Suppose I have a 4 dimensional system with 4 fixed points: $Q_1 = \left(p_1,0,0,0 \right)$, $Q_2 = \left(0,p_2,0,0 \right)$, $Q_3= \left(0,0,p_3,0 \right)$, and $Q_4 = \left(0,0,0,p_4 \right)$. Furthermore, I'd like there to be separatrices connecting them in this fashion:
$$Q_1 \rightarrow Q_2 \rightarrow Q_3 \rightarrow Q_1$$ and $$Q_1 \rightarrow Q_2 \rightarrow Q_4 \rightarrow Q_1$$
Essentially, it looks something like this:
I've seen this design used in a number of papers but not one of them has produced an explicit example of a dynamical system which has this behavior. I can easily find a dynamical system where each cycle exists separately, but not one where the 2 cycles share a joint edge (e.g. Competitive Lotka-Volterra equations, Guckenheimer-Holmes cycle, etc). Is it possible to come up with such a system that has this behavior?