Markus–Yamabe conjecture

Let $X=(P_1,\dots,P_n)$ be an autonomous $C^1$-vector field on $\mathbb{R}^n$ with a unique singularity at the origin. For each $x\in\mathbb{R}^n$, let $DX(x)$ denote its Jacobian matrix, and let $X$ be globally asymptotically stable (GAS) when every orbit tends to the origin in forward time. Markus–Yamabe conjecture. If every eigenvalue of $DX(x)$ has negative real part for every $x\in\mathbb{R}^n$, then $X$ is GAS. The conjecture asks whether pointwise linear stability of the Jacobian, together with uniqueness of the singularity, forces global asymptotic stability; its resolution depends on the dimension and the regularity or additional structure imposed on the vector field.