I am having a two dimensional autonomous system
$S' = 2S^3+2S^2+\frac{1}{2}SA-\frac{3}{2}S-\frac{3}{4}A$
$A' = 4AS^2+A^2+4AS$
which exhibits a critical point at the origin $(S,A)=(0,0)$, and others. The eigenvalues of the linearization of the system at the origin yields
an eigenvalue $\lambda_1 = -\frac{3}{2}$ corresponding to the eigenvector $v_1=\left[\begin{matrix}1 \\ 0\end{matrix}\right]$,
and an eigenvalue $\lambda_2=0$ corresponding to the eigenvector $v_2=\left[\begin{matrix}-1/2 \\ 1\end{matrix}\right]$.
Hence there is a center manifold tangent to the second vector. I am new to center manifold theory, but found some theorems which look easy to apply for simple cases in two dimensions in the book of Perko – Differential Equations and Dynamical systems (3rd ED), section 2.11. The theorem however requires the system to first be put in a specific form. I quote from the book on p. 150:
We first consider the case when the matrix $A$ [that is the linearization matrix at the fixed point] has one zero eigenvalue, i.e. when $\det A=0$, but $\mathrm{tr}\,A\neq0$. In this case, as in Chapter 1 and as is shown in [A-I] on p. 338, the system (1) can be put into the form
$\dot x=p_2(x,y)$
$\dot y=y+q_2(x,y)$
where $p_2$ and $q_2$ are analytic in a neighborhood of the origin and have expansions that begin with second-degree terms in $x$ and $y$.
So I am certainly looking at such a case with one zero eigenvalue. However I do not see, how to bring it into the required form, and I need it in this form to apply the theorem which follows in the book right after.
The source [A-I] he quotes is A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, Qualitative Theory of Second-Order Dynamical Systems, John Wiley and Sons, New York 1973. My library doesn't have it, it seems out of print, and I also couldn't find it anywhere on the internet.
The reference to Chapter 1 in his own book didn't help me either so far, but I will check that in greater detail.
Does anyone here have an idea how to bring the system into the required form?
Cheers!
You use the eigenvalues and eigenvectors to diagonalise the system. So, you choose your coordinate axes along the eigenvectors, and write $$ \begin{pmatrix} S \\ A \end{pmatrix} = y\,v_1 + x\,v_2 = \begin{pmatrix} y-\frac{1}{2}x \\ x \end{pmatrix}, $$ which gives you the linear coordinate transformation $x = A$, $y = S + \frac{1}{2} A$. Hence, you obtain the system \begin{align} \dot{x} &= \text{second degree terms},\\ \dot{y} &= -\frac{3}{2} y + \text{second degree terms}. \end{align} Now, rescaling time by introducing $\tau = \lambda_1 t = -\frac{3}{2} t$, you transform the system into the required standard form.