Drawing the Trace Determinant Diagram on LaTeX
% Poincaré Diagram: Classification of Phase Portaits in the (det A,Tr A)-plane
% Author: Gernot Salzer
% Based on a drawing by Douglas R. Hundley, people.whitman.edu/~hundledr/courses/M244/Poincare.pdf
\documentclass[border=1mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{decorations.markings,arrows}
\tikzset
{every pin/.style={pin edge={<-}}
,>=stealth
,flow/.style=
{decoration=
{markings
,mark=at position #1 with {\arrow{>}}
}
,postaction={decorate}
}
,flow/.default=0.5
}
\newcommand\inlayscale{}
\newcommand\inlaycaption[1]{{\sffamily\scriptsize#1}}
\newcommand\newinlay[4][0.18]%
{\renewcommand\inlayscale{#1}%
\newsavebox#2%
\savebox#2%
{\begin{tabular}{@{}c@{}}
#4\\[-1ex]
\inlaycaption{#3}\\[-1ex]
\end{tabular}%
}%
}
\newcommand\inlay[1]{\usebox{#1}}
\newcommand\Tr{\mathop{\mathrm{Tr}}}
\newinlay\saddle{saddle}%
{\begin{tikzpicture}[scale=\inlayscale]
\foreach \sx in {+,-}
{\draw[flow] (\sx4,0) -- (0,0);
\draw[flow] (0,0) -- (0,\sx4);
\foreach \sy in {+,-}
\foreach \a/\b/\c/\d in {2.8/0.3/0.7/0.6,3.9/0.4/1.3/1.1}
\draw[flow] (\sx\a,\sy\b)
.. controls (\sx\c,\sy\d) and (\sx\d,\sy\c)
.. (\sx\b,\sy\a);
}
\end{tikzpicture}%
}
\newinlay\sink{sink}%
{\begin{tikzpicture}[scale=\inlayscale]
\foreach \sx in {+,-}
{\draw[flow] (\sx4,0) -- (0,0);
\draw[flow] (0,\sx4) -- (0,0);
\foreach \sy in {+,-}
\foreach \a/\b in {2/1,3/0.44}
\draw[flow,domain=\sx\a:0] plot (\x, {\sy\b*\x*\x});
}
\end{tikzpicture}%
}
\newinlay\source{source}%
{\begin{tikzpicture}[scale=\inlayscale]
\foreach \sx in {+,-}
{\draw[flow] (0,0) -- (\sx4,0);
\draw[flow] (0,0) -- (0,\sx4);
\foreach \sy in {+,-}
\foreach \a/\b in {2/1,3/0.44}
\draw[flow,domain=0:\sx\a] plot (\x, {\sy\b*\x*\x});
}
\end{tikzpicture}%
}
\newinlay\stablefp{line of stable fixed points}%
{\begin{tikzpicture}[scale=\inlayscale]
\draw (-4,0) -- (4,0);
\foreach \s in {+,-}
{\draw[flow] (0,\s4) -- (0,0);
\foreach \x in {-3,-2,-1,1,2,3}
\draw[flow] (\x,\s3) -- (\x,0);
}
\end{tikzpicture}%
}
\newinlay\unstablefp{line of unstable fixed points}%
{\begin{tikzpicture}[scale=\inlayscale]
\draw (-4,0) -- (4,0);
\foreach \s in {+,-}
{\draw[flow] (0,0) -- (0,\s4);
\foreach \x in {-3,-2,-1,1,2,3}
\draw[flow] (\x,0) -- (\x,\s3);
}
\end{tikzpicture}%
}
\newinlay\spiralsink{spiral sink}%
{\begin{tikzpicture}[scale=\inlayscale]
\draw (-4,0) -- (4,0);
\draw (0,-4) -- (0,4);
\draw[samples=100,smooth,domain=27:7] plot ({\x r}: {0.005*\x*\x});
\draw[->] ({26 r}: {0.005*26*26}) -- +(0.01,-0.01);
\end{tikzpicture}%
}
\newinlay\spiralsource{spiral source}%
{\begin{tikzpicture}[scale=\inlayscale]
\draw (-4,0) -- (4,0);
\draw (0,-4) -- (0,4);
\draw [samples=100,smooth,domain=10:28] plot ({-\x r}: {0.005*\x*\x});
\draw[<-] ({-27.5 r}: {0.005*27.5*27.5}) -- +(0.01,-0.008);
\end{tikzpicture}%
