Suppose we have a matrix equation $$ \frac{d}{dx}\mathbf{u} = \mathbf{A}(x) \mathbf{u} + \mathbf{b}. $$ If $\mathbf{A}(x)=\mathbf{A}$ were constant, then one can inspect the eigenvalues of the Jacobian of this equation for local bifurcations. However, how does one best go about bifurcation analysis when the system becomes non-autonomous?
It seems there are numerous angles one may take, but the first step it seems would be to try transform this system into an autonomous one defining a new variable $h = x$, giving $dh/dx=1$. Then we can define a new state vector $\mathbf{v}:=(\mathbf{u},h)$, so that our new set of equations looks like \begin{equation} \frac{d}{dx}\mathbf{v} = \frac{d}{dx} \begin{pmatrix} \mathbf{u} \\ h \end{pmatrix} = \begin{pmatrix} \mathbf{A}(h) & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} \mathbf{u} \\ h \end{pmatrix} + \begin{pmatrix} \mathbf{b} \\ 1 \end{pmatrix}. \end{equation} However I'm not sure if this is how one would best rewrite the equations. Supposing $\mathbf{A}$ is a 2x2 matrix, then the Jacobian then becomes \begin{equation} \begin{pmatrix} \frac{\partial}{\partial u_1} \mathbf{A_{1j}}^T \mathbf{u} & \frac{\partial}{\partial u_2} \mathbf{A_{1j}}^T \mathbf{u} & \frac{\partial}{\partial h} \mathbf{A_{1j}}^T \mathbf{u} \\ \frac{\partial}{\partial u_1} \mathbf{A_{2j}}^T \mathbf{u} & \frac{\partial}{\partial u_2} \mathbf{A_{2j}}^T \mathbf{u} & \frac{\partial}{\partial h} \mathbf{A_{2j}}^T \mathbf{u} \\ 0 & 0 & 0 \end{pmatrix}. \end{equation} This will have at least one zero eigenvalue. Moreover, one would evaluate the eigenvalues at steady-state values, however, I find it hard to understand what this steady-state value is for $x$ if $x$ is time or another variable e.g. position. This feels like I have gone wrong in how I approached this. I am new to this kind of analysis, and I would expect this kind of system to be rather common, but am struggling to find insight online. Any help would be greatly appreciated.
In general, it is not helpful to add the time as a new variable. However, an up-to-date account to bifurcations in nonautonomous equations can be found in https://link.springer.com/book/10.1007/978-3-031-29842-4