File:Lorenz System Lyapunov time.webm
| Uploaded by | Berto |
|---|---|
| Upload date | 2024-02-22T11:53:27Z |
| MIME type | video/webm |
| Dimensions | 480 × 522 px |
| File size | 199.0 KB |
Summary
| Description |
English: A chaotic system is extremely sensitive to the initial condition, so very close initial conditions will evolve into very different trajectories. The Lyapunov time is the characteristic time before this happens.
This simulation shows this happening for a Lorenz system. |
| Date | |
| Source | https://mathstodon.xyz/@j_bertolotti/111969456691906891 |
| Author | Berto |
| Permission (Reusing this file) |
https://mathstodon.xyz/@j_bertolotti/111363365323269417 |
| WEBM development |  This diagram was created with Mathematica. |
Mathematica 14.0 code
\[Sigma] = 10; \[Rho] = 28; \[Beta] = 8/3; Tmax = 50;
eqn = {x'[t] == \[Sigma] (-x[t] + y[t]), y'[t] == \[Rho]*x[t] - y[t] - x[t]*z[t], z'[t] == x[t]*y[t] - \[Beta]*z[t]};
trange = 0.1; npoints = 30;
p0 = RandomPoint[Sphere[{20, -20, 10}, 0.01], npoints];
soln = Table[
NDSolveValue[ Join[eqn, {x[0] == p0[[j, 1]], y[0] == p0[[j, 2]], z[0] == p0[[j, 3]]}], {x[t], y[t], z[t]}, {t, 0, Tmax}], {j, 1, npoints}];
frames = Table[
Graphics3D[{RGBColor[1, 215/255, 0], Table[ Tube[Table[soln[[j]], {t, Max[0, \[Tau] - 1], \[Tau], 0.01}]], {j, 1, npoints}]}, PlotRange -> {{-30, 30}, {-30, 30}, {0, 60}}, Axes -> False, Boxed -> False, Background -> Black], {\[Tau], 0.001, 20, 0.05}];
ListAnimate[frames]
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. |
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