I am a bit confused about the connection between linear stability analysis, bifurcation points and amplitude expansions. I have a non-linear system, given as, for $1\leq i \leq n$, $$ \frac{d}{dt}\begin{pmatrix} x_i\\ y_i \end{pmatrix}=\begin{pmatrix} f_1(x_1,...,x_n,y_1,...,y_n)\\ f_2(x_1,...,x_n,y_1,...,y_n) \end{pmatrix}=\begin{pmatrix} f_1(\mathbf{x},\mathbf{y})\\ f_2(\mathbf{x},\mathbf{y}) \end{pmatrix} $$ and I want to estimate how much a linear or weakly nonlinear solution explains the numerical solution of $(\mathbf{x},\mathbf{y})$.
One thing I can try to do first is to perform standard linear stability analysis, around a homogeneous steady-state $(\mathbf{x}^*,\mathbf{y}^*)$. After linearising the system, I may decouple it by applying a discrete Fourier transform (changing variables to $(\xi_i, \chi_i)$), so that I end up with a system in the form $$ \frac{d}{dt}\begin{pmatrix} \xi_i\\ \chi_i \end{pmatrix}=L\begin{pmatrix} \xi_i \\ \chi_i \end{pmatrix} $$ for some matrix $L$. While this is all covered in the literature, I now struggle understanding some of the conclusions I may extract from this approach.
First, I can work out solutions to the linearised system and explore the Fourier mode selection based on that (dependent on the eigenvalues of $L$). Second, I was asked to perform an amplitude (mode) expansion close to the bifurcation point with a few relevant modes, but I am struggling to understand this, given what I have achieved thus far. How would I determine the bifurcation in this case?
In general, I am aware of multiple scale methods and weakly nonlinear analysis performed in reaction-diffusion equations. In the first, for example, considered solutions are of the form $$ \mathbf{x}=\mathbf{x}_0+\epsilon \mathbf{x}_1 + \epsilon^2 \mathbf{x}_2 + \epsilon^3 \mathbf{x}_3 +\cdots $$ and similar for $\mathbf{y}$. Note that the case $\mathbf{x}=\mathbf{x}_0+\epsilon \mathbf{x}_1$. However, I am not too sure how to implement this in a differential system such as this one, given I decouple the system after linearisation. On a personal note, I often find it hard to track some of the papers/textbooks on these topics, as they tend to focus on specific equations, but I wonder if there is more general literature where I could learn more about this.