Zeichnen einer zufälligen Flugbahn

Question

Ich versuche, ein Trajektorien-Diagramm zu erstellen, um Stabilität zu demonstrieren. Das Bild, an das ich denke, ähnelt der folgenden Abbildung:

Das Problem ist, dass ich nicht wirklich weiß, wie die Linienbahn erzeugt werden soll, da ich keine numerischen Daten habe, auf die ich mich verlassen kann. Die Form der Flugbahn ist irrelevant, solange sie asymptotisches Verhalten zeigt. Vorzugsweise sollte es wie in der Abbildung aussehen. Ich werde meinen aktuellen Fortschritt im folgenden Code verfolgen:

\documentclass[tikz]{standalone}
\usepackage{geometry}
\usepackage{pgfplots}
\usepackage{bm}
\usetikzlibrary{3d,calc}
\usepgfplotslibrary{fillbetween}

\begin{document}
   \begin{tikzpicture}%[show grid=all]
      % background
      \path[name path=C] (-1.5,1.75) -- (1.5,3.25);
      \path[name path=D] (-1.5,-1.5) -- (1.5,0.5);
      \tikzfillbetween[of=C and D]{top color=gray!10!white, bottom color=white, opacity=0.02};
      % axis
      \draw[gray!50!white,->,very thick,>=latex] (0,-0.5) -- (0,2.5);
      \draw[gray!50!white,->,very thick,>=latex] (1.5,1.75)  -- (-1.5,0.25);
      \draw[gray!50!white,->,very thick,>=latex] (0,1)  -- (7,1);
      \filldraw[black] (0,1) circle (1.2pt);
      \filldraw[black] (6,1) circle (1.2pt);
      \draw[blue!50!white,dashed,very thick] (0,0) arc (-90:90:0.6cm and 1cm);
      \draw[blue!50!white,thick] (0,0) arc (-90:90:-0.6cm and 1cm);
      \draw[blue!50!white,thick] (6,0.5) arc (-90:90:-0.4cm and 0.5cm);
      \draw[blue!50!white,thick] (6,0.5) arc (-90:90:0.4cm and 0.5cm);
      \draw[blue!50!white,thick,name path = A] (0,2) to[in=180,out=-12] (6,1.5);
      \draw[blue!50!white,thick,name path = B] (0,0) to[in=180,out=12] (6,0.5);
      \tikzfillbetween[of=A and B]{blue!50!white, opacity=0.2};
      \shade[left color=blue!50!white,right color=blue!50!white,opacity=0.2] (0,0) arc (-90:90:-0.6cm and 1cm);
      \shade[left color=blue!50!white,right color=blue!50!white,opacity=0.2] (6,0.5) arc (-90:90:0.4cm and 0.5cm);
      % axis labels
      \node at (0,2.9) {$\varphi_{1}(t)$};
      \node at (-2,0.21) {$\varphi_{2}(t)$};
      \node at (7.15,1) {$t$};
      \draw[thick,->,>=latex] (0,1) -- (0,2) node [midway,scale=0.9,right]{$\delta$};
      \draw[thick,->,>=latex] (6,1) -- (6,1.5) node [midway,scale=0.9,right]{$\varepsilon$};
      \draw[dashed,blue] (-0.3,1.35) -- (-2,2.5) node [pos=1.2,text=black,scale=0.9]{$\bm{\varphi}(0)$};
      % trajectory
      \filldraw[blue!90!white] (-0.3,1.35) circle (1.2pt);
      % border adjustment (standalone crop problems)
      \node at (0,-2.3) {$$}; \node at (8,0) {$$};
   \end{tikzpicture}
\end{document}

Answer 1

Ich konnte die Antwort aus Plotten von Schleifen verwenden: Trajektorien und Richtungen , um die folgende Trajektorie zu zeichnen:

Hier ist der Code:

\documentclass[tikz]{standalone}
\usepackage{geometry}
\usepackage{pgfplots}
\usepackage{bm}
\usetikzlibrary{3d,calc}
\usepgfplotslibrary{fillbetween}
\usetikzlibrary{arrows.meta,hobby}

\begin{document}
   \begin{tikzpicture}[use Hobby shortcut]%[show grid=all]
      \tikzset{axisArrow/.style={gray!50!white,very thick,>=stealth,arrows={->[fill=black]}}}
      % background
      \path[name path=C] (-1.5,1.75) -- (1.5,3.25);
      \path[name path=D] (-1.5,-1.5) -- (1.5,0.5);
      \tikzfillbetween[of=C and D]{top color=gray!10!white, bottom color=white, opacity=0.02};
      % axis
      \draw[axisArrow] (0,-0.5) -- (0,2.5);
      \draw[axisArrow] (1.5,1.75)  -- (-1.5,0.25);
      \draw[axisArrow] (0,1)  -- (7,1);
      \filldraw[black] (0,1) circle (1.2pt);
      \filldraw[black] (6,1) circle (1.2pt);
      \draw[blue!50!white,dashed, thick] (0,0) arc (-90:90:0.6cm and 1cm);
      \draw[blue!50!white,thick] (0,0) arc (-90:90:-0.6cm and 1cm);
      \draw[blue!50!white,thick] (6,0.5) arc (-90:90:-0.4cm and 0.5cm);
      \draw[blue!50!white,thick] (6,0.5) arc (-90:90:0.4cm and 0.5cm);
      \draw[blue!50!white,thick,name path = A] (0,2) to[in=180,out=-12] (6,1.5);
      \draw[blue!50!white,thick,name path = B] (0,0) to[in=180,out=12] (6,0.5);
      \tikzfillbetween[of=A and B]{blue!50!white, opacity=0.2};
      \shade[left color=blue!50!white,right color=blue!50!white,opacity=0.2] (0,0) arc (-90:90:-0.6cm and 1cm);
      \shade[left color=blue!50!white,right color=blue!50!white,opacity=0.2] (6,0.5) arc (-90:90:0.4cm and 0.5cm);
      % axis labels
      \node at (0,2.9) {$\varphi_{1}(t)$};
      \node at (-2,0.21) {$\varphi_{2}(t)$};
      \node at (7.15,1) {$t$};
      \node[scale=0.7] at (5,1.65) {$t\rightarrow \infty$};
      \draw[thick,->,>=stealth] (0,1) -- (0,2) node [midway,scale=0.9,right]{$\delta$};
      \draw[thick,->,>=stealth] (6,1) -- (6,1.5) node [midway,scale=0.9,right]{$\varepsilon$};
      \draw[dashed,blue] (-0.3,1.35) -- (-2,2.5) node [pos=1.2,text=black,scale=0.9]{$\bm{\varphi}(0)$};
      % trajectory
      \filldraw[blue!90!white] (-0.3,1.35) circle (1.2pt);
     \draw[-stealth,blue!90!white,thick] (-0.3,1.35) .. (1,0.5) .. (1.75,1.4) .. (1.5,1.2) .. (2,1.2) .. (2.5,1.1) .. (3,0.9) .. (3,1) .. (4,1) .. (4.3,1) .. (4.8,1) -- (5.55,1);
      % border adjustment (standalone crop problems)
      \node at (0,-2.3) {$$}; \node at (8,0) {$$};
   \end{tikzpicture}
\end{document}