Plotting streams of vector fields

Question

So I have the vector field ${x^2+2*x*y,y^2+2*x*y} and I would like to plot 'streams.' quiver from pgfplots does produce arrows (which I normalise) but I think that streams would make things easier to understand. Here is a StreamPlot that mathematica creates.

I would like to be able to make something like this with pgfplots to be able to maintain style consistency. Perhaps if I could extract the points that produce the curves then that could be plotted with pgfplots.

Answer 1

I used this: https://mathematica.stackexchange.com/questions/19859/plot-extract-data-to-a-file initially. The first element in the plot dynamicalStream was an empty string plus an object GraphicsComplex. I extracted the points from the plot with points = dynamicalStream[[1]][[2]][[1]]. It may be different for someone else. The Line objects from this GraphicsComplex object were extracted with (and I apologise for the way I did this, there are probably better methods)

lines = dynamicalStream[[1]][[2]][[2]][[2]][[3]][[2 ;; Length[ dynamicalStream[[1]][[2]][[2]][[2]][[3]]]] ].

I'd say this is unlikely to be useful, so either you find a more general method or do what I did which is manually find the correct part of the list to extract from.

I then created a list where each element is itself a list of points for each line. This was done with pointstable = Table[points[[#[[1]][[i]]]], {i, 1, Length[#[[1]]]}] & /@ lines. The inner list was created by going through each element in the Line object (this is a position of a point in the points list) and extracting the correct point. Thus you make a list of points corresponding to the Line. This is then mapped over each Line object in lines so the final list is of the described form.

I then saved each list in a separate text file: Export["line" <> IntegerString[#2] <> ".txt", #,"Table"] &~MapIndexed~pointstable as described in the link.

I plotted with:

\documentclass[border=5pt]{standalone}
\usepackage{pgfplots}
\usetikzlibrary{decorations.markings}
\tikzset{->-/.style={decoration={
  markings,
  mark=at position #1 with {\arrow{>}}},postaction={decorate}}}
  \tikzset{-<-/.style={decoration={
  markings,
  mark=at position #1 with {\arrow{<}}},postaction={decorate}}}

\begin{document}
    \begin{tikzpicture}[scale=1.5]
        \begin{axis}[axis lines=none]
            \foreach \i in {1,2,3,4,...,107}{
                \addplot[black,->-=0.5] table[]{line\i.txt};
            }
        \end{axis}
    \end{tikzpicture}
\end{document}

After looking at that link properly, an easier way to do it would be:

points  = Cases[dynamicalStream, GraphicsComplex[data__] :> data, -3, 1][[1]]
lines = Cases[dynamicalStream, Line[data__] :> data, -3];
pointstable = Table[points[[#[[i]]]], {i, 1, Length[#]}] & /@ lines