2D Strange Attractors

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.

Clifford Attractor

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

Clifford Attractor

A=-1.7 B=1.3 C=-0.1 D=-1.21

Clifford Attractor

A=-1.7 B=1.8 C=-0.9 D=-0.4

Clifford Attractor

A=1.5 B=-1.8 C=1.6 D=2

Clifford Attractor

A=-2.239 B=-2.956 C=1.272 D=1.419

Clifford Attractor

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

Fractal Dream Attractor

A=-0.966918 B=2.879879 C=0.765145 D=0.744728

Fractal Dream Attractor

A=-2.9585 B=-2.2965 C=-2.8829 D=-0.1622

Fractal Dream Attractor

A=-2.8276 B=1.2813 C=1.9655 D=0.597

Fractal Dream Attractor

A=-1.1554 B=-2.3419 C=-1.9799 D=2.1828

Fractal Dream Attractor

A=-1.9956 B=-1.4528 C=-2.6206 D=0.8517

Gumowski-Mira Attractor

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.

Gumowski Mira Attractor

Initial X=0 Initial Y=0.5 A=0.008 B=-0.7

Gumowski Mira Attractor

Initial X=-0.723135391715914 Initial Y=-0.327585775405169 A=0.79253300698474 B=0.345703079365194

Gumowski Mira Attractor

Initial X=-0.312847771216184 Initial Y=-0.710899183526635 A=0.579161538276821 B=-0.820410779677331

Gumowski Mira Attractor

Initial X=-0.325819793157279 Initial Y=0.48573582014069 A=0.062683217227459 B=-0.436713613104075

Gumowski Mira Attractor

Initial X=0.78662442881614 Initial Y=0.919355855789036 A=0.900278024375439 B=0.661233567167073

Hopalong Attractor

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.

Hopalong Attractor

A=7.16878197155893 B=8.43659746693447 C=2.55983412731439

Hopalong Attractor

A=7.7867514709942 B=0.132189802825451 C=8.14610984409228

Hopalong Attractor

A=9.74546888144687 B=1.56320227775723 C=7.86818214459345

Hopalong Attractor

A=9.8724800767377 B=8.66862616268918 C=8.66950439289212

Hopalong Attractor

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

Jason Rampe 1 Attractor

A=2.6 B=-2.5995 C=-2.9007 D=0.3565

Jason Rampe 1 Attractor

A=1.8285 B=-1.8539 C=0.3816 D=1.9765

Jason Rampe 1 Attractor

A=2.5425 B=2.8358 C=-0.8721 D=2.7044

Jason Rampe 1 Attractor

A=-1.8669 B=1.2768 C=-2.9296 D=-0.4121

Jason Rampe 1 Attractor

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

Jason Rampe 2 Attractor

A=1.546 B=1.929 C=1.09 D=1.41

Jason Rampe 2 Attractor

A=2.907 B=-1.9472 C=1.2833 D=1.3206

Jason Rampe 2 Attractor

A=0.8875 B=0.7821 C=-2.3262 D=1.5379

Jason Rampe 2 Attractor

A=-2.4121 B=-1.0028 C=-2.2386 D=0.274

Jason Rampe 2 Attractor

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

Jason Rampe 3 Attractor

A=2.0246 B=-1.323 C=-2.8151 D=0.2277

Jason Rampe 3 Attractor

A=1.4662 B=-2.3632 C=-0.4167 D=2.4162

Jason Rampe 3 Attractor

A=-2.7564 B=-1.8234 C=2.8514 D=-0.8745

Jason Rampe 3 Attractor

A=-2.218 B=1.4318 C=-0.3346 D=2.4993

Jason Rampe 3 Attractor

A=1.2418 B=-2.4174 C=-0.7112 D=-1.9802

Johnny Svensson Attractor

See here.


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

Johnny Svensson Attractor

A=1.40 B=1.56 C=1.40 D=-6.56

Johnny Svensson Attractor

A=-2.538 B=1.362 C=1.315 D=0.513

Johnny Svensson Attractor

A=1.913 B=2.796 C=1.468 D=1.01

Johnny Svensson Attractor

A=-2.337 B=-2.337 C=0.533 D=1.378

Johnny Svensson Attractor

A=-2.722 B=2.574 C=1.284 D=1.043

Peter DeJong Attractor

See here.


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

Peter DeJong Attractor

A=0.970 B=-1.899 C=1.381 D=-1.506

Peter DeJong Attractor

A=1.4 B=-2.3 C=2.4 D=-2.1

Peter DeJong Attractor

A=2.01 B=-2.53 C=1.61 D=-0.33

Peter DeJong Attractor

A=-2.7 B=-0.09 C=-0.86 D=-2.2

Peter DeJong Attractor

A=-0.827 B=-1.637 C=1.659 D=-0.943

Peter DeJong Attractor

A=-2 B=-2 C=-1.2 D=2

Peter DeJong Attractor

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.

Symmetric Icon - Chaotic Flower

L=-2.5 A=5 B=-1.9 G=1 O=0.188 D=5

Symmetric Icon - Clam Triple

L=1.56 A=-1 B=0.1 G=-0.82 O=0.12 D=3

Symmetric Icon - Emporer's Cloak

L=-1.806 A=1.806 B=0 G=1 O=0 D=5

Symmetric Icon - Fish and Eye

L=-2.195 A=10 B=-12 G=1 O=0 D=3

Symmetric Icon - Flintstone

L=2.5 A=-2.5 B=0 G=0.9 O=0 D=3

Symmetric Icon - French Glass

L=-2.05 A=3 B=-16.79 G=1 O=0 D=9

Symmetric Icon - Halloween

L=-2.7 A=5 B=1.5 G=1.0 O=0 D=6

Symmetric Icon - Kachina Dolls

L=2.409 A=-2.5 B=0 G=0.9 O=0 D=23

Symmetric Icon - Mayan Bracelet

L=-2.08 A=1 B=-0.1 G=0.167 O=0 D=7

Symmetric Icon - Pentacle

L=-2.32 A=2.32 B=0 G=0.75 O=0 D=5

Symmetric Icon - Pentagon

L=2.6 A=-2 B=0 G=-0.5 O=0 D=5

Symmetric Icon - Sanddollar

L=-2.34 A=2 B=0.2 G=0.1 O=0 D=5

Symmetric Icon - Swirling Streamers

L=-1.86 A=2 B=0 G=1 O=0.1 D=4

Symmetric Icon - Trampoline

L=1.56 A=-1 B=0.1 G=-0.82 O=0 D=3

Symmetric Icon - Trinity

L=1.5 A=-1 B=0.1 G=-0.805 O=0 D=3

Symmetric Icon - Untitled 01

L=1.455 A=-1 B=0.03 G=-0.8 O=0 D=3

Symmetric Icon - Unititled 02

L=2.39 A=-2.5 B=-0.1 G=0.9 O=-0.15 D=16

3D Alternatives

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.

Jason.

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