How do I test stability of a MIMO system?
Let's try to make your system into a state space representation with state:
$$
\mathbf{x}(t) = \left [ 
\begin{array}{c}
x_1(t)\\
x_2(t)\\
x_1(t-1)\\
x_2(t-1)\\
\end{array}
\right ]
$$
so that
$$
\mathbf{x}(t) = \mathbf{A} \mathbf{x}(t-1) + \mathbf{y}(t)
$$
where
$$
\mathbf{A} =
\left [ \begin{array}{cccc}
a_{11} &amp; a_{12} &amp; a_{13} &amp; a_{14}\\
a_{21} &amp; a_{22} &amp; a_{23} &amp; a_{24}\\
1   &amp; 0 &amp; 0 &amp; 0\\
0   &amp; 1 &amp; 0 &amp; 0\\
\end{array}
\right ]
$$
and
$$
\mathbf{y}(t) = \left [ 
\begin{array}{c}
y_1(t)\\
y_2(t)\\
0\\
0
\end{array}
\right ]
$$
or you could make $\mathbf{y}$ $2 \times 1$ and introduce a $\mathbf{B}$ matrix.

The system may be unstable if the eigenvalues of $\mathbf{A}$ are outside the unit circle.