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

The asymptotic behaviour of the solutions of a real two-dimensional system x ′ = A ( t ) x ( t ) + B ( t ) x ( t - r ) + h ( t , x ( t ) , x ( t - r ) ) , where r > 0 is a constant delay, is studied under the assumption of instability. Here A , B and h are matrix functions and a vector function,...

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Bibliographic Details
Published in:Nonlinear analysis 2005-07, Vol.62 (2), p.207-224
Main Author: Kalas, Josef
Format: Article
Language:English
Subjects:
Asymptotic behaviour
Boundedness of solutions
Delayed differential equation
Exact sciences and technology
Lyapunov method
Mathematical analysis
Mathematics
Ordinary differential equations
Sciences and techniques of general use
Two-dimensional systems
Ważewski topological principle
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Summary:The asymptotic behaviour of the solutions of a real two-dimensional system x ′ = A ( t ) x ( t ) + B ( t ) x ( t - r ) + h ( t , x ( t ) , x ( t - r ) ) , where r > 0 is a constant delay, is studied under the assumption of instability. Here A , B and h are matrix functions and a vector function, respectively. The conditions for the existence of bounded solutions or solutions tending to the origin as t → ∞ are given. The method of investigation is based on the transformation of the considered real system 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 Ważewski topological principle. The results supplement those of Kalas and Baráková [J. Math. Anal. Appl. 269(1) (2002) 278–300], where the stability and asymptotic behaviour were investigated for the stable case.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2005.03.015