File:Double pendulum predicting dynamics.gif
Summary
| Description |
English: In a chaotic system, like a double pendulum, small changes in the initial conditions become large changes in the long-term evolution of the system, but on the short/medium term, how well we can predict the dynamics is not constant. Sometimes small perturbations will not change much, sometimes small perturbation make even short-term predictions impossible.
At each frame the position of the second mass is perturbed slightly, and the grey lines show the predicted dynamics for all the perturbed conditions. |
| Date | |
| Source | https://twitter.com/j_bertolotti/status/1411987574109913092 |
| Author | Jacopo Bertolotti |
| Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
Mathematica 12.0 code
p1[t_] := {l Sin[\[Theta][t]], -l Cos[\[Theta][t]]};
p2[t_] := p1[t] + {l Sin[\[Phi][t]], -l Cos[\[Phi][t]]};
T[t_] := FullSimplify[1/2 m Norm[D[p1[t], t]]^2 + 1/2 m Norm[D[p2[t], t]]^2, Assumptions -> {m > 0, l > 0, \[Theta][t] > 0, \[Theta]'[t] > 0, \[Phi]'[t] > 0, \[Phi][t] > 0}];
V[t_] := FullSimplify[m g p1[t][[2]] + m g p2[t][[2]], Assumptions -> {m > 0 , l > 0}];
L[t_] := T[t] - V[t];
e1[t_] := FullSimplify[D[D[L[t], \[Theta]'[t]], t] - D[L[t], \[Theta][t]]];
e2[t_] := FullSimplify[D[D[L[t], \[Phi]'[t]], t] - D[L[t], \[Phi][t]]];
(**)
tmax = 1000; m1 = 1; m2 = 1; g = 1; l = 1;
s1 = NDSolve[{e1[t] == 0, e2[t] == 0, \[Theta][0] == \[Pi], \[Phi][0] == \[Pi]/2, \[Theta]'[0] == 0, \[Phi]'[0] == 0}, {\[Theta], \[Phi]}, {t, 0, tmax}, MaxSteps -> Infinity]
frames = Table[
prediction = Table[NDSolve[{e1[t1] == 0, e2[t1] == 0, \[Theta][0] == Evaluate[(\[Theta][t] /. s1)][[1]], \[Phi][0] == Evaluate[(\[Phi][t] /. s1)][[1]] + \[Delta], \[Theta]'[0] == Evaluate[(\[Theta]'[t] /. s1)][[1]], \[Phi]'[0] == Evaluate[(\[Phi]'[t] /. s1)][[1]]}, {\[Theta], \[Phi]}, {t1, 0, 10}, MaxSteps -> Infinity]
, {\[Delta], -0.002, 0.002, 0.00025}];
Show[
ParametricPlot[p2[t] /. prediction, {t, 0, 10}, PlotStyle -> Directive[Gray, Opacity[0.1]] , Axes -> False]
,
Graphics[{
Black, Thickness[0.01], Line[{{0, 0}, (p1[t] /. s1)[[1]], (p2[t] /. s1)[[1]]}], Orange, Disk[(p1[t] /. s1)[[1]], 0.15], Disk[(p2[t] /. s1)[[1]], 0.15], Black, Thick, Circle[(p1[t] /. s1)[[1]], 0.15],
Circle[(p2[t] /. s1)[[1]], 0.15]
,
Black, Disk[{0, 0}, 0.1]
}, PlotRange -> {{-3, 3}, {-3, 3}}], PlotRange -> {{-3, 3}, {-3, 3}}]
, {t, 0, 100, 0.3}];
ListAnimate[frames, 10]
Licensing
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.
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