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.

__Bedhead Attractor__

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
```

A=0.65343 B=0.7345345

A=-0.81 B=-0.92

A=-0.64 B=0.76

A=0.06 B=0.98

A=-0.67 B=0.83

__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
```

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

__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.
```

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

__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.
```

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__

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
```

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__

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
```

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

__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.

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).