The Lambda fractal is generated from the logistic map, a classic example of how chaotic behavior can arise from a simple non-linear equation. The iteration formula is:

z_{n+1} = \lambda z_n (1 - z_n)
    

This fractal is the Julia set for the quadratic map $f(z) = \lambda z (1-z)$. The parameter $\lambda$ is a fixed complex number, and the initial value $z_0$ is the point on the complex plane being tested.

For more information on the general principles of Julia sets, see the description of the Julia set.

Parameters

GUI Parameters

• Iterations: This integer value determines the maximum number of times the formula is applied for each point.

Complex Parameters (Control Points)

• Lambda: This is a fixed complex number, $\lambda$, that defines the specific shape of the fractal. It can be moved around on the complex plane to explore different variations.