Asymptotic properties of an unstable two-dimensional differential system with delay

Josef Kalas

Mathematica Bohemica (2006)

  • Volume: 131, Issue: 3, page 305-319
  • ISSN: 0862-7959

Abstract

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The asymptotic behaviour of the solutions is studied for a real unstable two-dimensional system , where is a constant delay. It is supposed that , and are matrix functions and a vector function, respectively. Our results complement those of Kalas [Nonlinear Anal. 62(2) (2005), 207–224], where the conditions for the existence of bounded solutions or solutions tending to the origin as are given. The method of investigation is based on the transformation of the real system considered to one equation with complex-valued coefficients. Asymptotic properties of this equation are studied by means of a suitable Lyapunov-Krasovskii functional and by virtue of the Wa.zewski topological principle. Stability and asymptotic behaviour of the solutions for the stable case of the equation considered were studied in Kalas and Baráková [J. Math. Anal. Appl. 269(1) (2002), 278–300].

How to cite

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Kalas, Josef. "Asymptotic properties of an unstable two-dimensional differential system with delay." Mathematica Bohemica 131.3 (2006): 305-319. <http://eudml.org/doc/249901>.

@article{Kalas2006,
abstract = {The asymptotic behaviour of the solutions is studied for a real unstable two-dimensional system $x^\{\prime \}(t)=\{\mathsf \{A\}\}(t)x(t)+\{\mathsf \{B\}\}(t)x(t-r)+h(t,x(t),x(t-r))$, where $r>0$ is a constant delay. It is supposed that $\mathsf \{A\}$, $\mathsf \{B\}$ and $h$ are matrix functions and a vector function, respectively. Our results complement those of Kalas [Nonlinear Anal. 62(2) (2005), 207–224], where the conditions for the existence of bounded solutions or solutions tending to the origin as $t\rightarrow \infty $ are given. The method of investigation is based on the transformation of the real system considered to one equation with complex-valued coefficients. Asymptotic properties of this equation are studied by means of a suitable Lyapunov-Krasovskii functional and by virtue of the Wa.zewski topological principle. Stability and asymptotic behaviour of the solutions for the stable case of the equation considered were studied in Kalas and Baráková [J. Math. Anal. Appl. 269(1) (2002), 278–300].},
author = {Kalas, Josef},
journal = {Mathematica Bohemica},
keywords = {delayed differential equation; asymptotic behaviour; boundedness of solutions; two-dimensional systems; Lyapunov method; Wa.zewski topological principle; delayed differential equation; asymptotic behaviour; boundedness of solutions; Lyapunov method},
language = {eng},
number = {3},
pages = {305-319},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Asymptotic properties of an unstable two-dimensional differential system with delay},
url = {http://eudml.org/doc/249901},
volume = {131},
year = {2006},
}

TY - JOUR
AU - Kalas, Josef
TI - Asymptotic properties of an unstable two-dimensional differential system with delay
JO - Mathematica Bohemica
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 131
IS - 3
SP - 305
EP - 319
AB - The asymptotic behaviour of the solutions is studied for a real unstable two-dimensional system $x^{\prime }(t)={\mathsf {A}}(t)x(t)+{\mathsf {B}}(t)x(t-r)+h(t,x(t),x(t-r))$, where $r>0$ is a constant delay. It is supposed that $\mathsf {A}$, $\mathsf {B}$ and $h$ are matrix functions and a vector function, respectively. Our results complement those of Kalas [Nonlinear Anal. 62(2) (2005), 207–224], where the conditions for the existence of bounded solutions or solutions tending to the origin as $t\rightarrow \infty $ are given. The method of investigation is based on the transformation of the real system considered to one equation with complex-valued coefficients. Asymptotic properties of this equation are studied by means of a suitable Lyapunov-Krasovskii functional and by virtue of the Wa.zewski topological principle. Stability and asymptotic behaviour of the solutions for the stable case of the equation considered were studied in Kalas and Baráková [J. Math. Anal. Appl. 269(1) (2002), 278–300].
LA - eng
KW - delayed differential equation; asymptotic behaviour; boundedness of solutions; two-dimensional systems; Lyapunov method; Wa.zewski topological principle; delayed differential equation; asymptotic behaviour; boundedness of solutions; Lyapunov method
UR - http://eudml.org/doc/249901
ER -

References

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