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Stability in the large of autonomous systems of two differential equations Mufti, Izhar-Ul Haq

Abstract

The object of this dissertation is to discuss the stability in the large of the trivial solution for systems of two differential equations using qualitative methods (of course in combination with the construction of Lyapunov function). The right hand sides of these systems do not contain the time t explicitly. First of all we discuss (Sec. 2.) the system of the type [equations omitted] These equations occur in automatic regulation. Using qualitative methods we determine sufficient conditions in order that the trivial solution of system (l) be asymptotically stable in the large. In this connection we note that a theorem proved by Aĭzerman for the systems of two equations (Sec. 3), namely, for the systems [equations omitted] In the case of system (2) we give a new proof of a theorem which asserts that if c² + ab ≠ o, then the trivial solution is asymptotically stable in the large under the generalized Hurwitz conditions. The theorem was first proved by Erugin [8]. For system (3) Malkin showed that the trivial solution is asymptotically stable in the large under the conditions a + c < o, (acy - bf(y)) y > o for y ≠ o and [formula omitted] We prove a similar theorem without the requirement of [formula omitted] We also discuss (Sec. 4) the stability in the large of the systems [equations omitted] We consider (Sec. 5) again the system of the type (l) but under assumptions as indicated by Ershov [6] who has discussed various cases where the asymptotic stability in the large holds. Not agreeing fully with the proofs of these theorems we give our own proofs. Finally we discuss (Sec. 6 and 7) the stability in the large of the systems [equations omitted] under suitable assumptions. As a sample case we prove that if ab > o,then the trivial solution of system (4) is asymptotically stable in the large under conditions h₁(y) + h₂(x) < o , h₁(y) h₂(x) - ab > o, for x ≠ o, y ≠ o

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