File:Transition to chaos.gif
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 |
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
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