Logistic Equation Variations

The colors represent Poincare Recurrance times for omega along the horizontal axis, and lambda along the vertical axis. Remember -- click on the small image to get a big view.

From left to right:
Sin
x[n+1] = lambda * sin (pi * (x[n]-omega)) mod 1
Gauss
x[n+1] = lambda * exp (pi * (x[n]-omega)^2) mod 1
Sinc
x[n+1] = lambda * sinc (pi * (x[n]-omega)) mod 1 where sinc(y) = sin(y) / y
Expand
Let y = x[n] - 1/ (omega-0.5) and z = x[n] + 1/ (omega-0.5) Then x[n+1] = lambda * y * z


Expansions and Distortions

All of the below represent the standard logistic equation; they differ merely in the coordinate mapping. That is, they are really the same picture, just distorted.

From left to right:
Squash
y = x[n] - omega, z = x[n] + omega, x[n+1] = lambda * y * (1-z) mod 1
Flip
y = x[n] - omega, z = x[n] + omega, x[n+1] = (1/lambda) * y * (1-z) mod 1
Sku
y = x[n] - (omega-0.5)/lambda + 0.5
z = x[n] + (omega-0.5)/lambda + 0.5
x[n+1] = y * (1-z) mod 1


From left to right:

Squirt
y = x[n] - (omega-0.5)/sqrt(lambda) + 0.5
z = x[n] + (omega-0.5)/sqrt(lambda) + 0.5
x[n+1] = y * (1-z) mod 1
Stret
y = x[n] - sqrt (omega / lambda) + 0.5
z = x[n] + sqrt (omega / lambda) + 0.5
x[n+1] = y * (1-z) mod 1
Tch
Same equation as Stret, different domain.