How can I make a similiar kind of diagram without solving the dynamical equation?

Question

This is the phase-space diagram of a system that is itself a modified Thomas system for c=5 and b=0.0.

However, I want to plot a similar figure without solving the equation. It does not need to be as accurate as the phase-space diagram above or the animation of the same below.

This is my code for the graphs:

soln = With[{b = 0.0, c = 5, tmax = 200},
   NDSolve[{x'[t] == -b x[t] + Sin[y[t]] + c y[t],
     y'[t] == -b y[t] + Sin[z[t]] - c x[t],
     z'[t] == -b z[t] + Sin[x[t]], x[0] == 1.0, y[0] == 0.0,
     z[0] == 1.0}, {x, y, z}, {t, 0, tmax, 0.1},
    MaxSteps -> \[Infinity]]];

Animate[ParametricPlot3D[
  Evaluate[{x[t], y[t], z[t]} /. soln], {t, 0, tmax},
  PlotPoints -> 500,
  Axes -> False,
  ColorFunction ->
   Function[{x, y, z}, ColorData[{"Rainbow", "Reverse"}][z]],
  AspectRatio -> 1, PlotRangePadding -> 1, ImageSize -> 800,
  PlotTheme -> "Scientific",
  PlotStyle -> Directive[Opacity[0.4]],
  PlotRange -> {{-5, 5}, {-5, 5}, {0, 5}}], {tmax, 0.1, 200},
 AnimationRate -> 5, AnimationRepetitions -> Infinity]

How can I make a conceptual diagram showing this phase-space portrait with magnetic field lines along the axis of the inner helical trajectories?

How can I plot something like the image below?

Answer 1

Using Arrow and Arrowheads is a nightmare especially in 3D.

n = 30;
ar = Table[
   RotationMatrix[
     fi, {0, 0, 1}] . {15/8 + 13/8 Cos[2 \[Pi] (t + 1/2)/2], 0, 
     7/2 Sin[2 \[Pi] (t + 1/2)/2]}, {fi, 0, 
    2 \[Pi] - \[Pi]/4, \[Pi]/4}];
a = ParametricPlot3D[{{5/4 Cos[2 n \[Pi] t] (3/2 + Cos[2 \[Pi] t]), 
    5/4 Sin[2 n \[Pi] t] (3/2 + Cos[2 \[Pi] t]), 
    2 Sin[2 \[Pi] t]}}, {t, 0, 1}, Boxed -> False, Axes -> False, 
  ColorFunction -> Hue, 
  PlotRange -> 
   All]; b = (ParametricPlot3D[ar, {t, 0, 1}, Boxed -> False, 
    Axes -> False, PlotRange -> All]) /. 
  Line[x_] :> {Gray, 
    Arrowheads[{0, -0.03, 0, 0, 0, 0, 0, 0, -0.03, 0}], 
    Arrow[Tube[x]]};
Show[a, b]
Clear[a, b, n, ar]

n = 30;
ar = Table[
   RotationMatrix[
     fi, {0, 0, 1}] . {15/8 + 13/8 Cos[2 \[Pi] (t + 1/2)/2], 0, 
     7/2 Sin[2 \[Pi] (t + 1/2)/2]}, {fi, 0, 
    2 \[Pi] - \[Pi]/4, \[Pi]/4}];
a = ParametricPlot3D[{{5/4 Cos[2 n \[Pi] t] (3/2 + Cos[2 \[Pi] t]), 
    5/4 Sin[2 n \[Pi] t] (3/2 + Cos[2 \[Pi] t]), 
    2 Sin[2 \[Pi] t]}}, {t, 0, 1}, Boxed -> False, Axes -> False, 
  ColorFunction -> 
   Function[{x, y, z}, ColorData[{"Rainbow", "Reverse"}][z]], 
  PlotRange -> 
   All]; b = (ParametricPlot3D[ar, {t, 0, 1}, Boxed -> False, 
    Axes -> False, PlotRange -> All]) /. 
  Line[x_] :> 
   Join[{Gray, Line[x[[32 ;; -1]]]}, {Red, 
     Arrowheads[{-0.03, 0, 0, 0, 0, 0, 0, 0, 0}], 
     Arrow[Tube[{x[[{1, 32}]], x[[-2 ;; -1]]}]]}];
Show[a, b]
Clear[a, b, n, ar]

Answer 2

One parameterization of a torus is $((2+\cos v)\cos u,(2+\cos v)\sin u,\sin v)$ for $u\in\{0,2\pi\}$ and $v\in\{0,2\pi\}$. Letting $u=50t$ and $v=t$ yields a curve along the surface like so:

With[{n = 50}, 
ParametricPlot3D[{Cos[2 n \[Pi] t] (2 + Cos[2 \[Pi] t]), 
Sin[2 n \[Pi] t] (2 + Cos[2 \[Pi] t]), Sin[2 \[Pi] t]}, {t, 0, 1}, 
ColorFunction -> Hue]]

Perhaps this animation is close enough to your goal

Export["~/Desktop/i.gif", 
Join[#, Most@Rest@Reverse@#] &@
Table[With[{n = 50}, 
Rasterize[
 Style[ParametricPlot3D[{Cos[2 n \[Pi] t] (2 + Cos[2 \[Pi] t]), 
    Sin[2 n \[Pi] t] (2 + Cos[2 \[Pi] t]), Sin[2 \[Pi] t]}, {t, 0,
     T}, ColorFunction -> ColorData@"DarkRainbow", 
   ViewPoint -> {0, 3, 1}, PlotTheme -> "Minimal", 
   PlotStyle -> Opacity@.5, 
   PlotRange -> {{-2.997721318162146`, 
      2.999999999938333`}, {-2.9962254214863`, 
      2.9982495368807554`}, {-0.9999998592131705`, 
      0.9999998782112116`}}], 
  RenderingOptions -> {"3DRenderingEngine" -> "OpenGL"}], 
 RasterSize -> 360]], {T, .02, 1, .02}], 
AnimationRepetitions -> \[Infinity], "DisplayDurations" -> 1/30]

Answer 3

I think we can do this all by StreamPlot3D,but I don't know what is the physical meaning of the stream lines in the center and what is its expression and what is the relation between the dynamic system and the sketch.

BTW, the stream lines is not the torus or revolution shape. Science is not art!

Clear[b, c, F, pt];
b = 0.0;
c = 5;
F = {-b*x + Sin[y] + c*y, -b*y + Sin[z] - c*x, -b*z + Sin[x]};
pt = {1, 0, 1};
StreamPlot3D[F, {x, -4, 4}, {y, -4, 4}, {z, -4, 4}, 
 StreamPoints -> {pt}]

Clear[b, c, F, pt];
b = 0;
c = 1.9;
F = {-b*x + Sin[y] + c*y, -b*y + Sin[z] - c*x, -b*z + Sin[x]};
pt = {1, 0, 1};
StreamPlot3D[F, {x, -5, 5}, {y, -5, 5}, {z, -5, 5}, 
 StreamPoints -> {pt}, AspectRatio -> Automatic]