# 3d phase portrait for a system of DEs

I'm trying to plot a phase portrait for a system of three differential equations
so could anybody help? example for :

          x'[t]=y[t]+x[t]
y'[t]=y[t]z[t]+x[t]
z'[t]=z[t]-x[t]-y[t]


I've tried using PhasePlot[] (package here) and ParametricPlot3D[] but couldn't achieve anything

• Welcome to Mathematica.SE! I suggest the following: 1) As you receive help, try to give it too, by answering questions in your area of expertise. 2) Read the faq! 3) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign! Dec 6, 2014 at 19:23
• Can you show what you've tried? Dec 6, 2014 at 19:24
• It seems that your system diverges, that is, all trajectories run away. Dec 6, 2014 at 19:38
• @jens I seem to remember that phase plots were velocity against position. Dec 6, 2014 at 20:43
• @SjoerdC.deVries Terminology always depends on the context of the field. What you mean is a phase-space portrait.
– Jens
Dec 6, 2014 at 20:44

Here is what I get using my answer to I'd like to display field lines for a point charge in 3 dimensions. You only have to copy the definitions from the first code block in that answer, and then enter this:

seedList =
With[{vertices = .1 N[PolyhedronData["Icosahedron"][[1, 1]]]},
Join[Map[{#, 2} &, vertices],
Map[{# + {1, 1, 1}, -2} &, vertices]]];

Show[fieldLinePlot[{y + x, y z + x, z - x - y}, {x, y, z}, seedList,
PlotStyle -> {Orange, Specularity[White, 16], Tube[.01]},
PlotRange -> All, Boxed -> False, Axes -> None],
Background -> Black]


The seed points in seedList can be adjusted to highlight different features, if desired.