I've been struggling with understanding some key concepts on long term behaviour of Fourier modes.
Imagine I have a $N\times M$ system of cells with two quantities being measured by $x_{i,j}(t)$ and $y_{i,j}(t)$ which are governed a coupled of non-linear ODEs. After linearizing the system and using the discrete Fourier transform, the solution of the linearized problem is given by $$ \begin{pmatrix} \tilde{x}_{j,k}(t)\\ \tilde{y}_{j,k}(t) \end{pmatrix} =\sum_{\bar{q}}\sum_{\bar{p}}e^{2\pi i (\bar{q}j+\bar{p}k)}\left( v_{\bar{q},\bar{p}}^+e^{\lambda_{\bar{q},\bar{p}}^+t}+v_{\bar{q},\bar{p}}^-e^{\lambda_{\bar{q},\bar{p}}^-t} \right) $$ where $v_{\bar{q},\bar{p}}^\pm$ and $\lambda_{\bar{q},\bar{p}}^\pm$ are the eigenvectors and eigenvalues associated to the Fourier modes $(\bar{q},\bar{p})=(q/N,p/M)$, respectively. These naturally depend on the linearizing matrix associated to the specific problem.
Now, I read that the dominant pattern for large $t$ is a superposition of modes and therefore, by looking at the fastest growing modes, the previous solutions asymptotically satisfy $$ \begin{pmatrix} \tilde{x}_{j,k}(t)\\ \tilde{y}_{j,k}(t) \end{pmatrix}\sim \begin{pmatrix} C_x\\C_y \end{pmatrix} \sum_{(\bar{q},\bar{p})} e^{2\pi i (\bar{q}j+\bar{p}k)} $$ where the sum is now made over the fastest growing modes $(\bar{q},\bar{p})$ and $C_x,C_y$ are constants.
What I don't understand is the following:
- How do I conclude that, as $t\to\infty$, I can approximate the solution by this?
- What is the relation between the constants $C_x$ and $C_y$ and the eigenvectors and eigenvalues?
- Can $C_x$ and $C_y$ be complex?
I tried searching for answers, but I might be missing some algebraic trick or something essential about the asymptotic behaviour of the Fourier transforms and fast growing modes. Any ideas?