File:Lorenz System Lyapunov time.webm
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Lorenz_System_Lyapunov_time.webm (WebM audio/video file, VP9, length 27 s, 480 × 522 pixels, 61 kbps overall, file size: 199 KB)
Captions
Captions
Evolution of a Lorenz system for different (but very close) initial conditions.
Summary[edit]
| 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 |
Mathematica 14.0 code[edit]
\[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[edit]
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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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 11:53, 22 February 2024 | 27 s, 480 × 522 (199 KB) | Berto (talk | contribs) | Uploaded own work with UploadWizard |
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