Suppose we have a system
$$\dot{x}=2x+y^2-1,$$ $$\dot{y}=6x-y^2+1.$$
The system has two equilibria at $(0,1)$ and $(0,-1)$. Now, when we linearise around these equilibria, we find the Jacobian
$$\begin{pmatrix} 2 & 2y \\ 6 & -2y \end{pmatrix} $$ and find the eigenvalues at each equilibrium.
Now consider a different system $$\dot{x}=y+2x+x^5,$$ $$\dot{y}=-x-y^5.$$
In this case, we first linearise the system to $$\dot{x}=y+2x,$$ $$\dot{y}=-x,$$ before finding the equilibria and finding the Jacobian.
My question is, why do we linearise first in the second equation? Is it because the Jacobian would have non-linear terms? But I have seen different systems with non-linear terms where the Jacobian had $y^2$.
If I had a system of equations like $$\dot{x}=x^3-3x^2+2x-y,$$ $$\dot{y}=x-y,$$ and want to analyse the stability of the equilibria, should I linearise first or just find the equilibria and find the Jacobian normally?