Publication detail

# Asymptotic behavior of solutions of systems of dynamic equations on time scales in a set whose boundary is a combination of strict egress and strict ingress points

DIBLÍK, J. VÍTOVEC, J.

Original Title

Asymptotic behavior of solutions of systems of dynamic equations on time scales in a set whose boundary is a combination of strict egress and strict ingress points

Czech Title

Asymptotické chování řešení systémů dynamických rovnic na časových škálách na množině, jejíž hranice je kombinací bodů ostrého výstupu a ostrého vstupu

English Title

Asymptotic behavior of solutions of systems of dynamic equations on time scales in a set whose boundary is a combination of strict egress and strict ingress points

Type

journal article

Language

en

Original Abstract

In this paper we study the asymptotic behavior of solutions of nonlinear dynamic systems on time scales of the form $$y^\Delta(t)=f(t,y(t)),$$ where $f\colon\mathbb{T}\times\mathbb{R}^n\rightarrow\mathbb{R}^n$ and $\mathbb{T}$ is a time scale. For a given set $\Omega\subset\mathbb{T}\times\mathbb{R}^{n}$, we formulate the conditions for function $f$, which guarantee that at least one solution $y$ of the above system stays in $\Omega$. The dimension of the space of initial data generating such solutions is discussed and perturbed linear systems are considered as well. A linear system with singularity at infinity is considered as an example.

Czech abstract

V tomto článku je studováno asymptotické chování řešení nelineárních dynamických systémů na časových škálách tvaru $$y^\Delta(t)=f(t,y(t)),$$ kde $f\colon\mathbb{T}\times\mathbb{R}^n\rightarrow\mathbb{R}^n$ a $\mathbb{T}$ je časová škála. Pro danou množinu $\Omega\subset\mathbb{T}\times\mathbb{R}^{n}$, formulujeme podmínky na funkci $f$, které zaručí, že alespoň jedno řešení, $y$ z výše uvedeného systému zůstane v $\Omega$. Rozměr prostoru řešení generovaného počáteční podmínkou je diskutován a pertrubovaný lineární systém je též uvažován. Lineární systém se singularitou v nekonečnu je uveden jako příklad.

English abstract

In this paper we study the asymptotic behavior of solutions of nonlinear dynamic systems on time scales of the form $$y^\Delta(t)=f(t,y(t)),$$ where $f\colon\mathbb{T}\times\mathbb{R}^n\rightarrow\mathbb{R}^n$ and $\mathbb{T}$ is a time scale. For a given set $\Omega\subset\mathbb{T}\times\mathbb{R}^{n}$, we formulate the conditions for function $f$, which guarantee that at least one solution $y$ of the above system stays in $\Omega$. The dimension of the space of initial data generating such solutions is discussed and perturbed linear systems are considered as well. A linear system with singularity at infinity is considered as an example.

Keywords

Time scale; Dynamic system; Asymptotic behavior of solution; Retract; Retraction; Lyapunov method

RIV year

2014

Released

04.06.2014

Pages from

289

Pages to

299

Pages count

11

URL

BibTex


@article{BUT107428,
author="Josef {Diblík} and Jiří {Vítovec}",
title="Asymptotic behavior of solutions of systems of dynamic equations on time scales in a set whose boundary is a combination of strict egress and strict ingress points",
annote="In this paper we study the asymptotic behavior of solutions of nonlinear dynamic systems on time scales of the form $$y^\Delta(t)=f(t,y(t)),$$ where
$f\colon\mathbb{T}\times\mathbb{R}^n\rightarrow\mathbb{R}^n$ and $\mathbb{T}$ is a time scale. For a given set $\Omega\subset\mathbb{T}\times\mathbb{R}^{n}$, we formulate the conditions for function $f$, which guarantee that at least one solution $y$ of the above system stays in $\Omega$. The dimension of the space of initial data generating such solutions is discussed and perturbed linear systems are considered as well. A linear system with singularity at infinity is considered as an example.",
chapter="107428",
doi="10.1016/j.amc.2014.04.021",
number="6",
volume="238",
year="2014",
month="june",
pages="289--299",
type="journal article"
}