File:Fixed Points.gif
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![File:Fixed Points.gif](https://upload.wikimedia.org/wikipedia/commons/thumb/0/00/Fixed_Points.gif/600px-Fixed_Points.gif?20230313152829)
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Original file (800 × 800 pixels, file size: 2.18 MB, MIME type: image/gif, looped, 40 frames, 4.0 s)
File information
Structured data
Captions
Captions
Schematic visualization of 4 of the most common kinds of fixed points.
Summary
[edit]DescriptionFixed Points.gif |
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
[edit](*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
[edit]I, the copyright holder of this work, hereby publish it under the following license:
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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.
http://creativecommons.org/publicdomain/zero/1.0/deed.enCC0Creative Commons Zero, Public Domain Dedicationfalsefalse |
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 15:28, 13 March 2023 | ![]() | 800 × 800 (2.18 MB) | Berto (talk | contribs) | Uploaded own work with UploadWizard |
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