Strange Attractors are plots of relatively simple formulas. They are created by repeating (or iterating) a formula over and over again and using the results at each iteration to plot a point. The result of each iteration is fed back into the equation. After millions of points have been plotted fractal structures appear. The repeated points fall within a basin of attraction (they are attracted to the points that make up these shapes).
I recently revisited my old strange attractor code in Visions of Chaos to add some new variations. This post will show many of the strange attractor formulas and some 4K resolution sample images they create. The images were created using over 1 billion points each. They have also been oversampled at least 3×3 pixels to reduce aliasing artifacts.
Discovered by Ivan Emrich.
x and y both start at 1.0 xnew=sin(x*y/b)*y+cos(a*x-y) ynew=x+sin(y)/b Variables a and b are floating point values bewteen -1 and +1
Discovered by Clifford A Pickover. I found them explained on Paul Bourke‘s page here.
x and y both start at 0.1 xnew=sin(a*y)+c*cos(a*x) ynew=sin(b*x)+d*cos(b*y) Variables a,b,c and d are floating point values bewteen -3 and +3
A=-1.7 B=1.3 C=-0.1 D=-1.21
A=-1.7 B=1.8 C=-0.9 D=-0.4
A=1.5 B=-1.8 C=1.6 D=2
A=-2.239 B=-2.956 C=1.272 D=1.419
A=-1.7 B=1.8 C=-1.9 D=-0.4
Fractal Dream Attractor
Discovered by Clifford A Pickover and discussed in his book “Chaos In Wonderland”.
x and y both start at 0.1 xnew=sin(y*b)+c*sin(x*b) ynew=sin(x*a)+d*sin(y*a) Variables a and b are floating point values bewteen -3 and +3 Variables c and d are floating point values between -0.5 and +1.5
A=-0.966918 B=2.879879 C=0.765145 D=0.744728
A=-2.9585 B=-2.2965 C=-2.8829 D=-0.1622
A=-2.8276 B=1.2813 C=1.9655 D=0.597
A=-1.1554 B=-2.3419 C=-1.9799 D=2.1828
A=-1.9956 B=-1.4528 C=-2.6206 D=0.8517
The Gumowski-Mira equation was developed in 1980 at CERN by I. Gumowski and C. Mira to calculate the trajectories of sub-atomic particles. It can also be used to create attractor images.
x and y both start at any floating point value between -20 and +20 t=x xnew=b*y+w w=a*x+(1-a)*2*x*x/(1+x*x) ynew=w-t The a and b parameters can be any floating point value between -1 and +1.
Initial X=0 Initial Y=0.5 A=0.008 B=-0.7
Initial X=-0.723135391715914 Initial Y=-0.327585775405169 A=0.79253300698474 B=0.345703079365194
Initial X=-0.312847771216184 Initial Y=-0.710899183526635 A=0.579161538276821 B=-0.820410779677331
Initial X=-0.325819793157279 Initial Y=0.48573582014069 A=0.062683217227459 B=-0.436713613104075
Initial X=0.78662442881614 Initial Y=0.919355855789036 A=0.900278024375439 B=0.661233567167073
The Hopalong attractor was discovered by Barry Martin.
x and y both start at 0 xnew=y-1-sqrt(abs(b*x-1-c))*sign(x-1) ynew=a-x-1 The parameters a, b and c can be any floating point value between 0 and +10.
A=7.16878197155893 B=8.43659746693447 C=2.55983412731439
A=7.7867514709942 B=0.132189802825451 C=8.14610984409228
A=9.74546888144687 B=1.56320227775723 C=7.86818214459345
A=9.8724800767377 B=8.66862616268918 C=8.66950439289212
A=9.7671244922094 B=4.10973468795419 C=3.78332691499963
Jason Rampe 1
A variation I discovered while trying random formula changes.
x and y both start at 0.1 xnew=cos(y*b)+c*sin(x*b) ynew=cos(x*a)+d*sin(y*a) Variables a, b, c and d are floating point values between -3 and +3
A=2.6 B=-2.5995 C=-2.9007 D=0.3565
A=1.8285 B=-1.8539 C=0.3816 D=1.9765
A=2.5425 B=2.8358 C=-0.8721 D=2.7044
A=-1.8669 B=1.2768 C=-2.9296 D=-0.4121
A=-2.7918 B=2.1196 C=1.0284 D=0.1384
Jason Rampe 2
Another variation I discovered while trying random formula changes.
x and y both start at 0.1 xnew=cos(y*b)+c*cos(x*b) ynew=cos(x*a)+d*cos(y*a) Variables a, b, c and d are floating point values between -3 and +3
A=1.546 B=1.929 C=1.09 D=1.41
A=2.907 B=-1.9472 C=1.2833 D=1.3206
A=0.8875 B=0.7821 C=-2.3262 D=1.5379
A=-2.4121 B=-1.0028 C=-2.2386 D=0.274
A=-2.9581 B=0.927 C=2.7842 D=0.6267
Jason Rampe 3
Yet another variation I discovered while trying random formula changes.
