Prove that if $\lambda_j$ are the eigen values of $Df(\bar x)$, and if $\lambda_j<1$, then $\bar x$ is assymptotically stable.

Question

We study the discrete dynamical system in $\mathbb{R^n}$ with differentiable function $f(x)$: $$x_{n+1}=f(x_n)$$ $1.$ Assume that $\bar x$ is a fixed point and consider small perturbations around $\bar x$ and define $x_n:=\bar x+y_n$ where $y_n$ is small.Derive the inearization of the above equation for $y_n$.

$2.$ Prove that if $\lambda_j$ are the eigen values of $Df(\bar x)$, and if $\lambda_j<1$, then $\bar x$ is assymptotically stable.

My Problem: I know only definations of stability,fixed point,jacobian but I do not know how to solve these.Please help me I need hints,I will follow them or give me some sketch of solution.

Thanks.

Answer 1

You need to look at $$x_{n+1}-\bar x=f(x_n) -f(\bar x) = Df_{\bar x}(x_n-\bar x)+o(x_n-\bar x). $$Let $A$ be the matrix $Df_{\bar x}$. You obtain a linearised system

$$y_{n+1}=Ay_n,$$which you will study by considering the eigenvalues of $A$. And then you will be able to conclude on the behavior of $x_n$.

If you need help with the above steps, ask in comments.