File:Arnold cat map.gif
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Arnold_cat_map.gif (600 × 235 pixels, file size: 6.45 MB, MIME type: image/gif, looped, 600 frames, 1 min 0 s)
Captions
Captions
Application of Arnold's cat map to two almost identical set of points, showing that the results quickly diverge from each other.
Summary
[edit]
| Description |
English: Arnold's cat map is a "stretch and fold" map from the plane (torus) onto itself. The initial points on the right image are shifted a part in a million with respect to the left one, but the two dynamics diverge quickly, showing that this map is chaotic. |
| Date | |
| Source | https://twitter.com/j_bertolotti/status/1240963578842296322 |
| Author | Jacopo Bertolotti |
| Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
Mathematica 12.0 code
[edit]
cat = 1 - ImageData[Import["<path to image>"], "Bit"][[All, All, 1]];
pos = Position[cat[[1 ;; -1 ;; 20, 1 ;; -1 ;; 20]], _?(# > 0.4 &)];
pos2 = (N[{#[[2]]/Max@Flatten@pos, -(#[[1]]/Max@Flatten@pos) + 1}] &) /@ pos; (*The image I used needed some reshuffling*)
nmax = 50;
evo = {pos2};
evo2 = {pos2 + RandomReal[{-10^-6, 10^-6}, {Dimensions[pos2][[1]], 2}]};
evoh = {};
evoh2 = {};
Do[
AppendTo[evoh, ({2 #[[1]] + #[[2]], #[[1]] + #[[2]]}) & /@ (evo[[-1]])];
AppendTo[evo, (Mod[#, 1] &) /@ evoh[[-1]] ];
, nmax + 1];
Do[
AppendTo[evoh2, ({2 #[[1]] + #[[2]], #[[1]] + #[[2]]}) & /@ (evo2[[-1]])];
AppendTo[evo2, (Mod[#, 1] &) /@ evoh2[[-1]] ];
, nmax + 1];
p0 = Join[Flatten@Table[
Join[Table[
Column[{
Style["Arnold's cat map: (x,y) \[Rule] (2x+y,x+y) mod 1 ", Black, Bold, 14],
GraphicsRow[{
Show[
Plot[0, {x, 0, 3}, PlotRange -> {{0, 3}, {0, 2}}, PlotStyle -> Opacity[0], Axes -> False, Frame -> True, AspectRatio -> 2/3],
Graphics[{Black, PointSize[0.009], Point[evo[[j]]*(1 - t) + evoh[[j]]*t ]}]
],
Show[
Plot[0, {x, 0, 3}, PlotRange -> {{0, 3}, {0, 2}},
PlotStyle -> Opacity[0], Axes -> False, Frame -> True, AspectRatio -> 2/3],
Graphics[{Black, PointSize[0.009], Point[evo2[[j]]*(1 - t) + evoh2[[j]]*t ]}]
]
}]}, Alignment -> Center]
, {t, 0, 1, 0.2}]
,
Table[
Column[{
Style["Arnold's cat map: (x,y) \[Rule] (2x+y,x+y) mod 1 ", Black, Bold, 14],
GraphicsRow[{
Show[
Plot[0, {x, 0, 3}, PlotRange -> {{0, 3}, {0, 2}},
PlotStyle -> Opacity[0], Axes -> False, Frame -> True, AspectRatio -> 2/3],
Graphics[{Black, PointSize[0.009], Point[evoh[[j]]*(1 - t) + evo[[j + 1]]*t ]}]
],
Show[
Plot[0, {x, 0, 3}, PlotRange -> {{0, 3}, {0, 2}},
PlotStyle -> Opacity[0], Axes -> False, Frame -> True, AspectRatio -> 2/3],
Graphics[{Black, PointSize[0.009], Point[evoh2[[j]]*(1 - t) + evo2[[j + 1]]*t ]}]
]
}]}, Alignment -> Center]
, {t, 0, 1, 0.2}]
], {j, 1, nmax}] ];
ListAnimate[p0]
Licensing
[edit]
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.
|
File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 15:57, 21 March 2020 | | 600 × 235 (6.45 MB) | Berto (talk | contribs) | Uploaded own work with UploadWizard |
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