I am being asked to find the Taylor expansion up to order two with respect to $(w, \mu)$ of the center manifold for the following ODE:
$\frac{dx^2}{dt^2}+\frac{dx}{dt}-\mu x + x^2=0$, where $\mu \in \mathbb{R}$ is a small parameter.
We usually denote $w$ the central variables of the system, which are the variables for which $Re(\lambda)=0$, where $\lambda$ is the corresponding eigenvalue of the linear part of the system.
I've represented the ode as the system of equations
$\begin{Bmatrix} \frac{dx}{dt}=y \\ \frac{dy}{dt}= -y +\mu x -x^2 \end{Bmatrix}$
Which has linearisation A= $\begin{pmatrix} 0 &1 \\ -1 & \mu \end{pmatrix}$
I then calculated the eigenvalues of the linearisation of the system, which are $\lambda_{1,2}=\frac{\mu \pm i \sqrt{4-\mu^2}}{2} $. I then deduced that we have one stable variable, and one unstable variable, but no central variable, unless $\mu=0$. After the corresponding change of variables, diagonalising A, the system becomes
$\begin{Bmatrix} \frac{du}{dt}= \lambda_1 u \\ \frac{dv}{dt}= \lambda_2 v \end{Bmatrix}$
What does it mean to calculate the Taylor expansions of the central manifold wrt $(w, \mu)$ in this case?
I've calculated such a Taylor expansion once before, for a system in which we had one stable variable $u$, and one central variable $w$. I used the differential equations in the system to write $u$ as a function of $w$. I'm confused here as there appears to be no central variable $w$ in the system.
I'm not asking for a solution to the problem, just for someone to clarify the question for me.
Thank you!