I am reading this article where the following non-linear ODE is considered: $$ \begin{cases} \frac{d\varphi_0}{dt}(t) & = \nu(t)( e^{-\nu(t)} - q_0 \varphi_0(t) + q_K \varphi_K(t) (1-e^{-\nu(t)}),\\ \frac{d\varphi_k}{dt}(t) & = q_{k-1}\varphi_{k-1}(t)(1-e^{-\nu(t)}) - q_k\varphi_k(t) \end{cases} $$ It is known that the function $\nu(t)$ converges to some point $\nu(\infty)$, moreover this $\nu$ depends ond $\varphi_0$ (in particular, $\nu(t) = \inf_{s \geq t} \varphi(s)$) making this ODE non-linear. The author states that, in order to have a global attractor, it suffices to look at the linear system: $$ \begin{cases} \frac{d\varphi_0}{dt}(t) & = \nu(\infty)( e^{-\nu(\infty)} - q_0 \varphi_0(t) + q_K \varphi_K(t) (1-e^{-\nu(\infty)}),\\ \frac{d\varphi_k}{dt}(t) & = q_{k-1}\varphi_{k-1}(t)(1-e^{-\nu(\infty)}) - q_k\varphi_k(t) \end{cases} $$ and show that this system has a global attractor (which is easy to do by applying Gershgorin's theorem).
Is this a general method for showing asymptotic stability, is it described somewhere?
First, you need to define stability. I assume you are using stability in the sense of Lyapunov.
As your ODE is not autonomous you need to apply Lyapunov's direct method for a nonautonomous system.
Let $\dot{\boldsymbol{x}}=\boldsymbol{F}(\boldsymbol{x},t)$ be a nonautonomuous system with the initial condition $\boldsymbol{x}(t=t_0)=\boldsymbol{x}_0$. Which has a trivial equilibrium point in $\boldsymbol{x}_{\text{eq}}=\boldsymbol{0}$. Note, that this is not a restriction on the generality because the stability analysis of any given solution $\boldsymbol{x}_{\text{solution}}(t)$ can be transformed into the stability analysis of the trivial solution by using $\boldsymbol{x}(t)=\boldsymbol{x}_{\text{solution}}(t)+\boldsymbol{z}(t)$ as a substitution, in which $\boldsymbol{z}_{\text{eq}}=\boldsymbol{0}$.
(Theorem 4.8 From Khalil - Nonlinear Systems) Let $\boldsymbol{x}_{\text{eq}}=\boldsymbol{0} \in D \subset \mathbb{R}^n$ be an equilibrium point for the nonautonomuous system $\dot{\boldsymbol{x}}=\boldsymbol{F}(\boldsymbol{x},t)$ with the initial condition $\boldsymbol{x}(t=t_0)=\boldsymbol{x}_0$. Let $V: [0,\infty) \times D \to R$ be ca countinuously differentiable function such that
$$W_1(\boldsymbol{x})\leq V(\boldsymbol{x},t) \leq W_2(\boldsymbol{x})$$ $$\dot{V}(\boldsymbol{x},t)=\dfrac{\partial V(\boldsymbol{x},t) }{\partial t}+\nabla V(\boldsymbol{x},t) \cdot \boldsymbol{F}(\boldsymbol{x},t)\leq 0 \qquad (*)$$
for all time $t\geq t_0$ and for all $\boldsymbol{x}\in D$, where $W_1(\boldsymbol{x})$ and $W_2(\boldsymbol{x})$ are continuous positive definite functions on D. Then, $\boldsymbol{x}_{\text{eq}}=\boldsymbol{0}$ is uniformly stable.
If we can replace inequaltiy $(*)$ by
$$\dfrac{\partial V(\boldsymbol{x},t) }{\partial t}+\nabla V(\boldsymbol{x},t) \cdot \boldsymbol{F}(\boldsymbol{x},t)\leq -W_3(\boldsymbol{x}) (*)$$
for all $t\geq t_0$ and for all $x \in D$, where $W_3(\boldsymbol{x})$ ia a continuous positive definite function on D. Then $\boldsymbol{x}_{\text{eq}}=\boldsymbol{0}$ is uniformly asymptotically stable.
Hassan Khalil's book on nonlinear systems is a very good reference for such problems. In chapter 4.6 also Linear Time-Varying Systems and Linearization is covered.