我怎样才能绘制如图所示的动力系统的轨道？

Question

我正在阅读有关 Latex 的文章，我想知道如何使用 tikZ 绘制以下图像

$x_0$ 应该是 $x$ 的一个 $\omega$ 极限点。我能够找到如何达到与我想要的类似的东西，但它还不一样

\documentclass[tikz,border=3.14mm]{standalone}
\usetikzlibrary{positioning, decorations.markings}
\begin{document}
\begin{tikzpicture}
  %\draw[-stealth] (-3,0) -- (3,0) node[below]{$x$};
  %\draw[-stealth] (0,-3) -- (0,3) node[left]{$y$};
  \draw [rotate=-45, postaction=decorate, decoration={markings,
     mark=at position 0.93 with {\arrow{latex}}}]
     plot[domain=32:0, variable=\t, smooth, samples=75] ({\t r}: {0.002*\t*\t} and {0.004*\t*\t} );

  % Draw rotated ellipse (90 degrees)
  \begin{scope}[rotate=-45+90]
     \draw[blue, dashed] (0,0) ellipse ({1.5} and {0.75});
  \end{scope}
\end{tikzpicture}

\end{document}

我得到的信息如下

但对我来说，这似乎并不是我想要的，因为这些曲线应该像蓝色椭圆一样渐近。另外，我该如何添加这些点并相应地命名它们？感谢您的耐心。

Answer 1

\documentclass[tikz, border=1cm]{standalone}
\usetikzlibrary{decorations.markings, arrows.meta}
\begin{document}
\begin{tikzpicture}
\draw[
rotate=45,
postaction=decorate,
decoration={markings,
mark=at position 0.010 with {\coordinate (A);},
mark=at position 0.020 with {\arrow{Latex}},
mark=at position 0.340 with {\coordinate (B);},
mark=at position 0.350 with {\arrow{Latex}},
mark=at position 0.615 with {\coordinate (C);},
mark=at position 0.625 with {\arrow{Latex}},
mark=at position 0.844 with {\coordinate (D);},
mark=at position 0.854 with {\arrow{Latex}},
}] plot[domain=1360:0, variable=\t, samples=200] ({exp(0.0005*\t)*1.5*cos(\t)},{exp(0.0005*\t)*0.75*sin(\t)});
\draw[blue, thick, dashed, rotate=45] (0,0) ellipse ({1.5} and {0.75});
\foreach \c in {A,B,C,D}
  \fill (\c) circle[radius=1pt];
\draw[densely dotted] (C) -- (D);
\node (E) at (3,-2) {$\phi(t_1,x)$};
\draw[-Latex, shorten >=2pt] (E) to[out=160, in=-90, looseness=0.6] +(-1,0.5) to[out=90, in=-20, looseness=0.6] (B);
\end{tikzpicture}
\end{document}

编辑：

\documentclass[tikz, border=1cm]{standalone}
\usetikzlibrary{decorations.markings, arrows.meta}
\begin{document}
\begin{tikzpicture}
\draw[
rotate=45,
postaction=decorate,
decoration={markings,
mark=at position 0.010 with {\coordinate (A);},
mark=at position 0.020 with {\arrow{Latex}},
mark=at position 0.340 with {\coordinate (B);},
mark=at position 0.350 with {\arrow{Latex}},
mark=at position 0.615 with {\coordinate (C);},
mark=at position 0.625 with {\arrow{Latex}},
mark=at position 0.844 with {\coordinate (D);},
mark=at position 0.854 with {\arrow{Latex}},
}] plot[domain=1360:0, variable=\t, samples=200] ({exp(0.0005*\t)*1.5*cos(\t)},{exp(0.0005*\t)*0.75*sin(\t)});
\draw[blue, thick, dashed, rotate=45] (0,0) ellipse ({1.5} and {0.75});
\path[rotate=45] ({1.5*cos(-75)},{0.75*sin(-75)}) coordinate (E);
\foreach \c in {A,B,C,D,E}
  \fill (\c) circle[radius=1.2pt];
\draw[densely dotted] (C) -- (D);
\node (F) at (3,-2) {$\phi(t_1,x)$};
\draw[-Latex, shorten >=2pt] (F) to[out=160, in=-90, looseness=0.6] +(-1,0.5) to[out=90, in=-20, looseness=0.6] (B);
\end{tikzpicture}
\end{document}