Some fun with strange attractors. Click on pictures.

Thursday, October 25, 2007

Black hole




x ↦ -0.9208898452045490y² + 0.8952066957450720xy + 1.29690105688198y - 0.6718008096375430x² - 1.27506260389526x - 0.3306488696530980

y ↦ -2.83239225043423y² + 1.63633718261426xy + 2.58028867963036y - 2.60270176665395x² - 1.93174765601573x + 0.2812448271223420

Starting point (0,1;0,5); 50000 iterations; all are drawn. Color version has point thickness increased. Click pictures to get 300 dpi versions — you want to do it.

You may want to see what's inside the central “black hole”, so here comes magnified version of the core. Click to see 300 dpi version.

Impulse



x ↦ 0.7280723536001950y² - 0.4285976376157210xy - 0.00504638252707146000y - 2.10138726728836x² - 1.20138428551700x + 0.1084001644604770


y ↦ -2.67993708147439y² + 0.6196384648658460xy - 0.5113698046489240y - 0.3187101296250590x² - 0.1522131699603020x + 0.3688420719297350

Starting point (0,1;0,5); 50000 iterations; first 200 are not drawn. Click to get 300 dpi version.

Dolphin plays



Red component corresponds to partial derivative by x, blue corresponds to partial derivative by y, green depends on them in a some strange illogical way and added to make picture more lovely.

x ↦ 0.3663914416345010y² - 1.03747828322561xy -
1.50178812723409y + 1.16992005362711x² + 1.99643560701994x +
0.3626745632492520

y ↦ -1.02282315224122y² + 1.76994066384234xy +
0.3215114959922850y - 0.4038307575891870x² - 0.1628996674407880x +
1.49434629375527

Starting point (0,1;0,5); 50000 iterations; first 200 are not drawn. Click to get 300 dpi version.

The March Hare



x ↦ -0.4814599171946820y² + 1.04099727230385xy +
0.7395358611379820y - 1.24737384814668x² - 1.96139561496028x -
0.3909564561259940

y ↦ -0.7294262704650580y² + 0.4968559173666280xy +
1.52447124466781y + 2.76842854599995x² - 2.47576632778457x -
0.7309209969132820

Starting point (0,1;0,5); 50000 iterations; first 200 are not drawn. Click to get 300 dpi version.

Feeling of Escher



This image is full of artistic expression.

x ↦ 0.650850850949240y² - 1.64884940553219xy -
0.6016011453529680y - 0.3473171828135250x² + 1.66145047715947x +
0.026518324295271800

y ↦ 0.414088889877770y² - 2.68030523991092xy +
0.1142930556175630y - 2.48282952418330x² + 1.48569354388800x +
0.8157087840835780

Starting point (0,1;0,5); 50000 iterations; first 200 are not drawn. Click to get 300 dpi version.

Möbius nightmare



Not really Möbius but looks pretty nice anyway.

x ↦ -0.104776680215230y² + 1.54876597480441xy +
0.043730911135073200y - 1.12780069118847x² - 1.63073613988159x +
0.4474983408487380

y ↦ -2.12534488089094y² - 1.85727434131060xy +
1.41227652801728y + 2.73903100990934x² + 0.597063987152260x +
0.2924226813599350

Starting point (0,1;0,5); 50000 iterations; first 200 are not drawn. Click to get 300 dpi version.

Paper



x ↦ 1.40936648408379 - 1.26965791645760x - 2.09733184273639y - 1.44917724644668xy - 1.49373403181735x² -0.397197354833836y²

y ↦ -0.444398110064565 - 0.527210197459999x + 1.57942039911374y + 2.51450650465324xy - 1.03619730603340x² + 0.478557808944680y²

Starting point (0,1;0,5); 50000 iterations; first 200 are not drawn. Click to get 300 dpi version.

Shells



x ↦ 0.895657150482777 - 0.982087149593454x + 0.433965843889041y +1.50122771166545xy - 0.895074503845995x² -1.24866149964234y²

y ↦ 1.06084640878010 + 0.564302052716122x - 0.00713820914694008y - 1.32887088405998xy - 0.324072346799950x² - 0.408963002365394y²

Starting point (0,1;0,5); 50000 iterations; first 20000 are not drawn. Click to get 300 dpi version.

Wednesday, October 24, 2007

Period 10 cycle



x ↦ 1.17976926117653y² - 1.99103109843869xy -
0.8836198223605760y + 2.92298762776119x² + 0.618287417078240x -
0.1005799820311670

y ↦ -0.8067594004089490y² + 2.70383794131581xy -
1.39375844017023y + 0.2668745290345870x² + 0.6331558790477710x +
0.1216386491005990

Starting point (0,1;0,5); 50000 iterations; first 200 are not drawn. Point thickness is increased. Click to get 300 dpi version.

On the image below all iterations are drawn, so you can see how it converge to the cycle.



The order in which nodes are visited is shown below.

How many dimensions?



This two-dimensional object looks like one-dimensional something in three-dimensional space...

x ↦ 0.3981160189010760y² - 0.4487560197417530xy -
0.8160548523878250y - 0.3099618651184830x² + 0.1783832997762290x +
0.5110094718235660

y ↦ -2.14401532324836y² + 1.95300655101845xy + 2.11158162596699y
- 0.7893644947161330x² - 1.65414134432414x + 0.7642494383138650

Starting point (0,1;0,5); 50000 iterations; first 200 are not drawn. Click to get 300 dpi version.

A Napkin



x ↦ -2.19411590687834y² + 2.80881480939661xy +
0.511146797188260y - 2.35554188445293x² + 0.4886705056756730x +
0.4298302148993460

y ↦ 0.9711709802741250y² + 1.99580291616684xy -
2.00863586550983y - 1.79440783747580x² + 0.7246467173026670x +
1.12762438782708

Starting point (0,1;0,5); 150000 iterations. I failed to colorize this image in a reasonable way. Click to get 300 dpi version.

The Book



x ↦ -2.05724294429887y² + 1.33517615789464xy + 1.94686506401586y
- 0.6093055167697250x² - 1.36855989242060x + 0.1635032756398870

y ↦ 0.032158171288221700y² + 1.42347739522981xy +
0.7837136763514730y + 2.72767392239642x² - 2.95922380870402x +
0.4080669690807120

Starting point (0,1;0,5); 50000 iterations; first 200 are not drawn. Click to get 300 dpi version.

Love for attractors



x ↦ -0.3192746660790910y² + 2.84377906233858xy +
0.2026062419937250y + 0.2364444454301320x² - 0.8027955662019190x -
0.2075389708169730

y ↦ 2.25020341463347y² - 1.54963497640821xy - 2.99352757179244y
+ 1.48930981816702x² + 0.6005834182716070x + 0.9743029327098640

Starting point (0,1;0,5); 50000 iterations; first 400 are not drawn. Click to get 300 dpi version.