Graphing System of ODE flows

Question

I'm currently trying to graph the flow applied to some points of the 2,3-dimensional euclidean space, and it has been quite hard to do.

What I want is to graph something like this

and this

using expresions like this

with this cleaner and better looking design

Is it possible to do so?

PS.: "sen" means sine.

---

Updates:

For the first example, the suggestion Daniel N. made mostly worked.

\documentclass[dvipsnames, margin = 5mm]{standalone}
\usepackage{pgfplots}
\usepackage{tikz-3dplot}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}
\pgfplotsset{compat=newest}
\def\Point{36.9}
\begin{document}
\tdplotsetmaincoords{80}{130}
\begin{tikzpicture}[tdplot_main_coords, scale = 1]
\begin{axis}[
 axis lines=middle,
 height=12cm,
 xtick=\empty,
 ytick=\empty,
]
\addplot[variable=\t, domain=.063:4, samples=80] plot (\t, 1/\t);
\addplot[variable=\t, domain=.063:4, samples=80] plot (-\t, 1/\t);
\addplot[variable=\t, domain=.063:4, samples=80] plot (\t, -1/\t);
\addplot[variable=\t, domain=.063:4, samples=80] plot (-\t, -1/\t);
\addplot[variable=\t, domain=.067:3.7, samples=80] plot (\t+0.3, 1/\t+1);
\addplot[variable=\t, domain=.067:3.7, samples=80] plot (-\t-0.3, 1/\t+1);
\addplot[variable=\t, domain=.067:3.7, samples=80] plot (\t+0.3, -1/\t-1);
\addplot[variable=\t, domain=.067:3.7, samples=80] plot (-\t-0.3, -1/\t-1);
\end{axis}
\end{tikzpicture}
\end{document}

This code wielded this image

which is okay. With a few tweaks it will be perfect.

The second one, however, it's tricky. I've found an example that resembles what I am looking for, but I'm still having trouble shaping it.

\documentclass[dvipsnames, margin = 5mm]{standalone}
\usepackage{pgfplots}
\usepackage{tikz-3dplot}
\usetikzlibrary{decorations.markings}
\pgfplotsset{compat=newest}
\def\Point{36.9}
\begin{document}
\tdplotsetmaincoords{80}{130}
\begin{tikzpicture}[tdplot_main_coords, scale = 1]
\begin{axis}[
 view={-5
 0}{-20},
 axis lines=middle,
 zmax=60,
 height=12cm,
 xtick=\empty,
 ytick=\empty,
 ztick=\empty
]
\addplot3+[,ytick=\empty,yticklabel=\empty,
  mark=none,
  thick,
  Black,
  domain=2:16.7pi,
  samples=400,
  samples y=0,
]
({(1/x)sin(0.28pideg(x))},{(1/x)cos(0.28pi*deg(x)},{x+20});
\addplot3+[,ytick=\empty,yticklabel=\empty,
  mark=none,
  thick,
  Black,
  domain=2:16.7pi,
  samples=400,
  samples y=0,
]
({(1/x)sin(0.28pideg(x))},{(1/x)cos(0.28pi*deg(x)},0);
\addplot3+[,ytick=\empty,yticklabel=\empty,
  mark=none,
  thick,
  Black,
  domain=2:16.7pi,
  samples=400,
  samples y=0,
]
({-(1/x)cos(0.28pideg(x))},{-(1/x)sin(0.28pi*deg(x)},{-x-20});
\end{axis}
\end{tikzpicture}
\end{document}

This code wields this image

which is almost what I want. The problem is the curve not approaching the z-axis quickly enough.

Answer 1

The sense on the trajectory should be given, I think, through a decoration. But I don't know how to include a decoration in \addplot such that only the curve be touched, and not the axes. So I reformulated your solution without using pgfplots

The decoration is called sense and depends on three arguments: the position on the curve, the slanted angle (to link it to the osculating plane of the curve), and the length (of the tip) in points.

If you need to revert the sense on the trajectory, you have to change \arrow into \arrowreversed.

Remarks.

1. I defined the coordinates such that the plane Oxy corresponds to the plane defined by the scree when the longitude is 0.
2. There are constants in the code to disconnect the curves' scaling from the coordinate system.
3. I tried to respect your choices concerning the upper and lower trajectories. They seem slightly strange to me. There should be some sort of continuity when crossing the horizontal plane.

The code

\documentclass[11pt, margin=5mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{math, calc}
\usetikzlibrary{decorations.markings, arrows.meta}
\begin{document}
\tikzset{%
  view/.style 2 args={%
    z={({-sin(#1)}, {-cos(#1)sin(#2)})},
    x={({cos(#1)}, {-sin(#1)sin(#2)})},
    y={(0, {cos(#2)})},
    evaluate={%
      \tox={sin(#1)cos(#2)};
      \toy={sin(#2)};
      \toz={cos(#1)cos(#2)};
    }
  },
  sense/.style n args={3}{ % position, slant, length, color
    decoration={
      markings,
      mark=at position #1 with {%
        \arrow{Stealth[length=#3, slant=#2]}
      }
    }, postaction=decorate
  }
}
\tikzmath{
  real \fc, \T;
  \fc = .28pi;
  \T = 16pi;
  function F(\t) {return {sin(\fcdeg(\t))/\t};};
  function G(\t) {return {.3(\t/\T +.2)};};
  function H(\t) {return {cos(\fc*deg(\t))/\t};};

}
\begin{tikzpicture}[view={30}{10}, scale=5]
  \begin{scope}[gray, very thin,
    evaluate={%
      real \a, \b, \c, \r;
      \a = .8; \b = .6; \c = 1;
      \r = .7;
    }]
    \draw[->] (-\r.6\a, 0, 0) -- (\r\a, 0, 0) node[scale=.7, pos=1.03] {$x$};
    \draw[->] (0, -\r\b, 0) -- (0, \r\b, 0);
    \draw[->] (0, 0, -\r\c) -- (0, 0, \r*\c);
  \end{scope}
\draw[variable=\t, domain=2:\T, samples=300,
  sense={.12}{1.5}{5}] plot ({F(\t)}, {G(\t)}, {H(\t)});
\draw[blue, variable=\t, domain=2:\T, samples=300,
  sense={.12}{1.5}{5}] plot ({F(\t)}, 0, {H(\t)});
\draw[variable=\t, domain=2:\T, samples=300,
  sense={.12}{-1.5}{5}] plot ({-H(\t)}, {-G(\t)}, {-F(\t)});

\end{tikzpicture}
\end{document}