Suppose we have a discrete dynamical system $$x_{k+1}=M(x_k)$$ where $M(x)$ is a diffeomorphism and $x\in \mathbb{R}^n$.
We have a fixed point of the system $\rho$, ($M(\rho)=\rho$) that satisfies: $DM(x)_{x=\rho}$ has no eigenvalues $\lambda=1$.
How would we proof that if we add a perturbation to the system: $$x_{k+1}=M(x_k)+\epsilon Q(x_k) $$ where $Q(x)$ is another diffeomorphism, the new system has a new fixed point $x(\epsilon)$ and $\left|x(\rho)-\rho \right|=\mathcal{O}(\epsilon) $.