}
\newinlay[0.15]\centre{center}%
{\begin{tikzpicture}[scale=\inlayscale]
\draw (-4,0) -- (4,0);
\draw (0,-4) -- (0,4);
\foreach \r in {1,2,3} \draw[flow=0.63] (\r,0) arc (0:-360:\r cm);
\end{tikzpicture}%
}
\newinlay\degensink{degenerate sink}%
{\begin{tikzpicture}[scale=\inlayscale]
\draw (0,-4) -- (0,4);
\draw[flow] (-4,0) -- (0,0);
\draw[flow] (4,0) -- (0,0);
\draw[flow] (-3.5,3.5) .. controls (4,1.5) and (4,1).. (0,0);
\draw[flow] (3.5,-3.5) .. controls (-4,-1.5) and (-4,-1) .. (0,0);
\draw[flow] (-3.5,2.5) .. controls (2,1) and (2,0.8).. (0,0);
\draw[flow] (3.5,-2.5) .. controls (-2,-1) and (-2,-0.8) .. (0,0);
\end{tikzpicture}%
}
\newinlay\degensource{degenerate source}%
{\begin{tikzpicture}[scale=\inlayscale]
\draw (0,-4) -- (0,4);
\draw[flow] (0,0) -- (-4,0);
\draw[flow] (0,0) -- (4,0);
\draw[flow] (0,0) .. controls (4,1.5) and (4,1).. (-3.5,3.5);
\draw[flow] (0,0) .. controls (-4,-1.5) and (-4,-1) .. (3.5,-3.5);
\draw[flow] (0,0) .. controls (2,1) and (2,0.8).. (-3.5,2.5);
\draw[flow] (0,0) .. controls (-2,-1) and (-2,-0.8) .. (3.5,-2.5);
\end{tikzpicture}%
}
\begin{document}
\begin{tikzpicture}[line cap=round,line join=round]
% Main diagram
\draw[line width=1pt,->] (0,-0.3) -- (0, 4.7) coordinate (+y);
\draw[line width=1pt,->] (-5,0) -- ( 5,0) coordinate (+x);
\draw[line width=1pt, domain=-4:4] plot (\x, {0.25*\x*\x});
\node at (+y) [label={[above,yshift=0.8cm]%
{\sffamily\large Poincar\'e Diagram: Classification of Phase Portaits
in the $(\det A,\Tr A)$-plane}}] {};
\node at (+x) [label={[right,yshift=-0.5ex]$\scriptstyle\Tr A$}] {};
\node at (+y) [label={[above]$\scriptstyle\det A$}] {};
\node at (-4,4) [pin={[above]$\scriptstyle\Delta=0$}] {};
\node at ( 4,4) [pin={[above,align=left]{%
$\scriptstyle\Delta=0$:\\
$\scriptstyle\det A=\frac{1}{4}(\Tr A)^2$}}] {};
% inlays
\node at (0,-1.4) {\inlay\saddle};
\node at (0,1.2)
[pin={[draw,right,xshift=0.3cm]\inlay\centre}] {};
\node at (0,0)
[pin={[draw,above left,align=center,xshift=-0.3cm]%
\inlaycaption{uniform}\\[-1ex]\inlaycaption{motion}}] {};
\node at (-4,1) {\inlay\sink};
\node at ( 4,1) {\inlay\source};
\node at (-3,0) [pin={[draw,below,yshift=-1cm]\inlay\stablefp}] {};
\node at (3,0) [pin={[draw,below,yshift=-1cm]\inlay\unstablefp}] {};
\node at (-1.8,3.7) {\inlay\spiralsink};
\node at ( 1.8,3.7) {\inlay\spiralsource};
\node at (-3.5,{0.25*3.5*3.5})
[pin={[draw,left,xshift=-1.15cm,yshift=-0.3cm]\inlay\degensink}] {};
\node at ( 3.5,{0.25*3.5*3.5})
[pin={[draw,right,xshift=0.9cm,yshift=-0.3cm]\inlay\degensource}] {};
\end{tikzpicture}
\end{document}
Before posting his final winning answer, gernot had posted a number of answers, each of which represented a single graphical component of the final picture. Once posting his final answer, he deleted the component answers.
