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

For a general 2-dimensional autonomous system View the MathML sourcex˙=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 ff is negative everywhere. We prove that under these three conditions global asymptotic stability is obtained when the components of ff 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.