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绘制随机轨迹

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绘制随机轨迹

我正在尝试重新创建轨迹图来展示稳定性。我脑海中的图像类似于下图：

渐近稳定性

问题是，我真的不知道如何生成线轨迹，因为我没有任何可以依赖的数值数据。只要轨迹表现出渐近行为，其形式就无关紧要。最好看起来像图中那样。我将在以下代码中附上我目前的进展：

\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}

现在的进展

答案1

我能够使用 绘制循环：轨迹和方向 绘制以下轨迹： 在此处输入图片描述

以下是代码：

\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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