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Original file (800 × 800 pixels, file size: 2.18 MB, MIME type: image/gif, looped, 40 frames, 4.0 s)

Summary

Description
English: Schematic visualization of 4 of the most common kinds of fixed points.
Date
Source https://twitter.com/j_bertolotti/status/1634148351296806914
Author Jacopo Bertolotti
Permission
(Reusing this file)
https://twitter.com/j_bertolotti/status/1030470604418428929

Mathematica 13.1 code

(*Generate blue noise to sample the plane*)
range = 5;
blue = {RandomReal[{-range, range}, {2}]};
Do[
  n = Length[blue];
  candidates = RandomReal[{-range, range}, {n + 1, 2}];
  bestcandidatepos = 
   Position[
     Table[Min[Norm[candidates[[j]] - #] & /@ blue], {j, 1, n}], Max[Table[Min[Norm[candidates[[j]] - #] & /@ blue], {j, 1, n}]] ][[1, 1]];
  AppendTo[blue, candidates[[bestcandidatepos]]];
  , 10^2];
(*definitions*)
eqs[matrix_] := ({q'[t], p'[t]} == matrix . {q[t], p[t]});
initialpoints = Select[blue, Norm[#] < 4.9 &];
initialcond = Table[{q[0] == initialpoints[[j, 1]], p[0] == initialpoints[[j, 2]]}, {j, 1, Length[initialpoints]}];
solutions[equations_] := Table[NDSolve[{equations, initialcond[[j]]}, {q[t], p[t]}, {t, -1, 10}], {j, 1, Length[initialcond]}]
plot[solution_, tmax_, plotlabel_] := Show[
  ParametricPlot[{q[t], p[t]} /. solution, {t, tmax - 0.5, tmax}, PlotStyle -> {Thick},Background -> White,  Axes -> False, PlotRange -> 5.1 {{-1, 1}, {-1, 1}}, PlotLabel -> plotlabel, LabelStyle -> {Black, Bold}, RegionFunction -> Function[{x, y, t}, Sqrt[x^2 + y^2] < 5], ColorFunction -> Function[{x, y, t}, Directive[ColorData["GrayTones"][t/\[Pi]] , Opacity[t^3] ] ]
   ]
  ,
  Graphics[{Black, PointSize[0.02], Point[Select[Flatten[{q[t], p[t]} /. solution /. {t -> tmax}, 1], Norm[#] < 5 &] ], Thick, Circle[{0, 0}, 5]}]
  ]
(*Solve the equations*)
solhyperbolic = solutions[eqs[DiagonalMatrix[{-1, 2}]]];
solelliptic = solutions[eqs[RotationMatrix[\[Pi]/2]]];
solspiralstable = solutions[eqs[-3 RotationMatrix[\[Pi]/5]]];
solspiralunstable = solutions[eqs[1.5*RotationMatrix[\[Pi]/5]]];
(*Plot and animate*)
frames = Table[
   GraphicsGrid[{{
      plot[solspiralstable, \[Tau], "Stable fixed point"], 
      plot[solspiralunstable, \[Tau], "Unstable fixed point"]
      }, {
      plot[solhyperbolic, \[Tau], "Hyperbolic fixed point"], 
      plot[solelliptic, \[Tau], "Elliptic fixed point"]
      }}]
   , {\[Tau], 10^-3, 2, 0.05}];
ListAnimate[frames]

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

Captions

Schematic visualization of 4 of the most common kinds of fixed points.

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depicts

10 March 2023

image/gif

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Date/TimeThumbnailDimensionsUserComment
current15:28, 13 March 2023Thumbnail for version as of 15:28, 13 March 2023800 × 800 (2.18 MB)BertoUploaded own work with UploadWizard

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