File:Attractors-Orbit.gif

Attractors-Orbit.gif (360 × 357 pixels, file size: 657 KB, MIME type: image/gif, looped, 91 frames, 9.1 s)

Summary

Description
English: Visualization of how a number of randomly chosen initial conditions are all attracted to a stable orbit for the f-2(1-f²)f'+f=0 differential equation (i.e. the Van Der Pol oscillator).
Date
Source https://twitter.com/j_bertolotti/status/1450378380193538049
Author Jacopo Bertolotti
Permission
(Reusing this file)
https://twitter.com/j_bertolotti/status/1030470604418428929

Mathematica 12.0 code

points1 = RandomReal[{-4, 4}, {100, 2}];
dim = Dimensions[points1][[1]];
psol1 = ParametricNDSolveValue[{x''[t] - 2 (1 - x[t]^2) x'[t] + x[t] == 0, 
  x[0] == x0, x'[0] == v0}, {x}, {t, 0, 50}, {x0, v0}]

frames1 = Table[
  ParametricPlot[ Table[{psol1[points1[[j, 1]], points1[[j, 2]] ][[1]][t], psol1[points1[[j, 1]], points1[[j, 2]]][[1]]'[t]}, {j, 1, dim}], {t, 0, \[Tau]}, ColorFunction -> Function[{x, y, t}, Directive[ColorData["AvocadoColors"][t/\[Pi]] , Opacity[t^3] ] ], Background -> Black, PlotRange -> {{-5, 5}, {-5, 5}}, AspectRatio -> 1, Axes -> True, LabelStyle -> {White, FontSize -> 12}, AxesStyle -> White, Ticks -> None, AxesLabel -> {"f(t)", "f'(t)"}, Epilog -> {White, Style[Text["Orbit: f''-2(1-\!\(\*SuperscriptBox[\(f\), \(2\)]\))f'+f=0", {-2.2, 4.7}], Bold, FontSize -> 14], PointSize[0.01], Point[Table[{psol1[points1[[j, 1]], points1[[j, 2]] ][[1]][\[Tau]], psol1[points1[[j, 1]], points1[[j, 2]]][[1]]'[\[Tau]]}, {j, 1,dim}]]
     }
   ]
  , {\[Tau], 10^-3, 7, 7/50}];
ListAnimate[Join[Table[frames1[[1]], {5}], frames1]]

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

Visualization of a stable orbit attractor for the VanDer Pol oscillator.

Items portrayed in this file

depicts

19 October 2021

image/gif

a7f1cbdbf11c82dbc88639a5429381653e8459a4

672,473 byte

9.099999999999984 second

357 pixel

360 pixel

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Date/TimeThumbnailDimensionsUserComment
current08:32, 19 October 2021Thumbnail for version as of 08:32, 19 October 2021360 × 357 (657 KB)BertoUploaded own work with UploadWizard

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