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

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

  1. Is there a periodic attractor for this 8 dimensional dynamical system?

2.Does $(*)$ have a periodic orbit which do not enter the diagonal space $$\{(A,B)\in M_{2}(\mathbb{R})\times M_{2}(\mathbb{R})\mid A, B \;\;\text{are diagonal matrices }\}$$

  1. 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 $(*)$.

  • 3
    $\begingroup$ Have you run a numerical test and plotted some slices? $\endgroup$ – AHusain Nov 8 '16 at 15:30
  • $\begingroup$ @AHusain Thanks for your comment. No I did not test. In this question I search for some theoretical methods which can be applied for consideration of a planar polynomial vector field as a vector field on matrix space. May be plotting can help us to find some ideas, but I did not try. $\endgroup$ – Ali Taghavi Nov 8 '16 at 18:57
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    $\begingroup$ It's "Van der Pol, not "Vander pol" $\endgroup$ – YCor Nov 9 '16 at 5:09

One thing I can tell you is that there are invariant manifolds $$ \eqalign{x_{21} &= a x_{12} \cr y_{21} &= a y_{12} \cr x_{11} &= x_{22} = y_{11} = y_{22} = 0\cr} $$ on which the system becomes $$ \eqalign{y_{12}' &= - x_{12}\cr x_{12}' &= x_{12} + y_{12} - a x_{12}^3 \cr}$$ which is similar to the van der Pol system if $a > 0$, but is unstable if $a < 0$.

In particular the answer to (2) is Yes.


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