Consider the following nonautonomous, nonlinear ODE:
$$y'(t)=\rho(W(t)y(t)+b(t)),$$
where $y(t),b(t)\in\mathbb{R}^{n},W(t)\in\mathbb{R}^{n,n}$ and $\rho:\mathbb{R}\to\mathbb{R}$ is some nonlinear function that is applied element-wise.
A paper I'm reading claims that this ODE is Ljapunov stable if
$$\max_{i\in\{1,\cdots,n\}}Re(\lambda_i(J(t)))\leq 0\hspace{6pt}\forall t,$$
where $Re(\cdot)$ denotes the real part and $\lambda_i(J(t))\in\mathbb{R}$ is the $i^{th}$ eigenvalue of the Jacobian $J(t)\in\mathbb{R}^{n,n}$ of the righthand side of the ODE. The paper does neither prove this nor provide a reference to any known theorem proving this, which leaves me with the question:
Why is this true? Can anybody point me to a result that shows this?
I know similar results for linear, autonomous ODEs with and without small nonlinear perturbations, but I know no such result for the above nonlinear, nonautonomous ODE.
For reference, the paper claiming this is this paper in equations (3.5) and (3.6).