Asymptotic behaviour of a two-dimensional differential system with a nonconstant delay under the conditions of instability

Josef Kalas; Josef Rebenda

Mathematica Bohemica (2011)

  • Volume: 136, Issue: 2, page 215-224
  • ISSN: 0862-7959

Abstract

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We present several results dealing with the asymptotic behaviour of a real two-dimensional system x ' ( t ) = 𝖠 ( t ) x ( t ) + ∑ k = 1 m 𝖡 k ( t ) x ( θ k ( t ) ) + h ( t , x ( t ) , x ( θ 1 ( t ) ) , ⋯ , x ( θ m ( t ) ) ) with bounded nonconstant delays t - θ k ( t ) ≥ 0 satisfying lim t → ∞ θ k ( t ) = ∞ , under the assumption of instability. Here 𝖠 , 𝖡 k and h are supposed to be matrix functions and a vector function, respectively. The conditions for the instable properties of solutions together with the conditions for the existence of bounded solutions are given. The methods are based on the transformation of the real system considered to one equation with complex-valued coefficients. Asymptotic properties are studied by means of a suitable Lyapunov-Krasovskii functional and the Ważewski topological principle. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one constant or nonconstant delay were studied.

How to cite

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Kalas, Josef, and Rebenda, Josef. "Asymptotic behaviour of a two-dimensional differential system with a nonconstant delay under the conditions of instability." Mathematica Bohemica 136.2 (2011): 215-224. <http://eudml.org/doc/196982>.

@article{Kalas2011,
abstract = {We present several results dealing with the asymptotic behaviour of a real two-dimensional system $x^\{\prime \}(t)=\{\mathsf \{A\}\}(t)x(t)+\sum _\{k=1\}^\{m\}\{\mathsf \{B\}\}_k(t)x(\theta _k(t)) +h(t,x(t),x(\theta _1(t)),\dots ,x(\theta _m(t)))$ with bounded nonconstant delays $t-\theta _k(t) \ge 0$ satisfying $\lim _\{t \rightarrow \infty \} \theta _k(t)=\infty $, under the assumption of instability. Here $\sf A$, $\{\mathsf \{B\}\}_k$ and $h$ are supposed to be matrix functions and a vector function, respectively. The conditions for the instable properties of solutions together with the conditions for the existence of bounded solutions are given. The methods are based on the transformation of the real system considered to one equation with complex-valued coefficients. Asymptotic properties are studied by means of a suitable Lyapunov-Krasovskii functional and the Ważewski topological principle. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one constant or nonconstant delay were studied.},
author = {Kalas, Josef, Rebenda, Josef},
journal = {Mathematica Bohemica},
keywords = {delayed differential equations; asymptotic behaviour; boundedness of solutions; Lyapunov method; Ważewski topological principle; delayed differential equation; asymptotic behaviour; boundedness of solutions; Lyapunov method; Wazewski topological principle},
language = {eng},
number = {2},
pages = {215-224},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Asymptotic behaviour of a two-dimensional differential system with a nonconstant delay under the conditions of instability},
url = {http://eudml.org/doc/196982},
volume = {136},
year = {2011},
}

TY - JOUR
AU - Kalas, Josef
AU - Rebenda, Josef
TI - Asymptotic behaviour of a two-dimensional differential system with a nonconstant delay under the conditions of instability
JO - Mathematica Bohemica
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 136
IS - 2
SP - 215
EP - 224
AB - We present several results dealing with the asymptotic behaviour of a real two-dimensional system $x^{\prime }(t)={\mathsf {A}}(t)x(t)+\sum _{k=1}^{m}{\mathsf {B}}_k(t)x(\theta _k(t)) +h(t,x(t),x(\theta _1(t)),\dots ,x(\theta _m(t)))$ with bounded nonconstant delays $t-\theta _k(t) \ge 0$ satisfying $\lim _{t \rightarrow \infty } \theta _k(t)=\infty $, under the assumption of instability. Here $\sf A$, ${\mathsf {B}}_k$ and $h$ are supposed to be matrix functions and a vector function, respectively. The conditions for the instable properties of solutions together with the conditions for the existence of bounded solutions are given. The methods are based on the transformation of the real system considered to one equation with complex-valued coefficients. Asymptotic properties are studied by means of a suitable Lyapunov-Krasovskii functional and the Ważewski topological principle. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one constant or nonconstant delay were studied.
LA - eng
KW - delayed differential equations; asymptotic behaviour; boundedness of solutions; Lyapunov method; Ważewski topological principle; delayed differential equation; asymptotic behaviour; boundedness of solutions; Lyapunov method; Wazewski topological principle
UR - http://eudml.org/doc/196982
ER -

References

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  7. 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
  8. Kalas, J., Osička, J., 10.1002/mana.19941700111, Math. Nachr. 170 (1994), 133-147. (1994) MR1302371DOI10.1002/mana.19941700111
  9. Ráb, M., Kalas, J., Stability of dynamical systems in the plane, Differ. Integral Equ. 3 (1990), 124-144. (1990) MR1014730
  10. Rebenda, J., Asymptotic behaviour of solutions of real two-dimensional differential system with nonconstant delay, Arch. Math., Brno 45 (2009), 223-236. (2009) Zbl1212.34235MR2591678
  11. Rebenda, J., Asymptotic behaviour of real two-dimensional differential system with a finite number of constant delays, Demonstr. Math. 41 (2008), 845-857. (2008) Zbl1169.34051MR2484509
  12. Rebenda, J., Asymptotic properties of solutions of real two-dimensional differential systems with a finite number of constant delays, Mem. Differ. Equ. Math. Phys. 41 (2007), 97-114. (2007) Zbl1157.34356MR2391945
  13. Rybakowski, K. P., 10.1016/0022-0396(81)90070-X, J. Differ. Equations 39 (1981), 131-150. (1981) Zbl0477.34048MR0607779DOI10.1016/0022-0396(81)90070-X

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