Drawing the phase portrait of two differential equations

Question

I am trying to make a phase portrait of two differential equations (second order) and the vector field of ODE. The equations are(I took the example from the following article):

x'=x-4x^2+2y^2+10xy
y'=y+4y^2+4xy

and the vector field is F(x,y)=(x-4x^2+2y^2+10xy,y+4y^2+4xy). I would like the result to be something like this. I have found many variants to model a phase diagram but none help me in this case (for example here I found this example, but I can't replicate it in latex). I would really appreciate your help because I have been trying to do something that works well regardless of the field I put in and nothing works for me.

Whereas the vector field should look something like this(I made this graph here):

Answer 1

A solution I often use to draw phase diagrams is this one from How to draw slope fields with all the possible solution curves in latex, which I added my version with two functions in quiver={ u={f(x,y)}, v={g(x,y)} ...}.

It lets me generate local quivers from functions f(x,y) and g(x,y) while keeping a predefined style. I may add new curves with \addplot such as \addplot +[blue] {-4*x};, which seems to be one of the the lines, the one with \addplot +[violet] {+x} I could visually find.

Improvements needed to achieve final result:

• Draw arrows correctly where I used \addplot to draw added functions.
• Draw arrows in quiver with curves.
• Automatically find equations for \addplot, as it is, one must do the math and then insert results.

\documentclass{article}
\usepackage{pgfplots}
\pgfplotsset{compat=1.8}

\usepackage{amsmath}
\usepackage{derivative}

\pgfplotsset{ % Define a common style, so we don't repeat ourselves
  MyQuiver2D/.style={
    width=0.6\textwidth, % Overall width of the plot
    axis equal image, % Unit vectors for both axes have the same length
    view={0}{90}, % We need to use "3D" plots, but we set the view so we look at them from straight up
    xmin=-2.1, xmax=2.1, % Axis limits
    ymin=-2.1, ymax=2.1,
    domain=-2:2, y domain=-2:2, % Domain over which to evaluate the functions
    xtick={-2,-1.5,...,2}, ytick={-2,-1.5,...,2}, % Tick marks %
    samples=21, % How many arrows?
    cycle list={    % Plot styles
      gray,
      quiver={
        u={f(x,y)}, v={g(x,y)}, % End points of the arrows
        scale arrows=0.015,
        every arrow/.append style={
          -latex % Arrow tip
        },
      }\\
      red, samples=31, smooth, very thick, no markers, domain=-2:2\\ % The plot style for the function
    }
  }
}

\begin{document}

\begin{tikzpicture}[
  declare function={f(\x,\y) = \x - 4*\x*\x + 2*\y*\y + 10*\x*\y;},
  declare function={g(\x,\y) = \y + 4*\y*\y + 4*\x*\y;}
  ]
  \begin{axis}[
      MyQuiver2D,
      title={$\displaystyle \odv{x}{t}=x-4x^2+2y^2+10xy; \odv{y}{t}=y+4y^2+4xy$},
      width=\textwidth
    ]
    \addplot3 (x,y,0);
    \addplot +[] {0};
    \addplot3 (x,y,0);
    \addplot +[magenta] {-4*x};
    \addplot3 (x,y,0);
    \addplot +[violet] {+x};
  \end{axis}
\end{tikzpicture}
\end{document}

---

Edit and Update

This solution improves:

• All vector are normalized and colored quiver, where colors represent the "strength" or "real size".
• Each plot has decorations with arrows.
• Better organization of styles to reuse and default settings.

The solution is based on:

• PGFPlots Parametric Plot, Arrow Spacing by Constant Time.
• Plot Vector Field in LaTeX TikZ.

While editing your graph, I realized it could be interesting show the fixed points of your EDO, despite they are not shown in the original paper. In this process, I noted I created the first solution with domain=-2:2, and I could not really see the vector field close to some fixed points. Therefore I added them with coordinates. I checked the fixed points with WolframAlpha:

• https://www.wolframalpha.com/input?i2d=true&i=%7B%7Bx-4Power%5Bx%2C2%5D%2B2Power%5By%2C2%5D%2B10xy%7D%2C%7By%2B4Power%5By%2C2%5D%2B4xy%7D%7D%3D%7B%7B0%7D%2C%7B0%7D%7D.

So I edited styles in order to better handles local defined domains and then I created a second figure with domain=-0.4:0.4.

A MWE follows:

\documentclass{article}
\usepackage{pgfplots}
\pgfplotsset{compat=1.8}
\usetikzlibrary{decorations.markings}

\usepackage{amsmath}
\usepackage{derivative}

% Define style to the axis
\pgfplotsset{MyQuiverAxis/.style={
    width=\textwidth, % Overall width of the plot
    xlabel={$x$}, ylabel={$y$},
    xmin=-2.1, xmax=2.1, % Axis limits
    ymin=-2.1, ymax=2.1,
    domain=-2:2, y domain=-2:2, % Domain over which to evaluate the functions
    axis equal image, % Unit vectors for both axes have the same length
    view={0}{90}, % We need to use "3D" plots, but we set the view so we look at them from straight up
    % colormap/viridis,
    colormap/hot,
    colorbar,
    colorbar style = {
      ylabel = {Vector Length}
    }
  }
}

