyl/workshop_2007/prispevky/Rebenda.pdf. (2007, 49–50 (abstract). Fulltext available at http://math.fce.vutbr.cz/~pribyl/workshop_2007/prispevky/Rebenda.pdf)
- Rebenda, J., Asymptotic behaviour of real two-dimensional differential system with a finite number of constant delays, Demonstratio Math. 41 (4) (2008), 845–857. (2008) Zbl1169.34051MR2484509

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