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Trazar una trayectoria aleatoria

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Trazar una trayectoria aleatoria

Estoy tratando de recrear un trazado de trayectoria para demostrar la estabilidad. La imagen que tengo en mente es algo similar a esta siguiente figura:

Estabilidad asintótica

El problema es que realmente no sé cómo generar la trayectoria de la línea, ya que no tengo datos numéricos en los que confiar. La forma de la trayectoria es irrelevante siempre que muestre un comportamiento asintótico. Preferiblemente debería verse como el de la figura. Adjuntaré mi progreso actual en el siguiente código:

\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}

Progreso actual

Respuesta1

Pude usar la respuesta de Trazar bucles: trayectorias y direcciones. para trazar la siguiente trayectoria: ingrese la descripción de la imagen aquí

Aquí está el código:

\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}

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