Weakened Markus-Yamabe conditions for 2-dimensional global asymptotic stability

Marc Chamberland, Jaume Llibre, Grzegorz Świrszcz

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6 Citations (Scopus)


For a general 2-dimensional autonomous system ẋ = f (x), it is difficult to find easily verifiable sufficient conditions guaranteeing global asymptotic stability of an equilibrium point. This paper considers three conditions which imply global asymptotic stability for a large class of systems, weakening the so-called Markus-Yamabe condition. The new conditions are: (1) the system admits a unique equilibrium point, (2) it is locally asymptotically stable, and (3) the trace of the Jacobian matrix of f is negative everywhere. We prove that under these three conditions global asymptotic stability is obtained when the components of f are polynomials of degree two or represent a Liénard system. However, we provide examples that global asymptotic stability is not obtained under these conditions for other classes of planar differential systems. © 2004 Elsevier Ltd. All rights reserved.
Original languageEnglish
Pages (from-to)951-958
JournalNonlinear Analysis, Theory, Methods and Applications
Publication statusPublished - 1 Dec 2004


  • Global asymptotic stability


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