I need to study the behavior of this discrete dynamical system:
$$\left\{ \begin{array}{c} x_{n+1}= y_n \\ y_{n+1}= \frac{1}{a}x_n-\frac{1}{a}y_n \\ \end{array} \right.$$
in function of the parameter $a>0$.
I studied before that the stability of the origin is given by the eigenvalues: $\lambda_1 , \lambda_2$
If $\lambda_1 , \lambda_2 < 1$, the origin is stable, hence, attractor.
If $\lambda_1 , \lambda_2 > 1$, the origin is unstable.
If $\lambda_1 > 1 > \lambda_2$ or $\lambda_2 > 1 > \lambda_1$, the origin is a saddle point.
First I found the matrix $A$ associated to this system
$A := \begin{bmatrix} 0 & 1 \\ \frac{1}{a} & \frac{-1}{a} \end{bmatrix}$
But I have problems when I'm studying the eigenvalues for stability. After analyzing the three cases, I determinated that the origin is a saddle point if $a \in (0,2)$ or the origin is stable if $a \in (2,+\infty)$. The problem is that if I graph this here, I always get a saddle point (at least in all the values of $a>0$ that i tested.
Any hints?