For this answer, I lifted those pieces-parts from gernot's [since deleted] components and the OP's question, to demonstrate that, once you have all the components, one can use \stackinset to overlay them on the base graph, perhaps.
His components included the tikz parts identified in the following MWE as \spiralA, \spiralB, \degensink, \degensource,
\nondegensink, and \nondegensource,
\documentclass[10pt]{article}
\usepackage{pgf,tikz}
\usetikzlibrary{arrows}
\usepackage{mathrsfs}
\usepackage{amssymb,fancyhdr,txfonts,pxfonts}
\pagestyle{empty}
\usetikzlibrary{decorations.markings}
\usepackage{stackengine}
\renewcommand\stacktype{L}
\setstackgap{L}{8pt}
\begin{document}
\savestack\mygraph{%
\begin{tikzpicture}[%
line cap=round,
line join=round,
>=triangle 45,
x=1.0cm,
y=1.0cm%
]
%
\clip(-5.8,-3.58) rectangle (5.56,5.64);
\draw [line width=1.2pt] (0.,4.68)-- (0.,-0.74);
\draw [line width=1.2pt] (-4.,0.)-- (4.3,0.);
\draw [thick, domain=-2:2] plot (\x, {\x*\x});
\draw (3.62,-0.12) node[anchor=north west] {$\mathrm{Tr}\,A$};
\draw (-.4,5.1) node[anchor=north west] {$\scriptstyle\det A$};
\draw (-2.35,4.7) node[anchor=north west] {%
\stackunder{$\scriptstyle\Delta=0$}{$\downarrow$}};
\draw (1.08,5.1) node[anchor=north west] {\ensurestackMath{\stackunder[11pt]{%
\Longstack[l]{\scriptstyle\Delta=0: \scriptstyle\det A=\frac{1}{4}(%
\mathrm{Tr}A)^2}}{\downarrow}}};
\draw (1.,-1.)-- (-1.,-3.);
\draw (-1.,-1.)-- (1.,-3.);
\draw [dash pattern=on 2pt off 2pt] (-1.,-2.)-- (1.,-2.);
\draw [dash pattern=on 2pt off 2pt] (0.,-1.)-- (0.,-3.);
\end{tikzpicture}%
}
\savestack\spiralA{%
\begin{tikzpicture}
[decoration=
{markings,
mark=at position 0.1 with {\arrow{stealth}}
}
,scale=0.5
]
\draw (-4,0) -- (4,0);
\draw (0,-4) -- (0,4);
\draw[postaction={decorate},domain=27:7,variable=\t,smooth,samples=100]
plot ({\t r}: {0.005*\t*\t});
\end{tikzpicture}%
}
\savestack\spiralB{%
\begin{tikzpicture}
[decoration=
{markings,
mark=at position 0.99 with {\arrow{stealth}}
}
,scale=0.5
]
\draw (-4,0) -- (4,0);
\draw (0,-4) -- (0,4);
\draw[postaction={decorate},domain=10:28.5,variable=\t,smooth,samples=100]
plot ({-\t r}: {0.005*\t*\t});
\end{tikzpicture}
}
\savestack\degensink{%
\begin{tikzpicture}
[decoration=
{markings,
mark=at position 0.5 with {\arrow{stealth}}
}
,scale=0.5
]
\draw (0,-4) -- (0,4);
\draw[postaction={decorate}] (-4,0) -- (0,0);
\draw[postaction={decorate}] (4,0) -- (0,0);
\draw[postaction={decorate}] (-3.5,3.5) .. controls (4,1.5) and (4,1).. (0,0);
\draw[postaction={decorate}] (3.5,-3.5) .. controls (-4,-1.5) and (-4,-1) .. (0,0);
\draw[postaction={decorate}] (-3.5,2.5) .. controls (2,1) and (2,0.8).. (0,0);
\draw[postaction={decorate}] (3.5,-2.5) .. controls (-2,-1) and (-2,-0.8) .. (0,0);
\end{tikzpicture}%
}
\savestack\degensource{%
\begin{tikzpicture}
[decoration=
{markings,
mark=at position 0.5 with {\arrow{stealth}}
}
,scale=0.5
]
\draw (0,-4) -- (0,4);
\draw[postaction={decorate}] (0,0) -- (-4,0);
