unique equilibrium in a perturbed system

Question

In today's lecture, I see that if $x_*$ is an asymptotic stable and hyperbolic equilibrium of the $\dot{x}=a(x), \, x\in\mathbb{R}^n$. But then prof said that "it's obvious" if we give a small $\epsilon$ s.t. $\dot{x}=a(x)+\epsilon b(x)$, then we can obtain another equilibrium $x_{**}$ near $x_*$ and it is unique. We assume $a,b$ are all smooth.

I feel this is obvious since we move the system a little bit by $\epsilon$, but I wonder how to write out a proof rigorously.

Answer 1

Hint: Implicit function theorem.

Let $f(x,\epsilon)=a(x)+\epsilon b(x)$. Then $Df_x(x_*,0)=Da(x_*).$ Why do the assumptions allow us to conclude $f(x(\epsilon),\epsilon)=0$ for small $\epsilon$?

Answer 2

Continuing on @zugzug answer, you need to argue that whether the $Da(x_*)$ is invertible or not(yes it is true), then using the Implicit Function Theorem, you can finish up your proof.