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

Abstract

For a general 2-dimensional autonomous system x ˙ = 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.