Global asymptotic stability for differentiable vector fields of \(\mathbb R^2\) (Q703823)

(a) Let \(X:\mathbb R^2\to\mathbb R^2\) be a differentiable map (not necessarily \(C^1\)) and let \(\text{Spec}(X)\) be the set of (complex) eigenvalues of the derivative \(DX_p\) when \(p\) varies in \(\mathbb E^2\). If, for some \(\varepsilon>0\), \(\text{Spec}(X)\cap [0,\varepsilon)= \emptyset\) then \(X\) is injective. (b) Let \(X:\mathbb R^2\to\mathbb R^2\) be a differentiable vector field such that \(X(0)=0\) and \(\text{Spec}(X)\subset \{z\in\mathbb C:\Re(z)<0\}\). Then, for all \(p\in\mathbb R^2\), there is a unique positive trajectory starting at \(p\); moreover the \(\omega\)-limit set of \(p\) is equal to \(\{0\}\).