Motivated by the classical Van der Pol equation which has a unique periodic attractor, we consider the following differential equation on $M_{2}(\mathbb{R})\times M_{2}(\mathbb{R}):$

$$(*)\;\;\;\begin{cases} X'=Y-(X^{3}-X)\\ Y'=-X\end{cases}$$

- Is there a periodic attractor for this 8 dimensional dynamical system?
2.Does $(*)$ have a periodic orbit which do not enter the diagonal space $$\{(A,B)\in M_{2}(\mathbb{R})\times M_{2}(\mathbb{R})\mid A, B \;\;\text{are diagonal matrices }\}$$

- Is it true to say that, similar as the classical 2 dimensional case, the infinity of $\mathbb{R}^{8}\simeq M_{2}(\mathbb{R})\times M_{2}(\mathbb{R})$ is structurally unstable , in the following sense?:

"For every compact set $K\subset \mathbb{R}^{8}$, there is a compact set $\tilde{K} \supseteq K$ such that whenever $p \in \tilde{K}^{c}$ we have $\phi_{t}(p)\in K^{c},\;\;\forall t \leq 0$."

Here $\phi_{t}$ is the flow of the corresponding vector field $(*)$.