I'm given a question dealing with the linearization of the lorenz system:
\begin{equation} \frac{dX}{dt} = Pr(Y-X), \hskip 0.1cm X(0) = X_{0} \end{equation}
\begin{equation} \hskip 0.63cm \frac{dY}{dt} = -XZ+rX-Y, \hskip 0.1cm Y(0) = Y_{0} \end{equation}
\begin{equation} \hskip -0.42cm \frac{dZ}{dt} = XY-bZ, \hskip 0.1cm Z(0) = Z_{0} \end{equation}
My hint tells me that: A nonlinear quantity,$YZ$, may be linearized by replacing with $Yn$ or $nZ$ where one of the original variables becomes a free parameter, $n$. Thus the above equations can be converted into a vector equation of the form
$$\dot x = A(n)x$$
Where $x=[X,Y,Z]^{T}$ and $\dot x$ means $\frac{dx}{dt}$. I have seen a few derivations throughout the internet, however non seem to explicitly employ this tactic. I'm not allowed to use "advanced" methods such as the Jacobian or anything like that - very basic knowledge of ODE's and linear algebra is all that's supposed to be required. Also, will this linearization allow me to find the critical points of the butterfly?