JULIA SET GENERATOR

Julia set generator is a web application programmed for visualization of the Julia sets by various methods and demonstration of their properties using computer graphics.
The application consists of four parts. The introduction part helps familiarize with the basics of the Julia sets and the Mandelbrot set. The second part is called the orbit and describes described the principle of whether the point lies in the (Julia or Mandelbrot) set or not and also demonstrates the connection in orbit behaviour between the Mandelbrot set and the Julia sets. The part named M vs. J focuses on one of the most famous results, their connection based on the fundamental dichotomy. J vs. J is the last part of the application which is useful for comparison of visualization methods from different points of view.

Iterating process

Let's have a function $z_{n+1} = f(z_{n})$. The iterating process of this function means that an initial value (input) $z_{0}$ is selected a then the function is applied on $z_{0}$, which generates a result (output) $z_{1}$. The output from the first iteration is used as the input to the next iteration and the cycle is repeated as long as needed. Then the entire iteration process produces a sequence of values from individual iterations $z_{0}, z_{1}, z_{2} \dots$, which is often called an Orbit of $z_{0}$.

Julia sets and Mandelbrot set

These sets are generated by the same function $$z_{n+1} = z^2_{n} + c \quad z,c \in \mathbb{C}.$$ However, there are a few differences. The main difference is in the iteration process itself. For the Mandelbrot set $z_{0}$ is always equal to $0$ and then every $c$ from the complex plane is tested, whether the orbit of $z_{0}$ for this particular $c$ after nn iterations, for $n \rightarrow \infty$, diverges or not. If not, $c$ lies in the Mandelbrot set. The orbit of $z_{0} = 0$ for the Julia and Mandelbrot set generating function is called the critical orbit. Generally, the critical orbit is an orbit of initial value where the function is non-differentiable.
In contrast to the Mandelbrot set, there are an infinite number of Julia sets. It means that each $c$ corresponds with a different Julia set. Therefore, in the iteration process $c$ is fixed and every initial value $z_{0}$ from the complex plane is tested by the same technique as for the Mandelbrot set. To be exact, Julia set is only the boundary of the non-diverging orbits initial values. However, in the following text, the Julia set is most of the time considered as a whole set, exactly called the filled Julia set.