File:Transition to chaos.gif
| Uploaded by | Berto |
|---|---|
| Upload date | 2023-03-24T16:23:09Z |
| MIME type | image/gif |
| Dimensions | 480 × 531 px |
| File size | 1.9 MB |
Summary
| Description |
English: A way to visualize the transition to chaos of a mechanical system is to look how its Poincaré section(s) changes when the perturbation increases.
In this example a damped simple pendulum with a sinusoidal forcing of amplitude "f", starting at rest. For small forcing, the motion is periodic, so the Poincaré section is just a single point. As the forcing increases we observe multiple bifurcations, aperiodic motion, a second regime of periodic motion, and then the Poincaré section becomes a fractal curve. |
| Date | |
| Source | https://twitter.com/j_bertolotti/status/1638866738082111488 |
| Author | Jacopo Bertolotti |
| Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
| GIF development |  This diagram was created with Mathematica. |
Mathematica 13.1 code
eqs = {\[Theta]''[t] == -(1/q) \[Theta]'[t] - Sin[\[Theta][t]] + g*Cos[\[Phi][t]], \[Phi]'[t] == \[Omega]};
data = Table[ Drop[Block[{q = 2, \[Omega] = 2/3}, First@Last@Reap@NDSolve[{eqs, \[Theta][0] == 0, \[Theta]'[0] == 0, \[Phi][1] == 2 \[Pi], WhenEvent[Mod[\[Phi][t], 2 \[Pi]] == 0, Sow[{g, Mod[\[Theta][t], 2 \[Pi], -\[Pi]], \[Theta]'[t]}]]}, {}, {t, 0, 100000}, MaxSteps -> \[Infinity]]], 100], {g, 1.05, 1.2, 0.001}];
frames = Table[
Show[
Graphics3D[{Gray, Point[Flatten[data[[1 ;; j - 1]], 1] ]}, BoxRatios -> {1, 1, 1}, Axes -> True, PlotRange -> {{1.05, 1.2}, {-\[Pi], \[Pi]}, {-\[Pi], \[Pi]}}, AxesLabel -> {"f", "\[Theta]", "p"},
LabelStyle -> Directive[Black, Bold ] ]
,
Graphics3D[{Point[data[[j]] ]}]
]
, {j, 2, 151}];
ListAnimate[frames]
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
 
|
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.
|