x and y both start at 0.1 xnew=sin(y*b)+c*cos(x*b) ynew=cos(x*a)+d*sin(y*a) Variables a, b, c and d are floating point values between -3 and +3
A=2.0246 B=-1.323 C=-2.8151 D=0.2277
A=1.4662 B=-2.3632 C=-0.4167 D=2.4162
A=-2.7564 B=-1.8234 C=2.8514 D=-0.8745
A=-2.218 B=1.4318 C=-0.3346 D=2.4993
A=1.2418 B=-2.4174 C=-0.7112 D=-1.9802
Johnny Svensson Attractor
x and y both start at 0.1 xnew=d*sin(x*a)-sin(y*b) ynew=c*cos(x*a)+cos(y*b) Variables a, b, c and d are floating point values between -3 and +3
A=1.40 B=1.56 C=1.40 D=-6.56
A=-2.538 B=1.362 C=1.315 D=0.513
A=1.913 B=2.796 C=1.468 D=1.01
A=-2.337 B=-2.337 C=0.533 D=1.378
A=-2.722 B=2.574 C=1.284 D=1.043
Peter DeJong Attractor
x and y both start at 0.1 xnew=sin(y*a)-cos(x*b) ynew=sin(x*c)-cos(y*d) Variables a, b, c and d are floating point values between -3 and +3
A=0.970 B=-1.899 C=1.381 D=-1.506
A=1.4 B=-2.3 C=2.4 D=-2.1
A=2.01 B=-2.53 C=1.61 D=-0.33
A=-2.7 B=-0.09 C=-0.86 D=-2.2
A=-0.827 B=-1.637 C=1.659 D=-0.943
A=-2 B=-2 C=-1.2 D=2
A=-0.709 B=1.638 C=0.452 D=1.740
Symmetric Icon Attractor
These attractors came from the book “Symmetry in Chaos” by Michael Field and Martin Golubitsky. They give symmetric results to the attractors formed.
x and y both start at 0.01 zzbar=sqr(x)+sqr(y) p=alpha*zzbar+lambda zreal=x zimag=y for i=1 to degree-2 do begin za=zreal*x-zimag*y zb=zimag*x+zreal*y zreal=za zimag=zb end zn=x*zreal-y*zimag p=p+beta*zn xnew=p*x+gamma*zreal-omega*y ynew=p*y-gamma*zimag+omega*x x=xnew y=ynew The Lambda, Alpha, Beta, Gamma, Omega and Degree parameters can be changed to create new plot shapes.
These sample images all come from paramters in the “Symmetry in Chaos” book.
L=-2.5 A=5 B=-1.9 G=1 O=0.188 D=5
L=1.56 A=-1 B=0.1 G=-0.82 O=0.12 D=3
L=-1.806 A=1.806 B=0 G=1 O=0 D=5
L=-2.195 A=10 B=-12 G=1 O=0 D=3
L=2.5 A=-2.5 B=0 G=0.9 O=0 D=3
L=-2.05 A=3 B=-16.79 G=1 O=0 D=9
L=-2.7 A=5 B=1.5 G=1.0 O=0 D=6
L=2.409 A=-2.5 B=0 G=0.9 O=0 D=23
L=-2.08 A=1 B=-0.1 G=0.167 O=0 D=7
L=-2.32 A=2.32 B=0 G=0.75 O=0 D=5
L=2.6 A=-2 B=0 G=-0.5 O=0 D=5
L=-2.34 A=2 B=0.2 G=0.1 O=0 D=5
L=-1.86 A=2 B=0 G=1 O=0.1 D=4
L=1.56 A=-1 B=0.1 G=-0.82 O=0 D=3
L=1.5 A=-1 B=0.1 G=-0.805 O=0 D=3
L=1.455 A=-1 B=0.03 G=-0.8 O=0 D=3
L=2.39 A=-2.5 B=-0.1 G=0.9 O=-0.15 D=16
Strange Attractors can also be extended into three dimensions. See here and here for my previous experiments with 3D Strange Attractors.
All of the images in this post were created using Visions of Chaos.
Hi, there’s a typo in your Gumowski-mira formulas. The final term should be 1/(1+x^2)^2 . The artwork, however, is ‘correct’ and beautiful.
Hi, I used the formulas from http://www.scipress.org/journals/forma/pdf/1502/15020121.pdf when creating these Gumowski-mira images.
Hi, and apologies for not doing more research — it seems many folks use your version and many others use the extra-squared version (Wolfram, for one).