I have been doing linear stability analysis (LSA) of systems of the autonomous form $$ \frac{du}{dt}=F(u)\Leftrightarrow \begin{cases} \frac{dy}{dt}&=f(y,z)\\ \frac{dz}{dt}&=g(y,z) \end{cases} $$ where $u=(y,z)$ and $F=(f,g)$, for nonlinear functions $f$ and $g$. I am now interested in doing weakly nonlinear stability analysis (WNSA), as I hope to get a better insight into the solutions of such a system.
However, I have been struggling to find good references for the main methods and tools to perform WNSA on autonomous systems. This paper, for example, looks at the case of Turing patterns, but the details are somewhat hard to follow. Turing's original paper on morphogens and reaction-diffusion also contains some explanation (see Section 9.4), but I was wondering if there are more references out there that I could look into, for a general space-independent system such as mine.
Any ideas?
What I've gathered so far: While in LSA we look for solutions of the type $$ u(t)=u^*+\tilde{u} $$ where $u^*$ is the homogeneous solution and $\tilde{u}$ is a small perturbation which is solution to the linearised problem $\frac{d\tilde{u}}{dt}=\mathbf{J}(u^*)\tilde{u}$, where $\mathbf{J}$ is the Jacobian matrix of $F$, given by, in general $$ \tilde{u}=C^+v^+e^{\lambda^+t}+C^-v^-e^{\lambda^-t} $$ where $\lambda^\pm$ and $v^\pm$ are the eigenvalues and eigenvectors of the matrix $\mathbf{J}$. $C^+$ and $C^-$ are constants that depend on the initial conditions of the system.
In WNSA, according to these papers [ref1,ref2,ref3], we aim to look for solutions given by the so-called Stuart-Watson formula given, for reaction-diffusion equations that also depend on the spatial coordinate $x$, by $$ u(x,t)=\sum_{m=0}^3 u_m(x)A^m(t)+O(A^4) $$ where $$ u_m(x)=\sum_{j=0}^m u_{mj}\cos(j\omega x) $$ where $\omega$ is the critical wavenumber, and $$ \frac{dA}{dt}(t)=\sum_{m=0}^3 b_m A^m(t)+O(A^4) $$ Finding coefficients $u_{mj}$ and $b_m$ relies then on substituting such candidate solution on the original system, leading to a sequence of vector problems.
In the autonomous system, however, there is no dependence on $x$, so I wonder how I can still apply this method to it. Intuitively I would just drop the cosine terms (or set $\omega=0$) to get $$ u(t)=\sum_{m=0}^3 u_m A^m(t)+O(A^4) $$ where $u_m$ are now constants. Is this fine? How do I then select the coefficients $u_m$ and $b_m$?
Notice also that this case mimics the LSA approach when the sums are up to $1$ and $u_0=u^*$, $u_1=1$, $b_0=0$ and $b_1=\lambda^+$, where $\lambda^+$ is the fastest growth rate (eigenvalue with highest real part). $A(t)$ would, in this case, correspond to $\tilde{u}$, I think.