% Define a common style to quivers
\pgfplotsset{MyQuiver2Dnorm/.style={
    %cycle list={% Plot styles
    samples=15, % How many arrows?
    quiver={
      u={f(x,y)/sqrt((f(x,y)^2+(g(x,y))^2))}, v={g(x,y)/sqrt((f(x,y)^2+(g(x,y))^2))}, % End points of the arrows
      scale arrows=0.2,
    },
    -latex,
    %},
    % domain=-0.5:0.5, y domain=-0.5:0.5, % Change if domain not equal to axis functions
    quiver/colored = {mapped color},
    point meta = {sqrt((f(x,y))^2+(g(x,y))^2)},
  }
}

\pgfplotsset{MyArrowDecorationPlot/.style n args={3}{
    decoration={
      markings,
      mark=between positions #1 and #2 step 2em with {\arrow [scale=#3]{latex}}
    }, postaction=decorate
  },
  MyArrowDecorationPlot/.default={0.1}{0.99}{1.5}
}

\begin{document}
\begin{tikzpicture}[
  declare function={f(\x,\y) = \x - 4*(\x)^2 + 2*(\y)^2 + 10*\x*\y;},
  declare function={g(\x,\y) = \y + 4*(\y)^2 + 4*\x*\y;}
  ]
  \begin{axis}[
    MyQuiverAxis,
    title={$\displaystyle \odv{x}{t} = x-4x^2+2y^2+10xy; \odv{y}{t} = y+4y^2+4xy$},
    ]
    \addplot3 [MyQuiver2Dnorm] (x,y,0);
    \addplot [thick, red, domain=2:-2, MyArrowDecorationPlot] {0};
    \addplot [thick, magenta, domain=2:-2, MyArrowDecorationPlot] {-4*x};
    \addplot [thick, violet, MyArrowDecorationPlot] {+x};
    \addplot [very thick, fill=white, only marks] coordinates {(0,0) (-1/8,-1/8) (1/12,-1/3) (1/4,0)};
  \end{axis}
\end{tikzpicture}

\begin{tikzpicture}[
  declare function={f(\x,\y) = \x - 4*\x*\x + 2*\y*\y + 10*\x*\y;},
  declare function={g(\x,\y) = \y + 4*\y*\y + 4*\x*\y;}
  ]
  \begin{axis}[
    MyQuiverAxis,
    title={$\displaystyle \odv{x}{t} = x-4x^2+2y^2+10xy; \odv{y}{t} = y+4y^2+4xy$},
    xmin=-0.4, xmax=0.4, % Axis limits
    ymin=-0.4, ymax=0.4,
    domain=-0.4:0.4, y domain=-0.4:0.4
    ]
    \addplot3 [MyQuiver2Dnorm, 
      domain=-0.35:0.35, y domain=-0.35:0.35,
      quiver={scale arrows=0.025}] (x,y,0);
      
    \addplot [thick, red, domain=0:-0.4, MyArrowDecorationPlot] {0};
    \addplot [thick, red, domain=0:1/4, MyArrowDecorationPlot] {0};
    \addplot [thick, red, domain=0.4:1/4, MyArrowDecorationPlot] {0};
    
    \addplot [thick, magenta, domain=0:-0.4,
      MyArrowDecorationPlot={0.05}{1}{1.25}] {-4*x};
    \addplot [thick, magenta, domain=0:1/12,
      MyArrowDecorationPlot={0.05}{1}{1.25}] {-4*x};
    \addplot [thick, magenta, domain=0.4:1/12, 
      MyArrowDecorationPlot={0.05}{1}{1.25}] {-4*x};
    
    \addplot [thick, violet, domain=-0.4:-1/8, MyArrowDecorationPlot] {+x};
    \addplot [thick, violet, domain=0:-1/8, MyArrowDecorationPlot] {+x};
    \addplot [thick, violet, domain=0:0.4, MyArrowDecorationPlot] {+x};
    
    \addplot [very thick, fill=white, only marks] coordinates {(0,0) (-1/8,-1/8) (1/12,-1/3) (1/4,0)};
  \end{axis}
\end{tikzpicture}
\end{document}

---

Figures

First figure with domain=-2:2.

Second figure with domain=-0.4:0.4. This new solution shows how to change the domain of the vector field and also how to show the decorated plots in the same direction as the vector field.

Answer 2

I guess this could be made with pgfplots, using a variation of this code? (Note that it must be compiled with lualatex.)

\documentclass[border=0.2cm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat = newest}

\begin{document}

\begin{tikzpicture}
\begin{axis}[
   view = {0}{90},
   domain = -0.2:0.2,
   domain y = -0.2:0.2,
   xmin=-0.2,
   xmax=0.2,
   ymin=-0.2,
   ymax=0.2,
   ]
   \addplot3[
      -stealth,
      black!20,
      samples=20,% Controls the number of arrows
      quiver = {
         u = x-4*x^2+2*y^2+10*x*y,
         v = y+4*y^2+4*x*y,
         scale arrows = 0.2,
      }
      ] {0};
   \addplot3[contour lua={number=10, labels=false}] {x-4*x^2+2*y^2+10*x*y};% !!!!! NEEEDS lualatex for compilation
\end{axis}
\end{tikzpicture}

\end{document}