The Markus–Yamabe conjecture was formulated as an attempt to give conditions for global stability of continuous dynamical systems in two dimensions.
However, the Markus–Yamabe conjecture does not hold for dimensions higher than two, a problem which autonomous convergence theorems attempt to address.
The autonomous convergence theorems by Russell Smith, Michael Li and James Muldowney work in a similar manner, but they rely on showing that the area of two-dimensional shapes in phase space decrease with time.
If the system is bounded, then according to Pugh's closing lemma there can be no chaotic behaviour either, so all trajectories must eventually reach an equilibrium.
Michael Li has also developed an extended autonomous convergence theorem which is applicable to dynamical systems containing an invariant manifold.