# Visualize fixed points and stable points in 4D

I have a list of 4 dimensional data points which are sum to one and always positive. These points are the fixed points of the 4 dimensional nonlinear ODE. I also found the stable points of the system using fixed point analysis. (I found the Jacobian at the fixed points and looked at the eigenvalues to decide the stability). Now, I am trying to visualize the fixed points and stable points in 4D. I am not sure whether this is doable but at least I am trying to get 3 dimensional phase planes, trajectories or basin of attractions type graphs if it is possible. Let say as an example, I have the following data points:

{{ 0,0,0,1}, {0,1,0,0}, {0,0,1,0}, {0.0740741, 0.925926, 0, 0},
{1,0,0,0}, {0.444444, 0, 0, 0.555556}, {0.333333, 0, 0.154762, 0.511905},
{0, 0.483592, 0.491029, 0.0253783}, { 0.10009, 0.431624, 0.468287, 0},
{0.137688, 0.283838, 0.389616, 0.188858},
{0, 0.5, 0.5, 0}}


After the fixed point analysis I found that {0,0,1,0} and { 0.0740741, 0.925926, 0, 0} are only stable fixed points. For now, I would like to plot these points as I explained above.

Any help will be greatly appreciated. Thank you so much.

Your points lay on an "affine subspace".

If you were in $R^2$:

p = {#[], 1 - Tr@##} & /@ RandomReal[{0, 1}, {10, 1}];
p1 = RotationMatrix[{{1, 1}, {0, 1}}].# - {0, Norm@{1, 1}/2} & /@ p;
GraphicsRow[{ListPlot@p, ListPlot@p1}] In $R^3$

p = {#[], #[], 1 - Tr@##} & /@ RandomReal[{0, 1}, {10, 2}];
p1 = RotationMatrix[{{1, 1, 1}, {0, 0, 1}}].# - {0, 0, Norm@{1, 1, 1}/3} & /@ p;
Framed@GraphicsRow[{ListPlot3D@p, ListPointPlot3D[p1]}] And in $R^4$

p = {#[], #[], #[], 1 - Tr@##} & /@ RandomReal[{0, 1}, {10, 3}];
p1 = RotationMatrix[{{1, 1, 1, 1}, {0, 0, 0, 1}}].# - {0, 0, 0, Norm@{1, 1, 1, 1}/4} & /@ p;
Framed@ListPointPlot3D[p1[[All, 1 ;; 3]]] (The other plot (in $R^4$) is rather difficult to see in SE)

• Okay...How about trajectories or direction fields in 4D ? I tried to do it using VectorPlot3D and that gave me some arrows, but still does not explain the system's behavior very well. Also, I want to have some trajectories which goes toward to stable points. – UTK Mar 18 '14 at 18:13
• @UTK Once you have the trajectory function, just transform it according to the affine transform I showed for p1 and plot it on R3 (first three components, the last one should be zero) – Dr. belisarius Mar 18 '14 at 18:23