# Drawing phase space curves in tikz

I would like to draw a graph like the one I attach to describe a family of solutions of some differential equations.

I managed to draw the extremal curves, but I have some difficulties with the internal ones, in particular I don't get how to make the lines following the "profile" of the external ones. I attach here also an example of what I get, with an internal line, which is not what I would like to get.

\documentclass[border=3pt]{standalone}
\usepackage{amsmath, amssymb, amsfonts, amscd, amsthm, bigints}
\usepackage{tikz}

\usetikzlibrary{decorations.markings}
%%%%%%%%Tikz definitions %%%%%%%%%%%%%%%%%
\tikzset{->-/.style={decoration={
markings,
mark=at position #1 with {\arrow{latex}}},postaction={decorate}}}
%
\begin{document}
\begin{tikzpicture}
%
\def\N{7}
\def\dDW4{5}
%
\node[label=below:$\mathrm{AdS}_2 \times \Sigma_\mathfrak{g}$, fill, draw, circle, minimum size=0.01, inner sep=1] (AdS2) at (0,0) {};
\node[fill, draw, circle, minimum size=0.01, inner sep=1] (DW4) at ($(AdS2) + (\thetaDW4:\dDW4)$) {};
\node[label=above:$\mathrm{AdS}_4$, fill, draw, circle, minimum size=0.01, inner sep=1] (AdS4) at ($(AdS2) + (\thetaAdS4:\dAdS4)$) {};
\node at (1.2,5.15) {$\textcolor{blue}{m^2 \varphi^2}$};
\node at (-.55, 3.5) {$\textcolor{red}{\bigintss\!\! F = q}$};

\def\outDW4{-75}
%

\end{tikzpicture}
\end{document}


You can define a decoration that help you mark the milestones and then use these milestones to calculate the homotopy

\documentclass[border=9,tikz]{standalone}
\usetikzlibrary{calc,decorations}

\pgfdeclaredecoration{mark milestone}{initial}{
\state{initial}[width=0pt,next state=mark]{
\xdef\markmilestoneindex{0}
\xdef\markmilestonename{\csname tikz@fig@name\endcsname}
\pgfmathsetmacro\markmilestonesep{\pgfdecoratedpathlength/10}
\xdef\markmilestonesep{\markmilestonesep}
}
\state{mark}[width=\markmilestonesep pt]{
\coordinate(\markmilestonename-\markmilestoneindex);
\pgfmathtruncatemacro\markmilestoneindex{\markmilestoneindex+1}
\xdef\markmilestoneindex{\markmilestoneindex}
}
\state{final}{
\pgftransformshift{\pgfpointdecoratedpathlast}
\coordinate(\markmilestonename-\markmilestoneindex);
}
}

\begin{document}
\begin{tikzpicture}
\def\N{7}
\def\dDW4{5}
\node(DW4)at($(AdS2)+(\thetaDW4:\dDW4)$){};
\node(AdS4)at($(AdS2)+(\thetaAdS4:\dAdS4)$){};
\def\outDW4{-75}
\path[name=pathA,decorate,decoration={mark milestone}]
($(pathA-0)!\r!(pathB-0)$)
--($(pathA-\i)!\r!(pathB-\i)$)