Stability and asymptotic behaviour of a two-dimensional differential system with delay

Kalas, J. Ellsworth; Baráková, Lenka

doi:10.1016/s0022-247x(02)00023-9

Cited by 16 publications

(16 citation statements)

References 6 publications

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“…with a constant delay r > 0 are investigated in Kalas and Baráková [6]. The results given in the present paper generalize the great part of the results of [6].…”

Section: Introductionsupporting

confidence: 78%

“…The results given in the present paper generalize the great part of the results of [6]. The used method, typical and effective for two-dimensional systems, is based on the transformation of (1.1) to an equation with complex conjugate coordinates and on the use of a convenient Lyapunov-Krasovskii functional.…”

Section: Introductionsupporting

confidence: 71%

“…For details see [6]. We shall use the following notation: R for the set of all real numbers, R + for the set of all positive real numbers, for the real part of z, Im z for the imaginary part of z, z for the complex conjugate of z.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic behaviour of a two‐dimensional differential system with non‐constant delay

Kalas

2010

Mathematische Nachrichten

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Key words Delayed differential system, asymptotic behaviour, stability, boundedness of solutions MSC (2000) Primary: 34K12; Secondary: 34K20The asymptotic behaviour and stability properties are studied for a real two-dimensional systemand h are matrix functions and a vector function, respectively. The method of investigation is based on the transformation of the considered real system to one equation with complex-valued coefficients. Stability and asymptotic properties of this equation are studied by means of a suitable Lyapunov-Krasovskii functional. The results generalize the great part of the results of J. Kalas and L. Baráková [J. Math. Anal. Appl. 269, No. 1, 278-300 (2002)] for two-dimensional systems with a constant delay.

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“…with a constant delay r > 0 are investigated in Kalas and Baráková [6]. The results given in the present paper generalize the great part of the results of [6].…”

Section: Introductionsupporting

confidence: 78%

Section: Introductionsupporting

confidence: 71%

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Asymptotic behaviour of a two‐dimensional differential system with non‐constant delay

Kalas

2010

Mathematische Nachrichten

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“…The asymptotic properties of the two-dimensional system with constant delay in the stable case were studied by J. Kalas and L. Baráová in [28], the asymptotic behavior of solutions under the conditions of instability was investigated by J. Kalas in [29], [30]. The equations with a finite system of constant delays were inspected by J. Rebenda in [49], [50].…”

Section: Perturbed Two-dimensional Linear Systemsmentioning

confidence: 99%

Eighty fifth anniversary of birthday and scientific legacy of Professor Miloš Ráb

Došlý¹,

Jansová²,

Kalas³

2014

Arch. Math. (Brno)

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“…This combination was successfully used in [2] for two-dimensional system of ODE's and in [1] for system with one constant delay and led to interesting results.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Behaviour of Real Two-Dimensional Differential System With a Finite Number of Constant Delays

Rebenda¹

2008

Demonstratio Mathematica

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Abstract. In this article stability and asymptotic properties of a real two-dimensional system x'(t) = A(t)x(t)+ Bj(t)x(t -r,) + h(t,x(t),x(t -ri),... ,x(t -r")) are studied, where r\ > 0, ...,rn > 0 are constant delays, A, Bi,...,B" are the matrix functions and h is the vector function. Generalization of results on stability of a twodimensional differential system with one constant delay is obtained using the methods of complexification and Lyapunov-Krasovskii functional and some new corollaries and an example are presented. IntroductionThe investigation of the problem is based on the combination of the method of complexification and the method of Lyapunov-Krasovskii functional, which is to a great extent effective for two-dimensional systems. This combination was successfully used in [2] for two-dimensional system of ODE's and in [1] for system with one constant delay and led to interesting results.This article is related to paper [3] where asymptotic properties of system with finite number of constant delays were studied. The aim is, under some special conditions, to improve the results presented in [3] and to illustrate the advancement with an example.The subject of our study is the real two-dimensional systemwhere A(t) = (aik(t)), Bj(t) = (bjik(t)) (i,k = 1,2) for j G {l,...,n} are real square matrices and 1991 Mathematics Subject Classification: 34K20, 34K25, 34K12.

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Stability and asymptotic behaviour of a two-dimensional differential system with delay

Cited by 16 publications

References 6 publications

Asymptotic behaviour of a two‐dimensional differential system with non‐constant delay

Asymptotic behaviour of a two‐dimensional differential system with non‐constant delay

Eighty fifth anniversary of birthday and scientific legacy of Professor Miloš Ráb

Asymptotic Behaviour of Real Two-Dimensional Differential System With a Finite Number of Constant Delays

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