I am interested in studying bifurcation in a specific class of dynamical systems, which follow a change of variables and decoupling by a Fourier transform.
In particular, consider the following dynamical system, which is given, at each index $1\leq j\leq N$, by \begin{align} \dot{x}_j&=F_x(x_j,\{x_{r}\},y_j,\{y_{r}\})\label{eq1gn}\\ \dot{y}_j&=F_y(x_j,\{x_{r}\},y_j,\{y_{r}\})\label{eq2gn}, \end{align} where $F_x$ and $F_y$ are general nonlinear functions. Here, $\{x_{r}\}$ and $\{y_{r}\}$ are the sets of all the variables such that $r\neq j$. Such a system is translationally invariant (dependence on other indexed variables is always relative to the focal index). Note that we have a total of $2N$ equations. A homogeneous steady state of the system, $(x^*,y^*)$, requires $\dot{x}_j=\dot{y}_j=0$ when $(x_j,y_j)=(x^*,y^*)$, for any cell $j$. This results in the homogeneous system \begin{align} 0&=F_x(x^*,\{x^*\},y^*,\{y^*\})\label{eq1g0n}\\ 0&=F_y(x^*,\{x^*\},y^*,\{y^*\})\label{eq2g0n}. \end{align} Next, we perturb this solution by setting $(x_j,y_j)=(x^*+\tilde{x}_j,y^*+\tilde{y}_j)$ and, following linearisation, we get \begin{equation}\label{eqlinsys1} \frac{d}{dt} \begin{pmatrix} \tilde{x}_j\\ \tilde{y}_j \end{pmatrix}\simeq \mathbf{J} \begin{pmatrix} \tilde{x}_j\\ \tilde{y}_j \end{pmatrix}+\sum_{r\neq j} \mathbf{J}^r \begin{pmatrix} \tilde{x}_r\\ \tilde{y}_r \end{pmatrix}, \end{equation} where \begin{equation}\label{eqJr} \mathbf{J}=\left. \begin{pmatrix} \frac{\partial F_x}{\partial x_j} & \frac{\partial F_x}{\partial y_j}\\ \frac{\partial F_y}{\partial x_j} & \frac{\partial F_y}{\partial y_j} \end{pmatrix}\right\rvert_{(x^*,y^*)}\text{ and }\mathbf{J}^r= \left.\begin{pmatrix} \frac{\partial F_x}{\partial x_r} & \frac{\partial F_x}{\partial y_r}\\ \frac{\partial F_y}{\partial x_r} & \frac{\partial F_y}{\partial y_r} \end{pmatrix}\right\rvert_{(x^*,y^*)}. \end{equation}
Now, we may decouple the system of $2N$ equations by performing a discrete Fourier transform with respect to $j$ and changing the variables as follows, for $1\leq q\leq N$, \begin{align} \tilde{x}_j&=\sum_{q=1}^{N} \xi_{q} e^{2\pi \text{i}qj/N}\label{ch01}\\ \tilde{y}_j&=\sum_{q=1}^{N} \eta_{q} e^{2\pi \text{i}qj/N}\label{ch02}, \end{align} which may also be written as \begin{align} \xi_{q}&=\frac{1}{N} \sum_{j=1}^N \tilde{x}_{j}e^{-2\pi \text{i}qj/N}\label{ch11}\\ \eta_{q}&=\frac{1}{N} \sum_{j=1}^N \tilde{y}_{j}e^{-2\pi \text{i}qj/N}\label{ch12}. \end{align} This leads to the linear system \begin{align}\label{eqsysone} \frac{d}{dt}\begin{pmatrix} \xi_q\\ \eta_q \end{pmatrix}\simeq \mathbf{L}_{q}\begin{pmatrix} \xi_q\\ \eta_q \end{pmatrix} \end{align} for some matrix $\mathbf{L}_{q}$.
My question: While studying bifurcation dynamics around steady-state, we are often interested in looking at non-hyperbolic equilibria, where the real part of an eigenvalue of the Jacobian matrix at such equilibrium is zero. Therefore, I wonder if, given the decoupling done here, for a translationally invariant system, it is possible to simply study bifurcation given this new linearized system. In other words, following transformation into the frequency domain, can the stability of each Fourier mode provide insights into the behaviour of the nonlinear system near the equilibrium point? Furthermore, could center manifold theory, for example, be applied to in the simpler decoupled form?
Example: A simple example to illustrate what I mean by translational invariance is \begin{align} \dot{x}_j&=f(y_{j-1}+y_{j+1})-x_j\\ \dot{y}_j&=g(x_j)-y_j, \end{align} where $f$ and $g$ are nonlinear and could, for example, incorporate bifurcation parameters.