Asymptotic behaviour of real two-dimensional differential system with a finite number of constant delays

Abstract

Abstract. In this article stability and asymptotic properties of a real two-dimensional system x ′ (t) = A(t)x(t) + n j=1 B j (t)x(t − r j ) + h(t, x(t), x(t − r 1 ), . . . , x(t − r n )) are studied, where r 1 > 0, . . . , r n > 0 are constant delays, A, B 1 , . . . , B n 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) = a ik (t) , B j (t) = b jik (t) (i, k = 1, 2) for j ∈ {1, . . . , n} are real square matrices and 1991 Mathematics Subject Classification: 34K20, 34K25, 34K12.

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

“…We study the equation Obviously, this case is included in the case liminft_>00(|a(i)| -|&(t)|) > 0 considered in [3], but in this special case we are able to derive more useful results as we will see later in an example. The idea is based upon the well known result that the condition \a\ > |6| in an autonomous equation z' = az + bz ensures that zero is a focus, a centre or a node while under the condition | Ima| > |6| zero can be just a focus or a centre.…”

“…and 9 are locally Lebesgue integrable on [T, 00). Moreover, if P E AC\ oc ([T, 00), R+), then in (iv) we may choose A(t) = max(*(t),^), m from which one can see that we slightly generalized the situation considered in [3], Notice that the condition (ii) implies that the functions Kj(t) are nonnegative on [T, oo) for j = 0,..., n, and due to this, il>{t) > 0 on [T, oo). Finally, if \{t) = 0 in (ii), then equation (1) has the trivial solution z(t) = 0.…”

“…The aim is, under some special conditions, to improve the results presented in [3] and to illustrate the advancement with an example.…”

“…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.…”

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

“…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]. The asymptotic properties of solutions of the two-dimensional system with nonconstant delays were investigated by J. Kalas and J. Rebenda in [31], [32], [34], [51].…”