The specific problem I meet is the following:
Suppose I have $$\dot{x} = (x-x_e)$$ with an equilibrium point at $x_e \neq 0$.
I want to study how does $x_e$ shift after add a perturbation nonlinear but analytic function $g(x)$: $$\dot{x}= (x-x_e)+\epsilon g(x),$$ where $\epsilon$ is small.
So I can assume my solution as $$x(\epsilon) = x_e+\epsilon x_1 + O(\epsilon^2)$$ and then plug in. And since $g(x)$ is analytic, I can do Taylor expansion of $g(x)$ at a point.
Where does this point should be? $0$ or $x_e$ why?
Please advise, thanks!
It seems that you're interested in finding a formal power series in $\epsilon$ for the value of $x$ in $$x-x_e+\epsilon g(x)=0\text{.}$$ But when $\epsilon=0$, $x=x_e$, so all your derivatives are going to be taken at $x_e$. In particular, the formal power series in question is given by the Lagrange–Bürmann formula
$$x(\epsilon)=x_e+\sum_{n=1}^{\infty}(-1)^n\frac{\epsilon^n}{n!}\left.\frac{\mathrm{d}^{n-1}}{\mathrm{d}x^{n-1}}(g(x)^{n})\right\rvert_{x=x_e}\text{.}$$