\draw[postaction={decorate}] (0,0) -- (4,0);
\draw[postaction={decorate}] (0,0) .. controls (4,1.5) and (4,1).. (-3.5,3.5);
\draw[postaction={decorate}] (0,0) .. controls (-4,-1.5) and (-4,-1) .. (3.5,-3.5);
\draw[postaction={decorate}] (0,0) .. controls (2,1) and (2,0.8).. (-3.5,2.5);
\draw[postaction={decorate}] (0,0) .. controls (-2,-1) and (-2,-0.8) .. (3.5,-2.5);
\end{tikzpicture}%
}
\savestack\nondegensink{%
\begin{tikzpicture}
[decoration=
{markings,
mark=at position 0.5 with {\arrow{stealth}}
}
,scale=0.5
,plotsettings/.style={postaction={decorate}}
]
\draw[postaction={decorate}] (0,-4) -- (0,0);
\draw[postaction={decorate}] (0,4) -- (0,0);
\draw[postaction={decorate}] (-4,0) -- (0,0);
\draw[postaction={decorate}] (4,0) -- (0,0);
\draw [plotsettings,domain=-2:0] plot (\x, {\x*\x});
\draw [plotsettings,domain= 0:2] plot (\x, {\x*\x});
\draw [plotsettings,domain=-2:0] plot (\x, {-\x*\x});
\draw [plotsettings,domain= 0:2] plot (\x, {-\x*\x});
\draw [plotsettings,domain=-3:0] plot (\x, {0.44*\x*\x});
\draw [plotsettings,domain= 0:3] plot (\x, {0.44*\x*\x});
\draw [plotsettings,domain=-3:0] plot (\x, {-0.44*\x*\x});
\draw [plotsettings,domain= 0:3] plot (\x, {-0.44*\x*\x});
\end{tikzpicture}%
}
\savestack\nondegensource{%
\begin{tikzpicture}
[decoration=
{markings,
mark=at position 0.5 with {\arrow{stealth}}
}
,scale=0.5
,plotsettings/.style={postaction={decorate}}
]
\draw[postaction={decorate}] (0,0) -- (-4,0);
\draw[postaction={decorate}] (0,0) -- (4,0);
\draw[postaction={decorate}] (0,0) -- (0,-4);
\draw[postaction={decorate}] (0,0) -- (0,4);
\draw [plotsettings,domain=0:-2] plot (\x, {\x*\x});
\draw [plotsettings,domain=2: 0] plot (\x, {\x*\x});
\draw [plotsettings,domain=0:-2] plot (\x, {-\x*\x});
\draw [plotsettings,domain=2: 0] plot (\x, {-\x*\x});
\draw [plotsettings,domain=0:-3] plot (\x, {0.44*\x*\x});
\draw [plotsettings,domain=3: 0] plot (\x, {0.44*\x*\x});
\draw [plotsettings,domain=0:-3] plot (\x, {-0.44*\x*\x});
\draw [plotsettings,domain=3: 0] plot (\x, {-0.44*\x*\x});
\end{tikzpicture}%
}
\stackinset{c}{-55pt}{c}{2pt}{\stackunder[24pt]{$\vcenter{\hbox{\scalebox{.3}{%
\nondegensink}}}$}{\scriptsize\Longunderstack[l]{sink}}%
}{%
\stackinset{c}{60pt}{c}{2pt}{\stackunder[24pt]{$\vcenter{\hbox{\scalebox{.3}{%
\nondegensource}}}$}{\scriptsize\Longunderstack[l]{source}}%
}{%
\stackinset{c}{-71pt}{c}{56pt}{\stackunder[24pt]{$\vcenter{\hbox{\scalebox{.3}{%
\fbox{\degensink}}}}$}{\scriptsize\Longunderstack[l]{degenerate sink}}%
$\!\rightarrow$}{%
\stackinset{c}{78pt}{c}{56pt}{$\leftarrow\!$%
\stackunder[24pt]{$\vcenter{\hbox{\scalebox{.3}{%
\fbox{\degensink}}}}$}{\scriptsize\Longunderstack[r]{degenerate source}}%
}{%
\stackinset{c}{-20pt}{c}{62pt}{\stackunder{\scalebox{.35}{\spiralA}}{%
\scriptsize spiral sink}}{%
\stackinset{c}{27pt}{c}{62pt}{\stackunder{\scalebox{.35}{\spiralB}}{%
\scriptsize spiral source}}{%
\mygraph%
}}}}}}
\end{document}
