So I am trying to do a linear stability analysis for a very simple 2d discrete system:
\begin{equation} \begin{aligned} x_{n+1} &= y_{n}\\ y_{n+1} &= -\frac{x_{n}}{2} + ay_{n} + y_{n}^{3} \end{aligned} \end{equation}
One of the fixed points of this system is $(0,0)$ and I am interested in finding out the range of $a$ values in which this point is stable. I think there is something very wrong with what I am doing. The standard method of calculating eigenvalues of Jacobian gives me $a \in (-1.5, 1.5)$ but numerically I find that only between $(-0.5, 1.5)$ the point is stable and for $a\in (-1.5, -0.5)$ I numerically find that there exists a 2-cycle. I think this issue is not related to basin because I tried taking initial conditions quite close to $(0,0)$.
I would really appreciate if somebody comes up with the stability range and explain me what is wrong with my calculations. Thanks in advance.