Suppose I have a 4D dynamical system. Each axis has a fixed point, and there are orbits connecting the fixed points. It looks something like this:
Each $Q_i$ is a fixed point on each axis of a four-dimensional coordinate space. I believe I have found parameters of my dynamical system that leads to the behavior in the picture. However, I'd like to see if these parameters work numerically. I'd like to see if I start near the $Q_1-Q_2-Q_3$ plane, I stay there for all time, and if I start near the Each $Q_i$ is a fixed point on each axis of a four-dimensional coordinate space. I believe I have found parameters of my dynamical system that leads to the behavior in the picture. However, I'd like to see if these parameters work numerically. I'd like to see if I start near the $Q_2-Q_3-Q_4$ plane, I stay there for all time. Essentially, in the first case, the 4th dimension value tends to $0$ asymptotically, and in the latter, the 1st dimensional value tends to $0$. I should therefore have 2 basin of attractions for each plane. However, since this is a 4D system, this is not easy to plot and show numerically.
What is the best way to plot numerically the 4D system so that I can observe that I have 2 basin of attractions, each one exhibiting a heteroclinic cycle in each plane?